Package 'gamlss.dist'

Distributions for Generalized Additive Models for Location Scale and Shape

Description

A set of distributions which can be used for modelling the response variables in Generalized Additive Models for Location Scale and Shape, Rigby and Stasinopoulos (2005), <doi:10.1111/j.1467-9876.2005.00510.x>. The distributions can be continuous, discrete or mixed distributions. Extra distributions can be created, by transforming, any continuous distribution defined on the real line, to a distribution defined on ranges 0 to infinity or 0 to 1, by using a 'log' or a 'logit' transformation respectively.

Details

The DESCRIPTION file:

Package:	gamlss.dist
Title:	Distributions for Generalized Additive Models for Location Scale and Shape
Version:	6.1-8
Date:	2025-05-23
Authors@R:	c(person("Mikis", "Stasinopoulos", role = c("aut", "cre", "cph"), email = "[email protected]", comment = c(ORCID = "0000-0003-2407-5704")), person("Robert", "Rigby", role = "aut", email = "[email protected]", comment = c(ORCID = "0000-0003-3853-1707")), person("Calliope", "Akantziliotou", role = "ctb"), person("Vlasios", "Voudouris", role = "ctb"), person("Gillian", "Heller", role = "ctb", comment = c(ORCID = "0000-0003-1270-1499")), person("Fernanda", "De Bastiani", role = "ctb", comment = c(ORCID = "0000-0001-8532-639X")), person("Raydonal", "Ospina", role = "ctb", email= "[email protected]", comment = c(ORCID = "0000-0002-9884-9090")), person("Nicoletta", "Motpan", role = "ctb"), person("Fiona", "McElduff", role = "ctb"), person("Majid", "Djennad", role = "ctb"), person("Marco", "Enea", role = "ctb"), person("Alexios", "Ghalanos", role = "ctb"), person("Christos", "Argyropoulos", role = "ctb"), person("Almond", "Stöcker", role = "ctb", comment = c(ORCID = "0000-0001-9160-2397")), person("Jens", "Lichter", role = "ctb"), person("Stanislaus", "Stadlmann", role = "ctb", comment = c(ORCID = "0000-0001-6542-6342")), person("Achim", "Zeileis", role = "ctb", email = "[email protected]", comment = c(ORCID = "0000-0003-0918-3766")) )
Description:	A set of distributions which can be used for modelling the response variables in Generalized Additive Models for Location Scale and Shape, Rigby and Stasinopoulos (2005), <doi:10.1111/j.1467-9876.2005.00510.x>. The distributions can be continuous, discrete or mixed distributions. Extra distributions can be created, by transforming, any continuous distribution defined on the real line, to a distribution defined on ranges 0 to infinity or 0 to 1, by using a 'log' or a 'logit' transformation respectively.
License:	GPL-2 | GPL-3
URL:	https://www.gamlss.com/
BugReports:	https://github.com/gamlss-dev/gamlss.dist/issues
Depends:	R (>= 3.5.0)
Imports:	MASS, graphics, stats, methods, grDevices
Suggests:	distributions3 (>= 0.2.1)
Encoding:	UTF-8
Repository:	https://gamlss-dev.r-universe.dev
Date/Publication:	2025-05-23 15:05:19 UTC
RemoteUrl:	https://github.com/gamlss-dev/gamlss.dist
RemoteRef:	HEAD
RemoteSha:	c9bf8a51e95c5fe24e675b34557df1d82d569c8f
Author:	Mikis Stasinopoulos [aut, cre, cph] (ORCID: <https://orcid.org/0000-0003-2407-5704>), Robert Rigby [aut] (ORCID: <https://orcid.org/0000-0003-3853-1707>), Calliope Akantziliotou [ctb], Vlasios Voudouris [ctb], Gillian Heller [ctb] (ORCID: <https://orcid.org/0000-0003-1270-1499>), Fernanda De Bastiani [ctb] (ORCID: <https://orcid.org/0000-0001-8532-639X>), Raydonal Ospina [ctb] (ORCID: <https://orcid.org/0000-0002-9884-9090>), Nicoletta Motpan [ctb], Fiona McElduff [ctb], Majid Djennad [ctb], Marco Enea [ctb], Alexios Ghalanos [ctb], Christos Argyropoulos [ctb], Almond Stöcker [ctb] (ORCID: <https://orcid.org/0000-0001-9160-2397>), Jens Lichter [ctb], Stanislaus Stadlmann [ctb] (ORCID: <https://orcid.org/0000-0001-6542-6342>), Achim Zeileis [ctb] (ORCID: <https://orcid.org/0000-0003-0918-3766>)
Maintainer:	Mikis Stasinopoulos <[email protected]>

Index of help topics:

BB                      Beta Binomial Distribution For Fitting a GAMLSS
                        Model
BCCG                    Box-Cox Cole and Green distribution (or Box-Cox
                        normal) for fitting a GAMLSS
BCPE                    Box-Cox Power Exponential distribution for
                        fitting a GAMLSS
BCT                     Box-Cox t distribution for fitting a GAMLSS
BE                      The beta distribution for fitting a GAMLSS
BEINF                   The beta inflated distribution for fitting a
                        GAMLSS
BEOI                    The one-inflated beta distribution for fitting
                        a GAMLSS
BEZI                    The zero-inflated beta distribution for fitting
                        a GAMLSS
BI                      Binomial distribution for fitting a GAMLSS
BNB                     Beta Negative Binomial distribution for fitting
                        a GAMLSS
checklink               Set the Right Link Function for Specified
                        Parameter and Distribution
count_1_31              A set of functions to plot gamlss.family
                        distributions
DBI                     The Double binomial distribution
DBURR12                 The Discrete Burr type XII distribution for
                        fitting a GAMLSS model
DEL                     The Delaporte distribution for fitting a GAMLSS
                        model
DPO                     The Double Poisson distribution
EGB2                    The exponential generalized Beta type 2
                        distribution for fitting a GAMLSS
exGAUS                  The ex-Gaussian distribution
EXP                     Exponential distribution for fitting a GAMLSS
flexDist                Non-parametric pdf from limited information
                        data
GA                      Gamma distribution for fitting a GAMLSS
GAF                     The Gamma distribution family
GAMLSS                  Create a GAMLSS Distribution
gamlss.dist-package     Distributions for Generalized Additive Models
                        for Location Scale and Shape
gamlss.family           Family Objects for fitting a GAMLSS model
GB1                     The generalized Beta type 1 distribution for
                        fitting a GAMLSS
GB2                     The generalized Beta type 2 and generalized
                        Pareto distributions for fitting a GAMLSS
gen.Family              Functions to generate log and logit
                        distributions from existing continuous
                        gamlss.family distributions
GEOM                    Geometric distribution for fitting a GAMLSS
                        model
GG                      Generalized Gamma distribution for fitting a
                        GAMLSS
GIG                     Generalized Inverse Gaussian distribution for
                        fitting a GAMLSS
GPO                     The generalised Poisson distribution
GT                      The generalized t distribution for fitting a
                        GAMLSS
GU                      The Gumbel distribution for fitting a GAMLSS
hazardFun               Hazard functions for gamlss.family
                        distributions
IG                      Inverse Gaussian distribution for fitting a
                        GAMLSS
IGAMMA                  Inverse Gamma distribution for fitting a GAMLSS
JSU                     The Johnson's Su distribution for fitting a
                        GAMLSS
JSUo                    The original Johnson's Su distribution for
                        fitting a GAMLSS
LG                      Logarithmic and zero adjusted logarithmic
                        distributions for fitting a GAMLSS model
LNO                     Log Normal distribution for fitting in GAMLSS
LO                      Logistic distribution for fitting a GAMLSS
LOGITNO                 Logit Normal distribution for fitting in GAMLSS
LQNO                    Normal distribution with a specific mean and
                        variance relationship for fitting a GAMLSS
                        model
make.link.gamlss        Create a Link for GAMLSS families
MN3                     Multinomial distribution in GAMLSS
momentSK                Sample and theoretical Moment and Centile
                        Skewness and Kurtosis Functions
NBF                     Negative Binomial Family distribution for
                        fitting a GAMLSS
NBI                     Negative Binomial type I distribution for
                        fitting a GAMLSS
NBII                    Negative Binomial type II distribution for
                        fitting a GAMLSS
NET                     Normal Exponential t distribution (NET) for
                        fitting a GAMLSS
NO                      Normal distribution for fitting a GAMLSS
NO2                     Normal distribution (with variance as sigma
                        parameter) for fitting a GAMLSS
NOF                     Normal distribution family for fitting a GAMLSS
PARETO2                 Pareto distributions for fitting in GAMLSS
PE                      Power Exponential distribution for fitting a
                        GAMLSS
PIG                     The Poisson-inverse Gaussian distribution for
                        fitting a GAMLSS model
PO                      Poisson distribution for fitting a GAMLSS model
RG                      The Reverse Gumbel distribution for fitting a
                        GAMLSS
RGE                     Reverse generalized extreme family distribution
                        for fitting a GAMLSS
SEP                     The Skew Power exponential (SEP) distribution
                        for fitting a GAMLSS
SEP1                    The Skew exponential power type 1-4
                        distribution for fitting a GAMLSS
SHASH                   The Sinh-Arcsinh (SHASH) distribution for
                        fitting a GAMLSS
SI                      The Sichel dustribution for fitting a GAMLSS
                        model
SICHEL                  The Sichel distribution for fitting a GAMLSS
                        model
SIMPLEX                 The simplex distribution for fitting a GAMLSS
SN1                     Skew Normal Type 1 distribution for fitting a
                        GAMLSS
SN2                     Skew Normal Type 2 distribution for fitting a
                        GAMLSS
ST1                     The skew t distributions, type 1 to 5
TF                      t family distribution for fitting a GAMLSS
WARING                  Waring distribution for fitting a GAMLSS model
WEI                     Weibull distribution for fitting a GAMLSS
WEI2                    A specific parameterization of the Weibull
                        distribution for fitting a GAMLSS
WEI3                    A specific parameterization of the Weibull
                        distribution for fitting a GAMLSS
YULE                    Yule distribution for fitting a GAMLSS model
ZABB                    Zero inflated and zero adjusted Binomial
                        distribution for fitting in GAMLSS
ZABI                    Zero inflated and zero adjusted Binomial
                        distribution for fitting in GAMLSS
ZAGA                    The zero adjusted Gamma distribution for
                        fitting a GAMLSS model
ZAIG                    The zero adjusted Inverse Gaussian distribution
                        for fitting a GAMLSS model
ZANBI                   Zero inflated and zero adjusted negative
                        binomial distributions for fitting a GAMLSS
                        model
ZAP                     Zero adjusted poisson distribution for fitting
                        a GAMLSS model
ZIP                     Zero inflated poisson distribution for fitting
                        a GAMLSS model
ZIP2                    Zero inflated poisson distribution for fitting
                        a GAMLSS model
ZIPF                    The zipf and zero adjusted zipf distributions
                        for fitting a GAMLSS model

Author(s)

Mikis Stasinopoulos [aut, cre, cph] (ORCID: <https://orcid.org/0000-0003-2407-5704>), Robert Rigby [aut] (ORCID: <https://orcid.org/0000-0003-3853-1707>), Calliope Akantziliotou [ctb], Vlasios Voudouris [ctb], Gillian Heller [ctb] (ORCID: <https://orcid.org/0000-0003-1270-1499>), Fernanda De Bastiani [ctb] (ORCID: <https://orcid.org/0000-0001-8532-639X>), Raydonal Ospina [ctb] (ORCID: <https://orcid.org/0000-0002-9884-9090>), Nicoletta Motpan [ctb], Fiona McElduff [ctb], Majid Djennad [ctb], Marco Enea [ctb], Alexios Ghalanos [ctb], Christos Argyropoulos [ctb], Almond Stöcker [ctb] (ORCID: <https://orcid.org/0000-0001-9160-2397>), Jens Lichter [ctb], Stanislaus Stadlmann [ctb] (ORCID: <https://orcid.org/0000-0001-6542-6342>), Achim Zeileis [ctb] (ORCID: <https://orcid.org/0000-0003-0918-3766>)

Maintainer: Mikis Stasinopoulos <[email protected]>

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

# pdf plot
plot(function(y) dSICHEL(y, mu=10, sigma = 0.1 , nu=1 ), 
              from=0, to=30, n=30+1, type="h")
# cdf plot
PPP <- par(mfrow=c(2,1))
plot(function(y) pSICHEL(y, mu=10, sigma =0.1, nu=1 ), 
             from=0, to=30, n=30+1, type="h") # cdf
cdf<-pSICHEL(0:30, mu=10, sigma=0.1, nu=1) 
sfun1  <- stepfun(1:30, cdf, f = 0)
plot(sfun1, xlim=c(0,30), main="cdf(x)")
par(PPP)

---

Beta Binomial Distribution For Fitting a GAMLSS Model

Description

This function defines the beta binomial distribution, a two parameter distribution, for a gamlss.family object to be used in a GAMLSS fitting using the function gamlss()

Usage

BB(mu.link = "logit", sigma.link = "log")
dBB(x, mu = 0.5, sigma = 1, bd = 10, log = FALSE)
pBB(q, mu = 0.5, sigma = 1, bd = 10, lower.tail = TRUE,
      log.p = FALSE)
qBB(p, mu = 0.5, sigma = 1, bd = 10, lower.tail = TRUE, 
       log.p = FALSE)
rBB(n, mu = 0.5, sigma = 1, bd = 10)

Arguments

mu.link	Defines the mu.link, with "logit" link as the default for the mu parameter. Other links are "probit" and "cloglog"'(complementary log-log)
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" and "sqrt"
mu	vector of positive probabilities
sigma	the dispersion parameter
bd	vector of binomial denominators
p	vector of probabilities
x, q	vector of quantiles
n	number of random values to return
log, log.p	logical; if TRUE, probabilities p are given as log(p)
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

Details

Definition file for beta binomial distribution.

$f(y|\mu,\sigma)=\frac{\Gamma(n+1)} {\Gamma(y+1)\Gamma(n-y+1)} \frac{\Gamma(\frac{1}{\sigma}) \Gamma(y+\frac{\mu}{\sigma}) \Gamma[n+\frac{(1-\mu)}{\sigma}-y]}{\Gamma(n+\frac{1}{\sigma}) \Gamma(\frac{\mu}{\sigma}) \Gamma(\frac{1-\mu}{\sigma})}$

for $y=0,1,2,\ldots,n$, $0<\mu<1$ and $\sigma>0$, see pp. 523-524 of Rigby et al. (2019). . For $\mu=0.5$ and $\sigma=0.5$ the distribution is uniform.

Value

Returns a gamlss.family object which can be used to fit a Beta Binomial distribution in the gamlss() function.

Warning

The functions pBB and qBB are calculated using a laborious procedure so they are relatively slow.

Note

The response variable should be a matrix containing two columns, the first with the count of successes and the second with the count of failures. The parameter mu represents a probability parameter with limits $0 < \mu < 1$. $n \mu$ is the mean of the distribution where n is the binomial denominator. $\{n \mu (1-\mu)[1+(n-1) \sigma/(\sigma+1)]\}^{0.5}$ is the standard deviation of the Beta Binomial distribution. Hence $\sigma$ is a dispersion type parameter

Author(s)

Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape, (with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BI,

Examples

# BB()# gives information about the default links for the Beta Binomial distribution 
#plot the pdf
plot(function(y) dBB(y, mu = .5, sigma = 1, bd =40), from=0, to=40, n=40+1, type="h")
#calculate the cdf and plotting it
ppBB <- pBB(seq(from=0, to=40), mu=.2 , sigma=3, bd=40)
plot(0:40,ppBB, type="h")
#calculating quantiles and plotting them  
qqBB <- qBB(ppBB, mu=.2 , sigma=3, bd=40)
plot(qqBB~ ppBB)
# when the argument fast is useful
p <- pBB(c(0,1,2,3,4,5), mu=.01 , sigma=1, bd=5)
qBB(p, mu=.01 , sigma=1, bd=5)
#  0 1 1 2 3 5
qBB(p, mu=.01 , sigma=1, bd=5)
#  0 1 2 3 4 5
# generate random sample
tN <- table(Ni <- rBB(1000, mu=.2, sigma=1, bd=20))
r <- barplot(tN, col='lightblue')
# fitting a model 
# library(gamlss)
#data(aep)   
# fits a Beta-Binomial model
#h<-gamlss(y~ward+loglos+year, sigma.formula=~year+ward, family=BB, data=aep)

---

Box-Cox Cole and Green distribution (or Box-Cox normal) for fitting a GAMLSS

Description

The function BCCG defines the Box-Cox Cole and Green distribution (Box-Cox normal), a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dBCCG, pBCCG, qBCCG and rBCCG define the density, distribution function, quantile function and random generation for the specific parameterization of the Box-Cox Cole and Green distribution. [The function BCCGuntr() is the original version of the function suitable only for the untruncated Box-Cox Cole and Green distribution See Cole and Green (1992) and Rigby and Stasinopoulos (2003a, 2003b) for details. The function BCCGo is identical to BCCG but with log link for mu.

Usage

BCCG(mu.link = "identity", sigma.link = "log", nu.link = "identity")
BCCGo(mu.link = "log", sigma.link = "log", nu.link = "identity")
BCCGuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity")
dBCCG(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
pBCCG(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qBCCG(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rBCCG(n, mu = 1, sigma = 0.1, nu = 1)
dBCCGo(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
pBCCGo(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qBCCGo(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rBCCGo(n, mu = 1, sigma = 0.1, nu = 1)

Arguments

mu.link	Defines the mu.link, with "identity" link as the default for the mu parameter, other links are "inverse", "log" and "own"
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter, other links are "inverse", "identity" and "own"
nu.link	Defines the nu.link, with "identity" link as the default for the nu parameter, other links are "inverse", "log" and "own"
x, q	vector of quantiles
mu	vector of location parameter values
sigma	vector of scale parameter values
nu	vector of skewness parameter values
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required

Details

The probability distribution function of the untrucated Box-Cox Cole and Green distribution, BCCGuntr, is defined as

$f(y|\mu,\sigma,\nu)=\frac{1}{\sqrt{2\pi}\sigma}\frac{y^{\nu-1}}{\mu^\nu} \exp(-\frac{z^2}{2})$

where if $\nu \neq 0$ then $z=[(y/\mu)^{\nu}-1]/(\nu \sigma)$ else $z=\log(y/\mu)/\sigma$, for $y>0$, $\mu>0$, $\sigma>0$ and $\nu=(-\infty,+\infty)$.

The Box-Cox Cole and Green distribution, BCCG, adjusts the above density $f(y|\mu,\sigma,\nu)$ for the truncation resulting from the condition $y>0$. See Rigby and Stasinopoulos (2003a, 2003b) or pp. 439-441 of Rigby et al. (2019) for details.

Value

BCCG() returns a gamlss.family object which can be used to fit a Cole and Green distribution in the gamlss() function. dBCCG() gives the density, pBCCG() gives the distribution function, qBCCG() gives the quantile function, and rBCCG() generates random deviates.

Warning

The BCCGuntr distribution may be unsuitable for some combinations of the parameters (mainly for large $\sigma$) where the integrating constant is less than 0.99. A warning will be given if this is the case. The BCCG distribution is suitable for all combinations of the distributional parameters within their range [i.e. $\mu>0$, $\sigma>0$, $\nu=(-\infty, +\infty)$]

Note

$\mu$ is the median of the distribution $\sigma$ is approximately the coefficient of variation (for small values of $\sigma$), and $\nu$ controls the skewness.

The BCCG distribution is suitable for all combinations of the parameters within their ranges [i.e. $\mu>0,\sigma>0, {\rm and} \nu=(-\infty,\infty)$ ]

Author(s)

Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou

References

Cole, T. J. and Green, P. J. (1992) Smoothing reference centile curves: the LMS method and penalized likelihood, Statist. Med. 11, 1305–1319

Rigby, R. A. and Stasinopoulos, D. M. (2004). Smooth centile curves for skew and kurtotic data modelled using the Box-Cox Power Exponential distribution. Statistics in Medicine, 23: 3053-3076.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R.A. Stasinopoulos, D.M. (2006). Using the Box-Cox t distribution in GAMLSS to mode skewnees and and kurtosis. Statistical Modelling, 6(3) :209. doi:10.1191/1471082X06st122oa

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/9780429298547 An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCPE, BCT

Examples

BCCG()   # gives information about the default links for the Cole and Green distribution 
# library(gamlss)
#data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCCG, data=abdom) 
#plot(h)
plot(function(x) dBCCG(x, mu=5,sigma=.5,nu=-1), 0.0, 20, 
 main = "The BCCG  density mu=5,sigma=.5,nu=-1")
plot(function(x) pBCCG(x, mu=5,sigma=.5,nu=-1), 0.0, 20, 
 main = "The BCCG  cdf mu=5, sigma=.5, nu=-1")

---

Box-Cox Power Exponential distribution for fitting a GAMLSS

Description

This function defines the Box-Cox Power Exponential distribution, a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss().

The functions dBCPE, pBCPE, qBCPE and rBCPE define the density, distribution function, quantile function and random generation for the Box-Cox Power Exponential distribution.

The function checkBCPE (very old) can be used, typically when a BCPE model is fitted, to check whether there exit a turning point of the distribution close to zero. It give the number of values of the response below their minimum turning point and also the maximum probability of the lower tail below minimum turning point.

[The function Biventer() is the original version of the function suitable only for the untruncated BCPE distribution.] See Rigby and Stasinopoulos (2003) for details.

The function BCPEo is identical to BCPE but with log link for mu.

Usage

BCPE(mu.link = "identity", sigma.link = "log", nu.link = "identity",
          tau.link = "log")
BCPEo(mu.link = "log", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
BCPEuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
dBCPE(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCPE(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCPE(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)          
rBCPE(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
dBCPEo(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCPEo(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, 
       log.p = FALSE)
qBCPEo(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, 
       log.p = FALSE)          
rBCPEo(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
checkBCPE(obj = NULL, mu = 10, sigma = 0.1, nu = 0.5, tau = 2,...)

Arguments

mu.link	Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse", "log" and "own"
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" and "own"
nu.link	Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "inverse", "log" and "own"
tau.link	Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "logshifted", "identity" and "own"
x, q	vector of quantiles
mu	vector of location parameter values
sigma	vector of scale parameter values
nu	vector of nu parameter values
tau	vector of tau parameter values
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required
obj	a gamlss BCPE family object
...	for extra arguments

Details

The probability density function of the untrucated Box Cox Power Exponential distribution, (BCPE.untr), is defined as

$f(y|\mu,\sigma,\nu,\tau)=\frac{y^{\nu-1} \tau \exp[-\frac{1}{2}|\frac{z}{c}|^\tau]}{\mu^{\nu} \sigma c 2^{(1+1/\tau)} \Gamma(\frac{1}{\tau})}$

where $c = [ 2^{(-2/\tau)}\Gamma(1/\tau)/\Gamma(3/\tau)]^{0.5}$, where if $\nu \neq 0$ then $z=[(y/\mu)^{\nu}-1]/(\nu \sigma)$ else $z=\log(y/\mu)/\sigma$, for $y>0$, $\mu>0$, $\sigma>0$, $\nu=(-\infty,+\infty)$ and $\tau>0$ see pp. 450-451 of Rigby et al. (2019).

The Box-Cox Power Exponential, BCPE, adjusts the above density $f(y|\mu,\sigma,\nu,\tau)$ for the truncation resulting from the condition $y>0$. See Rigby and Stasinopoulos (2003) for details.

Value

BCPE() returns a gamlss.family object which can be used to fit a Box Cox Power Exponential distribution in the gamlss() function. dBCPE() gives the density, pBCPE() gives the distribution function, qBCPE() gives the quantile function, and rBCPE() generates random deviates.

Warning

The BCPE.untr distribution may be unsuitable for some combinations of the parameters (mainly for large $\sigma$) where the integrating constant is less than 0.99. A warning will be given if this is the case.

The BCPE distribution is suitable for all combinations of the parameters within their ranges [i.e. $\mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0$ ]

Note

$\mu$, is the median of the distribution, $\sigma$ is approximately the coefficient of variation (for small $\sigma$ and moderate nu>0), $\nu$ controls the skewness and $\tau$ the kurtosis of the distribution

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A. and Stasinopoulos, D. M. (2004). Smooth centile curves for skew and kurtotic data modelled using the Box-Cox Power Exponential distribution. Statistics in Medicine, 23: 3053-3076.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCT

Examples

# BCPE()   #
# library(gamlss) 
# data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCPE, data=abdom) 
#plot(h)
plot(function(x)dBCPE(x, mu=5,sigma=.5,nu=1, tau=3), 0.0, 15, 
 main = "The BCPE  density mu=5,sigma=.5,nu=1, tau=3")
plot(function(x) pBCPE(x, mu=5,sigma=.5,nu=1, tau=3), 0.0, 15, 
 main = "The BCPE  cdf mu=5, sigma=.5, nu=1, tau=3")

---

Box-Cox t distribution for fitting a GAMLSS

Description

The function BCT() defines the Box-Cox t distribution, a four parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

The functions dBCT, pBCT, qBCT and rBCT define the density, distribution function, quantile function and random generation for the Box-Cox t distribution.

[The function BCTuntr() is the original version of the function suitable only for the untruncated BCT distribution]. See Rigby and Stasinopoulos (2003) for details.

The function BCTo is identical to BCT but with log link for mu.

Usage

BCT(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
BCTo(mu.link = "log", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
BCTuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity", 
          tau.link = "log")
dBCT(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCT(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCT(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)          
rBCT(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
dBCTo(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCTo(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCTo(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)          
rBCTo(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)

Arguments

mu.link	Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse", "log" and "own"
sigma.link	Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse","identity", "own"
nu.link	Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "inverse", "log", "own"
tau.link	Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "inverse", "identity" and "own"
x, q	vector of quantiles
mu	vector of location parameter values
sigma	vector of scale parameter values
nu	vector of nu parameter values
tau	vector of tau parameter values
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required

Details

The probability density function of the untruncated Box-Cox t distribution, BCTuntr, is given by

$f(y|\mu,\sigma,\nu,\tau)=\frac{y^{\nu-1}}{\mu^{\nu}\sigma} \frac{\Gamma[(\tau+1)/2]}{\Gamma(1/2) \Gamma(\tau/2) \tau^{0.5}} [1+(1/\tau)z^2]^{-(\tau+1)/2}$

where if $\nu \neq 0$ then $z=[(y/\mu)^{\nu}-1]/(\nu \sigma)$ else $z=\log(y/\mu)/\sigma$, for $y>0$, $\mu>0$, $\sigma>0$, $\nu=(-\infty,+\infty)$ and $\tau>0$ see pp. 450-451 of Rigby et al. (2019).

The Box-Cox t distribution, BCT, adjusts the above density $f(y|\mu,\sigma,\nu,\tau)$ for the truncation resulting from the condition $y>0$. See Rigby and Stasinopoulos (2003) for details.

Value

BCT() returns a gamlss.family object which can be used to fit a Box Cox-t distribution in the gamlss() function. dBCT() gives the density, pBCT() gives the distribution function, qBCT() gives the quantile function, and rBCT() generates random deviates.

Warning

The use BCTuntr distribution may be unsuitable for some combinations of the parameters (mainly for large $\sigma$) where the integrating constant is less than 0.99. A warning will be given if this is the case.

The BCT distribution is suitable for all combinations of the parameters within their ranges [i.e. $\mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0$ ]

Note

$\mu$ is the median of the distribution, $\sigma(\frac{\tau}{\tau-2})^{0.5}$ is approximate the coefficient of variation (for small $\sigma$ and moderate nu>0 and moderate or large $\tau$), $\nu$ controls the skewness and $\tau$ the kurtosis of the distribution

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R.A. Stasinopoulos, D.M. (2006). Using the Box-Cox t distribution in GAMLSS to mode skewnees and and kurtosis. Statistical Modelling 6(3):200. doi:10.1191/1471082X06st122oa.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCPE, BCCG

Examples

BCT()   # gives information about the default links for the Box Cox t distribution
# library(gamlss)
#data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCT, data=abdom) # 
#plot(h)
plot(function(x)dBCT(x, mu=5,sigma=.5,nu=1, tau=2), 0.0, 20, 
 main = "The BCT  density mu=5,sigma=.5,nu=1, tau=2")
plot(function(x) pBCT(x, mu=5,sigma=.5,nu=1, tau=2), 0.0, 20, 
 main = "The BCT  cdf mu=5, sigma=.5, nu=1, tau=2")

---

The beta distribution for fitting a GAMLSS

Description

The functions BE() and BEo() define the beta distribution, a two parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). BE() has mean equal to the parameter mu and sigma as scale parameter, see below. BEo() is the original parameterizations of the beta distribution as in dbeta() with shape1=mu and shape2=sigma. The functions dBE and dBEo, pBE and pBEo, qBE and qBEo and finally rBE and rBE define the density, distribution function, quantile function and random generation for the BE and BEo parameterizations respectively of the beta distribution.

Usage

BE(mu.link = "logit", sigma.link = "logit")
dBE(x, mu = 0.5, sigma = 0.2, log = FALSE)
pBE(q, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
qBE(p, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
rBE(n, mu = 0.5, sigma = 0.2)
BEo(mu.link = "log", sigma.link = "log")
dBEo(x, mu = 0.5, sigma = 0.2, log = FALSE)
pBEo(q, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)
qBEo(p, mu = 0.5, sigma = 0.2, lower.tail = TRUE, log.p = FALSE)

Arguments

mu.link	the mu link function with default logit
sigma.link	the sigma link function with default logit
x, q	vector of quantiles
mu	vector of location parameter values
sigma	vector of scale parameter values
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required

Details

The standard parametrization of the beta distribution is given as:

$f(y|\alpha,\beta)=\frac{1}{B(\alpha, \beta)} y^{\alpha-1}(1-y)^{\beta-1}$

for $y=(0,1)$, $\alpha>0$ and $\beta>0$.

The first gamlss implementation the beta distribution is called BEo, and it is identical to the standard parametrization with $\alpha=\mu$ and $\beta = \sigma$, see pp. 460-461 of Rigby et al. (2019):

$f(y|\mu,\sigma)=\frac{1}{B(\mu, \sigma)} y^{\mu-1}(1-y)^{\sigma-1}$

for $y=(0,1)$, $\mu>0$ and $\sigma>0$. The problem with this parametrization is that with mean $E(y)=\mu/(\mu+\sigma)$ it is not convenient for modelling the response y as function of the explanatory variables. The second parametrization, BE see pp. 461-463 of Rigby et al. (2019), is using

$\mu=\frac{\alpha}{\alpha+\beta}$

$\sigma= \frac{1}{\alpha+\beta+1)^{1/2}}$

(of the standard parametrization) and it is more convenient because $\mu$ now is the mean with variance equal to $Var(y)=\sigma^2 \mu (1-\mu)$. Note however that $0<\mu<1$ and $0<\sigma<1$.

Value

BE() and BEo() return a gamlss.family object which can be used to fit a beta distribution in the gamlss() function.

Note

Note that for BE, mu is the mean and sigma a scale parameter contributing to the variance of y

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also https://www.gamlss.com/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BE, LOGITNO, GB1, BEINF

Examples

BE()# gives information about the default links for the beta distribution
dat1<-rBE(100, mu=.3, sigma=.5)
hist(dat1)        
#library(gamlss)
# mod1<-gamlss(dat1~1,family=BE) # fits a constant for mu and sigma 
#fitted(mod1)[1]
#fitted(mod1,"sigma")[1]
plot(function(y) dBE(y, mu=.1 ,sigma=.5), 0.001, .999)
plot(function(y) pBE(y, mu=.1 ,sigma=.5), 0.001, 0.999)
plot(function(y) qBE(y, mu=.1 ,sigma=.5), 0.001, 0.999)
plot(function(y) qBE(y, mu=.1 ,sigma=.5, lower.tail=FALSE), 0.001, .999)
dat2<-rBEo(100, mu=1, sigma=2)
#mod2<-gamlss(dat2~1,family=BEo) # fits a constant for mu and sigma 
#fitted(mod2)[1]
#fitted(mod2,"sigma")[1]

---

The beta inflated distribution for fitting a GAMLSS

Description

The function BEINF() defines the beta inflated distribution, a four parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The beta inflated is similar to the beta but allows zeros and ones as values for the response variable. The two extra parameters model the probabilities at zero and one.

The functions BEINF0() and BEINF1() are three parameter beta inflated distributions allowing zeros or ones only at the response respectively. BEINF0() and BEINF1() are re-parameterize versions of the distributions BEZI and BEOI contributed to gamlss by Raydonal Ospina (see Ospina and Ferrari (2010)).

The functions dBEINF, pBEINF, qBEINF and rBEINF define the density, distribution function, quantile function and random generation for the BEINF parametrization of the beta inflated distribution.

The functions dBEINF0, pBEINF0, qBEINF0 and rBEINF0 define the density, distribution function, quantile function and random generation for the BEINF0 parametrization of the beta inflated at zero distribution.

The functions dBEINF1, pBEINF1, qBEINF1 and rBEINF1 define the density, distribution function, quantile function and random generation for the BEINF1 parametrization of the beta inflated at one distribution.

plotBEINF, plotBEINF0 and plotBEINF1 can be used to plot the distributions. meanBEINF, meanBEINF0 and meanBEINF1 calculates the expected value of the response for a fitted model.

Usage

BEINF(mu.link = "logit", sigma.link = "logit", nu.link = "log", 
      tau.link = "log")
BEINF0(mu.link = "logit", sigma.link = "logit", nu.link = "log")
BEINF1(mu.link = "logit", sigma.link = "logit", nu.link = "log")
      
dBEINF(x, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1, 
       log = FALSE)
dBEINF0(x, mu = 0.5, sigma = 0.1, nu = 0.1, log = FALSE)
dBEINF1(x, mu = 0.5, sigma = 0.1, nu = 0.1, log = FALSE)       

pBEINF(q, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1, 
       lower.tail = TRUE, log.p = FALSE)
pBEINF0(q, mu = 0.5, sigma = 0.1, nu = 0.1, 
       lower.tail = TRUE, log.p = FALSE)       
pBEINF1(q, mu = 0.5, sigma = 0.1, nu = 0.1, 
        lower.tail = TRUE, log.p = FALSE)

qBEINF(p, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1, 
       lower.tail = TRUE, log.p = FALSE)
qBEINF0(p, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1, 
        lower.tail = TRUE, log.p = FALSE)
qBEINF1(p, mu = 0.5, sigma = 0.1, nu = 0.1, 
        lower.tail = TRUE, log.p = FALSE)
       
rBEINF(n, mu = 0.5, sigma = 0.1, nu = 0.1, tau = 0.1)
rBEINF0(n, mu = 0.5, sigma = 0.1, nu = 0.1)
rBEINF1(n, mu = 0.5, sigma = 0.1, nu = 0.1)

plotBEINF(mu = 0.5, sigma = 0.5, nu = 0.5, tau = 0.5, 
          from = 0.001, to = 0.999, n = 101, ...)
plotBEINF0(mu = 0.5, sigma = 0.5, nu = 0.5, 
          from = 1e-04, to = 0.9999, n = 101, ...)
plotBEINF1(mu = 0.5, sigma = 0.5, nu = 0.5, 
          from = 1e-04, to = 0.9999, n = 101, ...)

meanBEINF(obj)
meanBEINF0(obj)
meanBEINF1(obj)

Arguments

mu.link	the mu link function with default logit
sigma.link	the sigma link function with default logit
nu.link	the nu link function with default log
tau.link	the tau link function with default log
x, q	vector of quantiles
mu	vector of location parameter values
sigma	vector of scale parameter values
nu	vector of parameter values modelling the probability at zero
tau	vector of parameter values modelling the probability at one
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required
from	where to start plotting the distribution from
to	up to where to plot the distribution
obj	a fitted BEINF object
...	other graphical parameters for plotting

Details

The beta inflated distribution is defined as

$f(y)=p_0 \quad \textit{if} \quad y=0$

$f(y)=p_1 \quad \textit{if} \quad y=1$

$f(y|\alpha,\beta)=\frac{1}{B(\alpha, \beta)} y^{\alpha-1}(1-y)^{\beta-1} \quad \textit{otherwise}$

for $0 \le y \le 1$, $\alpha>0$, $\beta>0$, $0<p_0<1$, $0<p_1<1$ and $0<p_0+p_1<1$.

The GAMLSS function BEINF() re-parametrize the parameters as

$\mu=\frac{\alpha}{\alpha+\beta}$

$\sigma=\frac{1}{\alpha+\beta+1}$

$\nu=\frac{p_0}{p_2}$

$\tau=\frac{p_1}{p_2}$

where $p_2=1-p_0-p_1$ so $0<\mu<1$, $0<\sigma<1$, $\nu>0$ and $\tau>0$ see pp. 466-467 of Rigby et al. (2019).

The beta inflated at zero distribution is defined as

$f(y)=p_0 \quad \textit{if} \quad y=0$

$f(y|\alpha,\beta)=\frac{1}{B(\alpha, \beta)} y^{\alpha-1}(1-y)^{\beta-1} \quad \textit{otherwise}$

for $0 \le y < 1$, $\alpha>0$, $\beta>0$ and $0<p_0<1$.

The GAMLSS function BEINF0() re-parametrize the parameters as

$\mu=\frac{\alpha}{\alpha+\beta}$

$\sigma=\frac{1}{\alpha+\beta+1}$

$\nu=\frac{p_0}{1-P_0}$

so $0<\mu<1$, $0<\sigma<1$ and $\nu>0$ see pp. 467-468 of Rigby et al. (2019).

The beta inflated at 1 distribution is defined as

$f(y|\alpha,\beta)=\frac{1}{B(\alpha, \beta)} y^{\alpha-1}(1-y)^{\beta-1} \quad \textit{if} \quad 0<y<1$

$f(y)=p_1 \quad \quad \textit{if} \: \quad y=1$

for $0 < y \le 1$, $\alpha>0$, $\beta>0$ and $0<p_1<1$.

The GAMLSS function BEINF1() re-parametrize the parameters as

$\mu=\frac{\alpha}{\alpha+\beta}$

$\sigma=\frac{1}{\alpha+\beta+1}$

$\nu=\frac{p_1}{1-P_1}$

so $0<\mu<1$, $0<\sigma<1$ and $\nu>0$ see pp. 468-469 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a beta inflated distribution in the gamlss() function. ...

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Ospina R. and Ferrari S. L. P. (2010) Inflated beta distributions, Statistical Papers, 23, 111-126.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BE, BEo, BEZI, BEOI

Examples

BEINF()# gives information about the default links for the beta inflated distribution
BEINF0()
BEINF1()
# plotting the distributions
op<-par(mfrow=c(2,2)) 
plotBEINF( mu =.5 , sigma=.5, nu = 0.5, tau = 0.5, from = 0, to=1, n = 101)
plotBEINF0( mu =.5 , sigma=.5, nu = 0.5,  from = 0, to=1, n = 101)
plotBEINF1( mu =.5 , sigma=.5, nu = 0.5,  from = 0.001, to=1, n = 101)
curve(dBE(x, mu =.5, sigma=.5),  0.01, 0.999)
par(op)
# plotting the cdf
op<-par(mfrow=c(2,2)) 
plotBEINF( mu =.5 , sigma=.5, nu = 0.5, tau = 0.5, from = 0, to=1, n = 101, main="BEINF")
plotBEINF0( mu =.5 , sigma=.5, nu = 0.5,  from = 0, to=1, n = 101, main="BEINF0")
plotBEINF1( mu =.5 , sigma=.5, nu = 0.5,  from = 0.001, to=1, n = 101, main="BEINF1")
curve(dBE(x, mu =.5, sigma=.5),  0.01, 0.999, main="BE")
par(op)
#---------------------------------------------
op<-par(mfrow=c(2,2)) 
plotBEINF( mu =.5 , sigma=.5, nu = 0.5, tau = 0.5, from = 0, to=1, n = 101, main="BEINF")
plotBEINF0( mu =.5 , sigma=.5, nu = 0.5,  from = 0, to=1, n = 101, main="BEINF0")
plotBEINF1( mu =.5 , sigma=.5, nu = 0.5,  from = 0.001, to=1, n = 101, main="BEINF1")
curve(dBE(x, mu =.5, sigma=.5),  0.01, 0.999, main="BE")
par(op)
#---------------------------------------------
op<-par(mfrow=c(2,2)) 
curve( pBEINF(x, mu=.5 ,sigma=.5, nu = 0.5, tau = 0.5,), 0, 1, ylim=c(0,1), main="BEINF" )
curve(pBEINF0(x, mu=.5 ,sigma=.5, nu = 0.5), 0, 1, ylim=c(0,1), main="BEINF0")
curve(pBEINF1(x, mu=.5 ,sigma=.5, nu = 0.5), 0, 1, ylim=c(0,1), main="BEINF1")
curve(    pBE(x, mu=.5 ,sigma=.5), .001, .99, ylim=c(0,1), main="BE")
par(op)
#---------------------------------------------
op<-par(mfrow=c(2,2)) 
curve(qBEINF(x, mu=.5 ,sigma=.5, nu = 0.5, tau = 0.5), .01, .99, main="BEINF" )
curve(qBEINF0(x, mu=.5 ,sigma=.5, nu = 0.5), .01, .99, main="BEINF0" )
curve(qBEINF1(x, mu=.5 ,sigma=.5, nu = 0.5), .01, .99, main="BEINF1" )
curve(qBE(x, mu=.5 ,sigma=.5), .01, .99 , main="BE")
par(op)

#---------------------------------------------
op<-par(mfrow=c(2,2)) 
hist(rBEINF(200, mu=.5 ,sigma=.5, nu = 0.5, tau = 0.5))
hist(rBEINF0(200, mu=.5 ,sigma=.5, nu = 0.5))
hist(rBEINF1(200, mu=.5 ,sigma=.5, nu = 0.5))
hist(rBE(200, mu=.5 ,sigma=.5))
par(op)
# fit a model to the data 
# library(gamlss)
#m1<-gamlss(dat~1,family=BEINF)
#meanBEINF(m1)[1]

---

The one-inflated beta distribution for fitting a GAMLSS

Description

The function BEOI() defines the one-inflated beta distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The one-inflated beta is similar to the beta distribution but allows ones as y values. This distribution is an extension of the beta distribution using a parameterization of the beta law that is indexed by mean and precision parameters (Ferrari and Cribari-Neto, 2004). The extra parameter models the probability at one. The functions dBEOI, pBEOI, qBEOI and rBEOI define the density, distribution function, quantile function and random generation for the BEOI parameterization of the one-inflated beta distribution. plotBEOI can be used to plot the distribution. meanBEOI calculates the expected value of the response for a fitted model.

Usage

BEOI(mu.link = "logit", sigma.link = "log", nu.link = "logit")

dBEOI(x, mu = 0.5, sigma = 1, nu = 0.1, log = FALSE)

pBEOI(q, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qBEOI(p, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE,
        log.p = FALSE)
        
rBEOI(n, mu = 0.5, sigma = 1, nu = 0.1)

plotBEOI(mu = .5, sigma = 1, nu = 0.1, from = 0.001, to = 1, n = 101, 
    ...)
    
meanBEOI(obj)

Arguments

mu.link	the mu link function with default logit
sigma.link	the sigma link function with default log
nu.link	the nu link function with default logit
x, q	vector of quantiles
mu	vector of location parameter values
sigma	vector of precision parameter values
nu	vector of parameter values modelling the probability at one
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required
from	where to start plotting the distribution from
to	up to where to plot the distribution
obj	a fitted BEOI object
...	other graphical parameters for plotting

Details

The one-inflated beta distribution is given as

$f(y)=\nu \quad \textit{if} \quad y=1$

$f(y|\mu,\sigma)=(1-\nu)\frac{\Gamma(\sigma)}{\Gamma(\mu \sigma)\Gamma((1-\mu) \sigma)} y^{\mu \sigma}(1-y)^{((1-\mu)\sigma)-1} \quad \textit{otherwise}$

The parameters satisfy $0<\mu<0$, $\sigma>0$ and $0<\nu< 1$.

Here $E(y)=\nu+(1-\nu)\mu$ and $Var(y)=(1-\nu)\frac{\mu(1-\mu)}{\sigma+1}+\nu(1-\nu)(1-\mu)^2$.

Value

returns a gamlss.family object which can be used to fit a one-inflated beta distribution in the gamlss() function.

Note

This work is part of my PhD project at the University of Sao Paulo under the supervion of Professor Silvia Ferrari. My thesis is concerned with regression modelling of rates and proportions with excess of zeros and/or ones

Author(s)

Raydonal Ospina, Department of Statistics, University of Sao Paulo, Brazil.

[email protected]

References

Ferrari, S.L.P., Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31 (1), 799-815.

Ospina R. and Ferrari S. L. P. (2010) Inflated beta distributions, Statistical Papers, 23, 111-126.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape (with discussion). Applied Statistics, 54 (3), 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BEOI

Examples

BEOI()# gives information about the default links for the BEOI distribution
# plotting the distribution
plotBEOI( mu =0.5 , sigma=5, nu = 0.1, from = 0.001, to=1, n = 101)
# plotting the cdf
plot(function(y) pBEOI(y, mu=.5 ,sigma=5, nu=0.1), 0.001, 0.999)
# plotting the inverse cdf
plot(function(y) qBEOI(y, mu=.5 ,sigma=5, nu=0.1), 0.001, 0.999)
# generate random numbers
dat<-rBEOI(100, mu=.5, sigma=5, nu=0.1)
# fit a model to the data. 
# library(gamlss)
#mod1<-gamlss(dat~1,sigma.formula=~1, nu.formula=~1, family=BEOI) 
#fitted(mod1)[1]
#summary(mod1)
#fitted(mod1,"mu")[1]        #fitted mu
#fitted(mod1,"sigma")[1]     #fitted sigma
#fitted(mod1,"nu")[1]        #fitted nu
#meanBEOI(mod1)[1] # expected value of the response

---

The zero-inflated beta distribution for fitting a GAMLSS

Description

The function BEZI() defines the zero-inflated beta distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The zero-inflated beta is similar to the beta distribution but allows zeros as y values. This distribution is an extension of the beta distribution using a parameterization of the beta law that is indexed by mean and precision parameters (Ferrari and Cribari-Neto, 2004). The extra parameter models the probability at zero. The functions dBEZI, pBEZI, qBEZI and rBEZI define the density, distribution function, quantile function and random generation for the BEZI parameterization of the zero-inflated beta distribution. plotBEZI can be used to plot the distribution. meanBEZI calculates the expected value of the response for a fitted model.

Usage

BEZI(mu.link = "logit", sigma.link = "log", nu.link = "logit")

dBEZI(x, mu = 0.5, sigma = 1, nu = 0.1, log = FALSE)

pBEZI(q, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)

qBEZI(p, mu = 0.5, sigma = 1, nu = 0.1, lower.tail = TRUE,
        log.p = FALSE)
        
rBEZI(n, mu = 0.5, sigma = 1, nu = 0.1)

plotBEZI(mu = .5, sigma = 1, nu = 0.1, from = 0, to = 0.999, n = 101, 
    ...)
    
meanBEZI(obj)

Arguments

mu.link	the mu link function with default logit
sigma.link	the sigma link function with default log
nu.link	the nu link function with default logit
x, q	vector of quantiles
mu	vector of location parameter values
sigma	vector of precision parameter values
nu	vector of parameter values modelling the probability at zero
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required
from	where to start plotting the distribution from
to	up to where to plot the distribution
obj	a fitted BEZI object
...	other graphical parameters for plotting

Details

The zero-inflated beta distribution is given as

$f(y)=\nu$

if $(y=0)$

$f(y|\mu,\sigma)=(1-\nu)\frac{\Gamma(\sigma)}{\Gamma(\mu\sigma)\Gamma((1-\mu)\sigma)} y^{\mu\sigma}(1-y)^{((1-\mu)\sigma)-1}$

if $y=(0,1)$. The parameters satisfy $0<\mu<0$, $\sigma>0$ and $0<\nu< 1$.

Here $E(y)=(1-\nu)\mu$ and $Var(y)=(1-\nu)\frac{\mu(1-\mu)}{\sigma+1}+\nu(1-\nu)\mu^2$.

Value

returns a gamlss.family object which can be used to fit a zero-inflated beta distribution in the gamlss() function.

Note

This work is part of my PhD project at the University of Sao Paulo under the supervion of Professor Silvia Ferrari. My thesis is concerned with regression modelling of rates and proportions with excess of zeros and/or ones

Author(s)

Raydonal Ospina, Department of Statistics, University of Sao Paulo, Brazil.

[email protected]

References

Ferrari, S.L.P., Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31 (1), 799-815.

Ospina R. and Ferrari S. L. P. (2010) Inflated beta distributions, Statistical Papers, 23, 111-126.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape (with discussion). Applied Statistics, 54 (3), 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, BEZI

Examples

BEZI()# gives information about the default links for the BEZI distribution
# plotting the distribution
plotBEZI( mu =0.5 , sigma=5, nu = 0.1, from = 0, to=0.99, n = 101)
# plotting the cdf
plot(function(y) pBEZI(y, mu=.5 ,sigma=5, nu=0.1), 0, 0.999)
# plotting the inverse cdf
plot(function(y) qBEZI(y, mu=.5 ,sigma=5, nu=0.1), 0, 0.999)
# generate random numbers
dat<-rBEZI(100, mu=.5, sigma=5, nu=0.1)
# fit a model to the data. Tits a constant for mu, sigma and nu
# library(gamlss)
#mod1<-gamlss(dat~1,sigma.formula=~1, nu.formula=~1, family=BEZI) 
#fitted(mod1)[1]
#summary(mod1)
#fitted(mod1,"mu")[1]         #fitted mu   
#fitted(mod1,"sigma")[1]      #fitted sigma 
#fitted(mod1,"nu")[1]         #fitted nu  
#meanBEZI(mod1)[1] # expected value of the response

---

Binomial distribution for fitting a GAMLSS

Description

The BI() function defines the binomial distribution, a one parameter family distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dBI, pBI, qBI and rBI define the density, distribution function, quantile function and random generation for the binomial, BI(), distribution.

Usage

BI(mu.link = "logit")
dBI(x, bd = 1, mu = 0.5, log = FALSE)
pBI(q, bd = 1, mu = 0.5, lower.tail = TRUE, log.p = FALSE)
qBI(p, bd = 1, mu = 0.5, lower.tail = TRUE, log.p = FALSE)
rBI(n, bd = 1, mu = 0.5)

Arguments

mu.link	Defines the mu.link, with "logit" link as the default for the mu parameter. Other links are "probit" and "cloglog"'(complementary log-log)
x	vector of (non-negative integer) quantiles
mu	vector of positive probabilities
bd	vector of binomial denominators
p	vector of probabilities
q	vector of quantiles
n	number of random values to return
log, log.p	logical; if TRUE, probabilities p are given as log(p)
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

Details

Definition file for binomial distribution.

$f(y|\mu)=\frac{\Gamma(n+1)}{\Gamma(y+1) \Gamma{(n-y+1)}} \mu^y (1-\mu)^{(n-y)}$

for $y=0,1,2,...,n$ and $0<\mu< 1$ see pp. 521-522 of Rigby et al. (2019).

Value

returns a gamlss.family object which can be used to fit a binomial distribution in the gamlss() function.

Note

The response variable should be a matrix containing two columns, the first with the count of successes and the second with the count of failures. The parameter mu represents a probability parameter with limits $0 < \mu < 1$. $n\mu$ is the mean of the distribution where n is the binomial denominator.

Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family, ZABI, ZIBI

Examples

BI()# gives information about the default links for the Binomial distribution 
# data(aep)   
# library(gamlss)
# h<-gamlss(y~ward+loglos+year, family=BI, data=aep)  
# plot of the binomial distribution
curve(dBI(x, mu = .5, bd=10), from=0, to=10, n=10+1, type="h")
tN <- table(Ni <- rBI(1000, mu=.2, bd=10))
r <- barplot(tN, col='lightblue')

---

Beta Negative Binomial distribution for fitting a GAMLSS

Description

The BNB() function defines the beta negative binomial distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss().

The functions dBNB, pBNB, qBNB and rBNB define the density, distribution function, quantile function and random generation for the beta negative binomial distribution, BNB().

The functions ZABNB() and ZIBNB() are the zero adjusted (hurdle) and zero inflated versions of the beta negative binomial distribution, respectively. That is four parameter distributions.

The functions dZABNB, dZIBNB, pZABNB,pZIBNB, qZABNB qZIBNB rZABNB and rZIBNB define the probability, cumulative, quantile and random generation functions for the zero adjusted and zero inflated beta negative binomial distributions, ZABNB(), ZIBNB(), respectively.

Usage

BNB(mu.link = "log", sigma.link = "log", nu.link = "log")
dBNB(x, mu = 1, sigma = 1, nu = 1, log = FALSE)
pBNB(q, mu = 1, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qBNB(p, mu = 1, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE, 
     max.value = 10000)
rBNB(n, mu = 1, sigma = 1, nu = 1, max.value = 10000)

ZABNB(mu.link = "log", sigma.link = "log", nu.link = "log",
      tau.link = "logit")
dZABNB(x, mu = 1, sigma = 1, nu = 1, tau = 0.1, log = FALSE)
pZABNB(q, mu = 1, sigma = 1, nu = 1, tau = 0.1, lower.tail = TRUE, 
       log.p = FALSE)
qZABNB(p, mu = 1, sigma = 1, nu = 1, tau = 0.1, lower.tail = TRUE, 
       log.p = FALSE, max.value = 10000)
rZABNB(n, mu = 1, sigma = 1, nu = 1, tau = 0.1, max.value = 10000)

ZIBNB(mu.link = "log", sigma.link = "log", nu.link = "log", 
      tau.link = "logit")
dZIBNB(x, mu = 1, sigma = 1, nu = 1, tau = 0.1, log = FALSE)
pZIBNB(q, mu = 1, sigma = 1, nu = 1, tau = 0.1, lower.tail = TRUE, 
       log.p = FALSE)
qZIBNB(p, mu = 1, sigma = 1, nu = 1, tau = 0.1, lower.tail = TRUE, 
       log.p = FALSE, max.value = 10000)
rZIBNB(n, mu = 1, sigma = 1, nu = 1, tau = 0.1, max.value = 10000)

Arguments

mu.link	The link function for mu
sigma.link	The link function for sigma
nu.link	The link function for nu
tau.link	The link function for tau
x	vector of (non-negative integer)
mu	vector of positive means
sigma	vector of positive dispersion parameter
nu	vector of a positive parameter
tau	vector of probabilities
log, log.p	logical; if TRUE, probabilities p are given as log(p)
lower.tail	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
p	vector of probabilities
q	vector of quantiles
n	number of random values to return
max.value	a constant, set to the default value of 10000 for how far the algorithm should look for q

Details

The probability function of the BNB is

$P(Y=y|\mu,\sigma, \nu) = \frac{\Gamma(y+\nu^{-1}) B(y+\mu \sigma^{-1} \nu,\sigma^{-1}+\nu^{-1}+1)} {\Gamma(y+1) \Gamma(\nu^{-1}) B(\mu \sigma^{-1} \nu,\sigma^{-1}+1)}$

for $y=0,1,2,3,...$, $\mu>0$, $\sigma>0$ and $\nu>0$, see pp 502-503 of Rigby et al. (2019).

The distribution has mean $\mu$.

The definition of the zero adjusted beta negative binomial distribution, ZABNB and the the zero inflated beta negative binomial distribution, ZIBNB, are given in p. 517 and pp. 519 of of Rigby et al. (2019), respectively.

Value

returns a gamlss.family object which can be used to fit a Poisson distribution in the gamlss() function.

Author(s)

Bob Rigby and Mikis Stasinopoulos

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also https://www.gamlss.com/).

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

NBI, NBII

Examples

BNB()   # gives information about the default links for the beta negative binomial
# plotting the distribution
plot(function(y) dBNB(y, mu = 10, sigma = 0.5, nu=2), from=0, to=40, n=40+1, type="h")
# creating random variables and plot them 
tN <- table(Ni <- rBNB(1000, mu=5, sigma=0.5, nu=2))
r <- barplot(tN, col='lightblue')

ZABNB()
ZIBNB()
# plotting the distribution
plot(function(y) dZABNB(y, mu = 10, sigma = 0.5, nu=2, tau=.1),  
     from=0, to=40, n=40+1, type="h")
plot(function(y) dZIBNB(y, mu = 10, sigma = 0.5, nu=2, tau=.1),  
     from=0, to=40, n=40+1, type="h")
## Not run: 
library(gamlss)
data(species)
species <- transform(species, x=log(lake))
m6 <- gamlss(fish~ pb(x), sigma.fo=~1, data=species, family=BNB)

## End(Not run)

---

Set the Right Link Function for Specified Parameter and Distribution

Description

This function is used within the distribution family specification of a GAMLSS model to define the right link for each of the parameters of the distribution. This function should not be called by the user unless he/she specify a new distribution family or wishes to change existing link functions in the parameters.

Usage

checklink(which.link = NULL, which.dist = NULL, link = NULL, link.List = NULL)

Arguments

which.link	which parameter link e.g. which.link="mu.link"
which.dist	which distribution family e.g. which.dist="Cole.Green"
link	a repetition of which.link e.g. link=substitute(mu.link)
link.List	what link function are required e.g. link.List=c("inverse", "log", "identity")

Value

Defines the right link for each parameter

Author(s)

Calliope Akantziliotou

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape, (with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

---

A set of functions to plot gamlss.family distributions

Description

Those functions are used in the distribution book of gamlss, see Rigby et. al 2019.

Usage

binom_1_31(family = BI, mu = c(0.1, 0.5, 0.7), bd = NULL, miny = 0,
           maxy = 20, cex.axis = 1.2, cex.all = 1.5)
           
binom_2_33(family = BB, mu = c(0.1, 0.5, 0.8), sigma = c(0.5, 1, 2), 
           bd = NULL, miny = 0, maxy = 10, cex.y.axis = 1.5, 
           cex.all = 1.5)  
           
binom_3_33(family = ZIBB, mu = c(0.1, 0.5, 0.8), sigma = c(0.5, 1, 2),
           nu = c(0.01, 0.3), bd = NULL, miny = 0, maxy = 10, 
           cex.axis = 1.5, cex.all = 1.5, cols = c("darkgray", "black"), 
           spacing = 0.3, legend.cex=1, legend.x="topright",
          legend.where=c("left","right", "center"))

contR_2_12(family = "NO", mu = c(0, -1, 1), sigma = c(1, 0.5, 2), 
          cols=c(gray(.1),gray(.2),gray(.3)), 
          ltype = c(1, 2, 3), maxy = 7, 
          no.points = 201, y.axis.lim = 1.1,  
          cex.axis = 1.5, cex.all = 1.5,
          legend.cex=1, legend.x="topleft" )
          
contR_3_11(family = "PE", mu = 0, sigma = 1, nu = c(1, 2, 3), 
          cols=c(gray(.1),gray(.2),gray(.3)), maxy = 7, no.points = 201, 
          ltype = c(1, 2, 3), y.axis.lim = 1.1, cex.axis = 1.5, 
          cex.all = 1.5, legend.cex=1, legend.x="topleft")          

contR_4_13(family = "SEP3", mu = 0, sigma = 1, nu = c(0.5, 1, 2), 
         tau = c(1, 2, 5), cols=c(gray(.1),gray(.2),gray(.3)), maxy = 7, 
         no.points = 201, ltype = c(1, 2, 3), 
         y.axis.lim = 1.1, cex.axis = 1.5, cex.all = 1.5,
         legend.cex=1, legend.x="topleft", 
          legend.where=c("left","right"))

contRplus_1_1(family = EXP, mu = c(1, 2, 3), 
       cols = c(gray(0.1), gray(0.2), gray(0.3)), maxy = 4, 
       no.points = 201, y.axis.lim = 1.1, ltype = c(1, 2, 3), 
       cex.axis = 1.5, cex.all = 1.5, legend.cex = 1, 
       legend.x = "topright")
         
contRplus_2_11(family = GA, mu = 1, sigma = c(0.1, 0.6, 1), 
          cols=c(gray(.1),gray(.2),gray(.3)), 
          maxy = 4, no.points = 201, 
          y.axis.lim = 1.1, ltype = c(1, 2, 3),
           cex.axis = 1.5, cex.all = 1.5,
          legend.cex=1, legend.x="topright")         

contRplus_3_13(family = "BCCG", mu = 1, sigma = c(0.15, 0.2, 0.5), 
          nu = c(-2, 0, 4), 
          cols=c(gray(.1),gray(.2),gray(.3)), 
          maxy = 4, ltype = c(1, 2, 3), 
          no.points = 201, y.axis.lim = 1.1,
          cex.axis = 1.5, cex.all = 1.5,
          legend.cex=1, legend.x="topright",
          legend.where=c("left","right"))
          
contRplus_4_33(family = BCT, mu = 1, sigma = c(0.15, 0.2, 0.5), 
          nu = c(-4, 0, 2), tau = c(100, 5, 1), 
          cols=c(gray(.1),gray(.2),gray(.3)), 
          maxy = 4, ltype = c(1, 2, 3), 
          no.points = 201, y.axis.lim = 1.1, 
          cex.axis = 1.5, cex.all = 1.5,
          legend.cex=1, legend.x="topright",
          legend.where=c("left","right"))

contR01_2_13(family = "BE", mu = c(0.2, 0.5, 0.8), sigma = c(0.2, 0.5, 0.8), 
            cols=c(gray(.1),gray(.2),gray(.3)), 
            ltype = c(1, 2, 3), maxy = 7, no.points = 201, 
            y.axis.lim = 1.1, maxYlim = 10,  
            cex.axis = 1.5, cex.all = 1.5,
            legend.cex=1, legend.x="topright",
            legend.where=c("left","right", "center"))

contR01_4_33(family = GB1, mu = c(0.5), sigma = c(0.2, 0.5, 0.7),
            nu = c(1, 2, 5), tau = c(0.5, 1, 2), 
             cols=c(gray(.1),gray(.2),gray(.3)),  
             maxy = 0.999, ltype = c(1, 2, 3), 
             no.points = 201, y.axis.lim = 1.1, 
             maxYlim = 10,cex.axis = 1.5, cex.all = 1.5,
              legend.cex=1, legend.x="topright",
              legend.where=c("left","right", "center"))

count_1_31(family = PO, mu = c(1, 2, 5), miny = 0, maxy = 10, 
           cex.y.axis = 1.2, cex.all = 1.5)
           
count_1_22(family = PO, mu = c(1, 2, 5, 10), miny = 0, 
           maxy = 20, cex.y.axis = 1.2, cex.all = 1.5) 
           
count_2_32(family = NBI, mu = c(0.5, 1, 5), sigma = c(0.1, 2), 
           miny = 0, maxy = 10, cex.y.axis = 1.5, cex.all = 1.5)

count_2_32R(family = NBI, mu = c(1, 2), sigma = c(0.1, 1, 2), 
           miny = 0, maxy = 10, cex.y.axis = 1.5, cex.all = 1.5) 
           
count_2_33(family = NBI, mu = c(0.1, 1, 2), sigma = c(0.5, 1, 2), 
          miny = 0, maxy = 10, cex.y.axis = 1.5, cex.all = 1.5) 
          
count_3_32(family = SICHEL, mu = c(1, 5, 10), sigma = c(0.5, 1), 
           nu = c(-0.5, 0.5), miny = 0, maxy = 10, cex.axis = 1.5, 
           cex.all = 1.5, cols = c("darkgray", "black"), spacing = 0.2,
           legend.cex=1, legend.x="topright",
                  legend.where=c("left","right"))          
           
count_3_33(family = SICHEL, mu = c(1, 5, 10), sigma = c(0.5, 1, 2), 
           nu = c(-0.5, 0.5, 1), miny = 0, maxy = 10, cex.y.axis = 1.5,
           cex.all = 1.5, cols = c("darkgray", "black"), spacing = 0.3,
           legend.cex=1, legend.x="topright",
           legend.where=c("left","right", "center"))

Arguments

family	a gamlss family distribution
mu	the mu parameter values
sigma	The sigma parameter values
nu	the nu parameter values
tau	the tau parameter values
bd	the binomial denominator
miny	minimal value for the y axis
maxy	maximal value for the y axis
cex.axis	the size of the letters in the two axes
cex.y.axis	the size of the letters in the y axe
cex.all	the overall size of all plotting characters
cols	colours
spacing	spacing between plots
ltype	The type of lines used
no.points	the number of points in the curve
y.axis.lim	the maximum value for the y axis
maxYlim	the maximum permissible value for Y
legend.cex	the size of the legend
legend.x	where in the figure to put the legend
legend.where	where in the whole plot to put the legend

Details

Th function plot different types of continuous and discrete distributions: i) contR: continuous distribution defined on minus infinity to plus infinity, ii) contRplus: continuous distribution defined from zero to plus infinity, iii) contR01: continuous distribution defined from zero to 1, iv) bimom binomial type discrete distributions, v) count count type discrete distributions.

The first number after the first underline in the name of the function indicates the number of parameters in the distribution. The two numbers after the second underline indicate how may rows and columns are in the plot.

Value

The result is a plot

Note

more notes

Author(s)

Mikis Stasinopoulos, Robert Rigby, Gillian Heller, Fernada De Bastiani

References

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, doi:10.18637/jss.v023.i07.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC. doi:10.1201/b21973

(see also https://www.gamlss.com/).

See Also

gamlss.family

Examples

count_1_31()

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