Tools for a reparameterized beta regression model

Description

A set of functions related to the reparameterized beta regression model based on different measures of central tendency: mean, median, mode, geometric mean or harmonic mean.

Usage

BEAM(mu.link = "logit", sigma.link = "log") 
BEGM(mu.link = "logit", sigma.link = "log") 
BEHM(mu.link = "logit", sigma.link = "log") 
BEMD(mu.link = "logit", sigma.link = "log") 
BEMO(mu.link = "logit", sigma.link = "log") 
dBEAM(x, mu = 0.5, sigma = 1, log = FALSE) 
dBEGM(x, mu = 0.5, sigma = 1, log = FALSE) 
dBEHM(x, mu = 0.5, sigma = 1, log = FALSE) 
dBEMD(x, mu = 0.5, sigma = 1, log = FALSE) 
dBEMO(x, mu = 0.5, sigma = 1, log = FALSE) 
dRBE(x, mu=0.5, sigma=1, param="AM", log=FALSE)
fit.RBE(formula = formula(data), sigma.formula=~1, data, param="AM")
pBEAM(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
pBEGM(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
pBEHM(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
pBEMD(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
pBEMO(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE)
pRBE(q, mu=0.5, sigma=1, param="AM", lower.tail = TRUE, log.p = FALSE) 
qBEAM(p, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qBEGM(p, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qBEHM(p, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qBEMD(p, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qBEMO(p, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qRBE(p, mu=0.5, sigma=1, param="AM", lower.tail = TRUE, log.p = FALSE)
rBEAM(n, mu = 0.5, sigma = 1) 
rBEGM(n, mu = 0.5, sigma = 1) 
rBEHM(n, mu = 0.5, sigma = 1) 
rBEMD(n, mu = 0.5, sigma = 1) 
rBEMO(n, mu = 0.5, sigma = 1) 
rRBE(n, mu=0.5, sigma=1, param="AM")

Arguments

Details

The parameterization for the reparameterized beta distribution is given by

f(x; \mu, \sigma) = \frac{x^{\mu\,\sigma + \tau_1-1}(1 - x)^{(1-\mu)\sigma + \tau_2-\tau_1-1}}{B(\mu\,\sigma + \tau_1, (1-\mu)\sigma + \tau_2-\tau_1)}, \quad 0 < x < 1,

where 0 < \mu < 1, \sigma > 0 and \tau_1 and \tau_2 are constant. The following cases are highlighted:

- param="AM": \tau_1=\tau_2=0 and \mu represents the mean of the distribution.

- param="GM": \tau_1=\tau_2=1/2 and \mu represents the geometric mean of the distribution.

- param="HM": \tau_1=\tau_2=1 and \mu represents the harmonic mean of the distribution.

- param="MO": \tau_1=1 and \tau_2=2 and \mu represents the mode of the distribution.

- param="MD": \tau_1=1/2 and \tau_2=0 and \mu represents the median of the distribution.

Suppose the central tendency and the concentration parameter of Y_i satisfies the following functional relations

\mbox{logit}(\mu_i) = \mathbf{x}^\top_i\bm{\xi} \quad \textrm{and} \quad \log(\sigma_i) = \eta_{2i} = \mathbf{z}^\top_i\bm{\nu},

where \mbox{logit}(u)=\log(u/(1-u)) is the logit function, \bm{\xi} = (\xi_1, \ldots, \xi_p)^\top and \bm{\nu} = (\nu_1, \ldots, \nu_q)^\top are vectors of unknown regression coefficients which are assumed to be functionally independent, \bm{\xi} \in \mathbb{R}^p and \bm{\nu} \in \mathbb{R}^q, with p + q < n, and \mathbf{x}_i = (x_{i1}, \ldots, x_{ip})^\top and \mathbf{z}_i = (z_{i1}, \ldots, z_{iq})^\top are observations on p and q known regressors, for i = 1, \ldots, n. Furthermore, we assume that the covariate matrices \mathbf{X} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)^\top and \mathbf{Z} = (\mathbf{z}_1, \ldots, \mathbf{z}_n)^\top have rank p and q, respectively.

For this model, the Pearson's residuals are given by

r_i=\frac{y_i-m_i}{s_i}, \quad i=1,\ldots,n,

where

m_i=\frac{\mu_i \sigma_i+\tau_1}{\sigma_i+\tau_2} \quad \mbox{and} \quad s_i=\sqrt{\frac{(\mu_i \sigma_i+\tau_1)((1-\mu_i)\sigma_i+\tau_2-\tau_1)}{(\sigma_i+\tau_2)^2(\sigma_i+\tau_2+1)}}.

whereas the modified Pearson's residuals are given by

r_i^*=\frac{\mbox{logit}(y_i)-m_i^*}{s_i^*}, \quad i=1,\ldots,n,

where

m_i^*=\psi(\mu_i \sigma_i+\tau_1)-\psi((1-\mu_i)\sigma_i+\tau_2-\tau_1) \quad \mbox{and} \quad s_i^*=\sqrt{\psi'(\mu_i \sigma_i+\tau_1)+\psi'((1-\mu_i)\sigma_i+\tau_2-\tau_1)},

with \psi(\cdot) and \psi'(\cdot) denoting the digamma and trigamma functions, respectively. Finally, the quantile residuals are given by

r_i^q=\Phi^{-1}\left(I_{y_i}(\mu_i \sigma_i+\tau_1,(1-\mu_i)\sigma_i+\tau_2-\tau_1)\right), \quad i=1,\ldots,n,

where \Phi^{-1}(\cdot) denotes the inverse of the cumulative distribution function for the standard normal model and I_y(\alpha,\beta)=B_x(\alpha, \beta)/B(\alpha, \beta) is the incomplete beta function ratio, B_x(\alpha, \beta) = \int_{0}^{x}\omega^{\alpha-1}(1-\omega)^{\beta-1}\textrm{d} \omega is the incomplete beta function, B(\alpha, \beta) = \Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha + \beta) is the beta function and \Gamma(\alpha) = \int_{0}^{\infty}\omega^{\alpha-1}\textrm{e}^{-\omega}\textrm{d} \omega is the gamma function. dRBE gives the density, pRBE gives the distribution function, qRBE gives the quantile function, and rRBE generates random deviates from the beta distribution with the specified parameterization. In addition, dBEXX, pBEXX, qBEXX and rBEXX also provides the equivalent functions for a specified parameterization for XX: AM (mean), GM (geometric mean), HM (harmonic mean), MD (median) and MO (mode). For instance, dBEAM gives the density for the beta model parameterized in the mean, pBEGM gives the distribution function for the beta model parameterized in the geometric mean and so on. Finally, the functions BEAM, BEGM, BEHM, BEMD and BEMO also provide a framework to fit models with gamlss.

Value

an object of class "rregm" is returned. The object returned for this functions is a list containing the following components:

Author(s)

Diego Gallardo and Marcelo Bourguignon.

References

Bourguignon, M., Gallardo, D.I. (2025) A general and unified parameterization of the beta distribution: A flexible and robust beta regression model. Statistica Neerlandica, 79(2), e70007.

Examples

set.seed(2100)
n=100; x1=rnorm(max(n)) ##drawing covariates, the same for mu and sigma
mu=plogis(0.5-0.4*x1); sigma=exp(-0.1+0.05*x1)
y=rRBE(n, mu, sigma, param="MD") ## model parameterized in the median
data=list(y=y, x1=x1)
aux.RBE=fit.RBE(y~x1, sigma.formula=~x1, data=data, param="MD")
summary(aux.RBE)
qqnorm(res(aux.RBE, type="mod.pearson"))
#The beta model parameterized in the median also can be fitted using gamlss
#gamlss(y~x1, sigma.formula=~x1, data=data, family=BEMD)

Tools for a reparameterized beta prime regression model

Description

A set of functions related to the reparameterized beta prime regression model based on different measures of central tendency: mean, median, mode, geometric mean or harmonic mean.

Usage

fit.RBP(formula = formula(data), sigma.formula=~1, data, param="AM")
dRBP(x, mu=1, sigma=1.5, param="AM", log=FALSE)
pRBP(q, mu=1, sigma=1.5, param="AM", lower.tail = TRUE, log.p = FALSE) 
qRBP(p, mu=1, sigma=1.5, param="AM", lower.tail = TRUE, log.p = FALSE)
rRBP(n, mu=1, sigma=1.5, param="AM")

Arguments

Details

The parameterization for the reparameterized beta prime distribution is given by

f(x; \mu, \sigma) = \frac{x^{\mu\,\sigma + \tau_1 -1}(1 + x)^{-(\sigma(1+\mu) + \tau_1-\tau_2+1)}}{B(\mu\,\sigma + \tau_1, \sigma - \tau_2 + 1)}, \quad 0 < x < 1,

where \delta=\delta(\sigma)=(\sqrt{\sigma(\sigma+4)}+\sigma)/2, 0 < \mu < 1, \sigma > 0 and \tau is a constant. The following cases are highlighted:

- param="AM": \tau=0 and \mu represents the mean of the distribution.

- param="GM": \tau=1/2 and \mu represents the geometric mean of the distribution.

- param="MD": \tau=1/3 and \mu represents the median of the distribution.

- param="MO" or ="HM": \tau=1 and \mu represents the mode or the harmonic mean of the distribution.

Suppose the central tendency and the concentration parameter of Y_i satisfies the following functional relations

\log(\mu_i) = \mathbf{x}^\top_i\bm{\xi} \quad \textrm{and} \quad \log(\sigma_i) = \eta_{2i} = \mathbf{z}^\top_i\bm{\nu},

where \bm{\xi} = (\xi_1, \ldots, \xi_p)^\top and \bm{\nu} = (\nu_1, \ldots, \nu_q)^\top are vectors of unknown regression coefficients which are assumed to be functionally independent, \bm{\xi} \in \mathbb{R}^p and \bm{\nu} \in \mathbb{R}^q, with p + q < n, and \mathbf{x}_i = (x_{i1}, \ldots, x_{ip})^\top and \mathbf{z}_i = (z_{i1}, \ldots, z_{iq})^\top are observations on p and q known regressors, for i = 1, \ldots, n. Furthermore, we assume that the covariate matrices \mathbf{X} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)^\top and \mathbf{Z} = (\mathbf{z}_1, \ldots, \mathbf{z}_n)^\top have rank p and q, respectively.

For this model, the Pearson's residuals are given by

r_i=\frac{y_i-m_i}{s_i}, \quad i=1,\ldots,n,

where

m_i=\frac{\mu_i \sigma_i+\tau_1}{\sigma_i-\tau_2+1} \quad \mbox{and} \quad s_i=\frac{(\mu_i \sigma_i+\tau_1)((1+\mu_i)\sigma_i+\tau_1-\tau_2+1)}{(\sigma_i+\tau-1)}(\sigma_i-\tau_2)^{-1/2}.

Note that the Pearson's residuals are well defined as long as \sigma_i-\tau_2>1, \forall i=1,\ldots,n. On the other hand, the modified Pearson's residuals are given by

r_i^*=\frac{\mbox{logit}(y_i)-m_i^*}{s_i^*}, \quad i=1,\ldots,n,

where

m_i^*=\psi(\mu_i \sigma_i+\tau_1)-\psi((1+\mu_i)\sigma_i+\tau_1-\tau_2+1) \quad \mbox{and} \quad s_i^*=\sqrt{\psi'(\mu_i \sigma_i+\tau_1)-\psi'((1+\mu_i)\sigma_i+\tau_1-\tau_2+1)},

with \psi(\cdot) and \psi'(\cdot) denoting the digamma and trigamma functions, respectively. Finally, the quantile residuals are given by

r_i^q=\Phi^{-1}\left(I_{\frac{y_i}{1+y_i}}(\mu_i \sigma_i+\tau_1,(1+\mu_i)\sigma_i+\tau_1-\tau_2+1)\right), \quad i=1,\ldots,n,

where \Phi^{-1}(\cdot) denotes the inverse of the cumulative distribution function for the standard normal model and I_y(\alpha,\beta)=B_x(\alpha, \beta)/B(\alpha, \beta) is the incomplete beta function ratio, B_x(\alpha, \beta) = \int_{0}^{x}\omega^{\alpha-1}(1-\omega)^{\beta-1}\textrm{d} \omega is the incomplete beta function, B(\alpha, \beta) = \Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha + \beta) is the beta function and \Gamma(\alpha) = \int_{0}^{\infty}\omega^{\alpha-1}\textrm{e}^{-\omega}\textrm{d} \omega is the gamma function. dRBP gives the density, pRBP gives the distribution function, qRBP gives the quantile function, and rRBP generates random deviates from the beta distribution with the specified parameterization.

Value

an object of class "rregm" is returned. The object returned for this functions is a list containing the following components:

Author(s)

Diego Gallardo and Marcelo Bourguignon.

References

Bourguignon, M., Gallardo, D.I. (2025) A general and unified parameterization of the beta distribution: A flexible and robust beta regression model. Statistica Neerlandica, 79(2), e70007.

Examples

set.seed(2100)
n=100; x1=rnorm(max(n)) ##drawing covariates, the same for mu and sigma
mu=exp(0.5-0.4*x1); sigma=exp(-0.1+0.05*x1)
y=rRBP(n, mu, sigma, param="MD") ## model parameterized in the median
data=list(y=y, x1=x1)
aux.RBP=fit.RBP(y~x1, sigma.formula=~x1, data=data, param="MD")
summary(aux.RBP)
qqnorm(res(aux.RBP, type="mod.pearson"))

Tools for a reparameterized gamma regression model

Description

A set of functions related to the reparameterized gamma regression model based on different measures of central tendency: mean, median, mode, geometric mean or harmonic mean.

Usage

fit.RGA(formula = formula(data), sigma.formula=~1, data, param="AM")
dRGA(x, mu=1, sigma=1, param="AM", log=FALSE)
pRGA(q, mu=1, sigma=1, param="AM", lower.tail = TRUE, log.p = FALSE) 
qRGA(p, mu=1, sigma=1, param="AM", lower.tail = TRUE, log.p = FALSE)
rRGA(n, mu=1, sigma=1, param="AM")

Arguments

Details

The parameterization for the reparameterized beta distribution is given by

f(x; \mu, \sigma) = \frac{(\delta/\mu)^{\delta + \tau}}{\Gamma(\delta + \tau)}x^{\delta+\tau-1}\textrm{e}^{-\delta x/\mu}, \quad y > 0,

where \delta=\delta(\sigma)=(\sqrt{\sigma(\sigma+4)}+\sigma)/2, 0 < \mu < 1, \sigma > 0 and \tau is a constant. The following cases are highlighted:

- param="AM": \tau=0 and \mu represents the mean of the distribution.

- param="GM": \tau=1/2 and \mu represents the geometric mean of the distribution.

- param="MD": \tau=1/3 and \mu represents the median of the distribution.

- param="MO" or ="HM": \tau=1 and \mu represents the mode or the harmonic mean of the distribution.

Suppose the central tendency and the concentration parameter of Y_i satisfies the following functional relations

\log(\mu_i) = \mathbf{x}^\top_i\bm{\xi} \quad \textrm{and} \quad \log(\sigma_i) = \eta_{2i} = \mathbf{z}^\top_i\bm{\nu},

where \bm{\xi} = (\xi_1, \ldots, \xi_p)^\top and \bm{\nu} = (\nu_1, \ldots, \nu_q)^\top are vectors of unknown regression coefficients which are assumed to be functionally independent, \bm{\xi} \in \mathbb{R}^p and \bm{\nu} \in \mathbb{R}^q, with p + q < n, and \mathbf{x}_i = (x_{i1}, \ldots, x_{ip})^\top and \mathbf{z}_i = (z_{i1}, \ldots, z_{iq})^\top are observations on p and q known regressors, for i = 1, \ldots, n. Furthermore, we assume that the covariate matrices \mathbf{X} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)^\top and \mathbf{Z} = (\mathbf{z}_1, \ldots, \mathbf{z}_n)^\top have rank p and q, respectively.

For this model, the Pearson's residuals are given by

r_i=\frac{y_i-m_i}{s_i}, \quad i=1,\ldots,n,

where

m_i=\mu_i\left(1+\frac{\tau}{\xi(\sigma_i,\tau)}\right) \quad \mbox{and} \quad s_i=\frac{\mu}{\xi(\sigma_i,\tau)}\sqrt{\tau+\xi(\sigma_i,\tau)},

where \xi(\sigma_i,\tau)=(\sqrt{\sigma_i(\sigma_i+4\tau)}+\sigma_i)/2. On the other hand, the modified Pearson's residuals are given by

r_i^*=\frac{\log(y_i)-m_i^*}{s_i^*}, \quad i=1,\ldots,n,

where

m_i^*=\psi(\tau+\xi(\sigma_i,\tau))+\log \mu_i-\log \xi(\sigma_i,\tau)) \quad \mbox{and} \quad s_i^*=\sqrt{\psi'(\tau+\xi(\sigma_i,\tau))},

with \psi(\cdot) and \psi'(\cdot) denoting the digamma and trigamma functions, respectively. Finally, the quantile residuals are given by

r_i^q=\Phi^{-1}\left(\frac{\gamma(\tau+\xi(\sigma_i,\tau),\xi(\sigma_i,\tau) y_i/ \mu_i}{\Gamma(\tau+\xi(\sigma_i,\tau))}\right), \quad i=1,\ldots,n,

where \Phi^{-1}(\cdot) denotes the inverse of the cumulative distribution function for the standard normal model and \gamma(a,z) = \int_{0}^{z}t^{a-1}\textrm{e}^{-t}\textrm{d}t is the lower incomplete gamma function and \Gamma(\alpha) = \int_{0}^{\infty}\omega^{\alpha-1}\textrm{e}^{-\omega}\textrm{d} \omega is the gamma function. dRGA gives the density, pRGA gives the distribution function, qRGA gives the quantile function, and rRGA generates random deviates from the beta distribution with the specified parameterization.

Value

an object of class "rregm" is returned. The object returned for this functions is a list containing the following components:

Author(s)

Diego Gallardo and Marcelo Bourguignon.

References

Bourguignon, M., Gallardo, D.I. (2025) A general and unified class of gamma regression models. Chemometrics and Intelligent Laboratory Systems, 261, 105382.

Examples

set.seed(2100)
n=100; x1=rnorm(max(n)) ##drawing covariates, the same for mu and sigma
mu=exp(0.5-0.4*x1); sigma=exp(-0.1+0.05*x1)
y=rRGA(n, mu, sigma, param="MD") ## model parameterized in the median
data=list(y=y, x1=x1)
aux.RGA=fit.RGA(y~x1, sigma.formula=~x1, data=data, param="MD")
summary(aux.RGA)
qqnorm(res(aux.RGA, type="mod.pearson"))

Tools for a reparameterized inverse gamma regression model

Description

A set of functions related to the reparameterized inverse gamma regression model based on different measures of central tendency: mean, median, mode, geometric mean or harmonic mean.

Usage

IGAM(mu.link = "log", sigma.link = "log") 
IGGM(mu.link = "log", sigma.link = "log") 
IGHM(mu.link = "log", sigma.link = "log") 
IGMD(mu.link = "log", sigma.link = "log") 
IGMO(mu.link = "log", sigma.link = "logshiftto1") 
dIGAM(x, mu = 1, sigma = 1, log = FALSE) 
dIGGM(x, mu = 1, sigma = 1, log = FALSE) 
dIGHM(x, mu = 1, sigma = 1, log = FALSE) 
dIGMD(x, mu = 1, sigma = 1, log = FALSE) 
dIGMO(x, mu = 1, sigma = 1.5, log = FALSE) 
dRIG(x, mu=1, sigma=1.5, param="AM", log=FALSE)
fit.RIG(formula = formula(data), sigma.formula=~1, data, param="AM")
pIGAM(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
pIGGM(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
pIGHM(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
pIGMD(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
pIGMO(q, mu = 1, sigma = 1.5, lower.tail = TRUE, log.p = FALSE)
pRIG(q, mu=1, sigma=1.5, param="AM", lower.tail = TRUE, log.p = FALSE) 
qIGAM(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qIGGM(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qIGHM(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qIGMD(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) 
qIGMO(p, mu = 1, sigma = 1.5, lower.tail = TRUE, log.p = FALSE) 
qRIG(p, mu=1, sigma=1.5, param="AM", lower.tail = TRUE, log.p = FALSE)
rIGAM(n, mu = 1, sigma = 1) 
rIGGM(n, mu = 1, sigma = 1) 
rIGHM(n, mu = 1, sigma = 1) 
rIGMD(n, mu = 1, sigma = 1) 
rIGMO(n, mu = 1, sigma = 1.5) 
rRIG(n, mu=1, sigma=1.5, param="AM")

Arguments

Details

The parameterization for the reparameterized inverse gamma distribution is given by

f(y; \mu, \sigma) = \frac{(\sigma \mu)^{\sigma + \tau}}{\Gamma(\sigma + \tau)}y^{-\sigma - \tau-1}\textrm{e}^{-\frac{\sigma \mu}{y}}, \quad y > 0,

where 0 < \mu < 1, \sigma > 0 and \tau is a constant. The following cases are highlighted:

- param="AM": \tau=1 and \mu represents the mean of the distribution.

- param="GM": \tau=1/2 and \mu represents the geometric mean of the distribution.

- param="HM": \tau=0 and \mu represents the harmonic mean of the distribution.

- param="MO": \tau=-1 and \mu represents the mode of the distribution.

- param="MD": \tau=1/2 and \mu represents the median of the distribution.

Suppose the central tendency and the concentration parameter of Y_i satisfies the following functional relations

\log(\mu_i) = \mathbf{x}^\top_i\bm{\xi} \quad \textrm{and} \quad \log(\sigma_i) = \eta_{2i} = \mathbf{z}^\top_i\bm{\nu},

\bm{\xi} = (\xi_1, \ldots, \xi_p)^\top and \bm{\nu} = (\nu_1, \ldots, \nu_q)^\top are vectors of unknown regression coefficients which are assumed to be functionally independent, \bm{\xi} \in \mathbb{R}^p and \bm{\nu} \in \mathbb{R}^q, with p + q < n, and \mathbf{x}_i = (x_{i1}, \ldots, x_{ip})^\top and \mathbf{z}_i = (z_{i1}, \ldots, z_{iq})^\top are observations on p and q known regressors, for i = 1, \ldots, n. Furthermore, we assume that the covariate matrices \mathbf{X} = (\mathbf{x}_1, \ldots, \mathbf{x}_n)^\top and \mathbf{Z} = (\mathbf{z}_1, \ldots, \mathbf{z}_n)^\top have rank p and q, respectively.

For this model, the Pearson's residuals are given by

r_i=\frac{y_i-m_i}{s_i}, \quad i=1,\ldots,n,

where

m_i=\frac{\mu_i \sigma_i}{\sigma_i+\tau_2} \quad \mbox{and} \quad s_i=\frac{\mu_i \sigma_i}{(\sigma_i+\tau-1)}(\sigma_i+\tau-2)^{-1/2}.

Note that the Pearson's residuals are well defined as long as \sigma_i+\tau>2, \forall i=1,\ldots,n. On the other hand, the modified Pearson's residuals are given by

r_i^*=\frac{-\log(y_i)-m_i^*}{s_i^*}, \quad i=1,\ldots,n,

where

m_i^*=\psi(\sigma_i+\tau)-\log(\mu_i \sigma_i) \quad \mbox{and} \quad s_i^*=\sqrt{\psi'(\sigma_i+\tau)},

with \psi(\cdot) and \psi'(\cdot) denoting the digamma and trigamma functions, respectively. Finally, the quantile residuals are given by

r_i^q=\Phi^{-1}\left(\frac{\gamma(\sigma_i+\tau,\mu_i \sigma_i/y_i)}{\Gamma(\sigma_i+\tau)}\right), \quad i=1,\ldots,n,

where \Phi^{-1}(\cdot) denotes the inverse of the cumulative distribution function for the standard normal model and \gamma(a,z) = \int_{0}^{z}t^{a-1}\textrm{e}^{-t}\textrm{d}t is the lower incomplete gamma function and \Gamma(\alpha) = \int_{0}^{\infty}\omega^{\alpha-1}\textrm{e}^{-\omega}\textrm{d} \omega is the gamma function. dRIG gives the density, pRIG gives the distribution function, qRIG gives the quantile function, and rRIG generates random deviates from the inverse gamma distribution with the specified parameterization. In addition, dIGXX, pIGXX, qIGXX and rIGXX also provides the equivalent functions for a specified parameterization for XX: AM (mean), GM (geometric mean), HM (harmonic mean), MD (median) and MO (mode). For instance, dIGAM gives the density for the inverse gamma model parameterized in the mean, pIGGM gives the distribution function for the inverse gamma model parameterized in the geometric mean and so on. Finally, the functions IGAM, IGGM, IGHM, IGMD and IGMO also provide a framework to fit models with gamlss.

Value

an object of class "rregm" is returned. The object returned for this functions is a list containing the following components:

Author(s)

Diego Gallardo and Marcelo Bourguignon.

References

Bourguignon, M., Gallardo, D.I. (2025) A general and unified class of gamma regression models. Chemometrics and Intelligent Laboratory Systems, 261, 105382.

Examples

set.seed(2100)
n=100; x1=rnorm(max(n)) ##drawing covariates, the same for mu and sigma
mu=exp(0.5-0.4*x1); sigma=exp(-0.1+0.05*x1)
y=rRIG(n, mu, sigma, param="MD") ## model parameterized in the median
data=list(y=y, x1=x1)
aux.RIG=fit.RIG(y~x1, sigma.formula=~x1, data=data, param="MD")
summary(aux.RIG)
qqnorm(res(aux.RIG, type="mod.pearson"))
#The inverse gamma model parameterized in the median also can be fitted using gamlss
#gamlss(y~x1, sigma.formula=~x1, data=data, family=IGMD)

Print a summary for a object of the "rregm" class.

Description

Tools for a objects of the "rregm" class.

Usage

res(object, type="pearson")
## S3 method for class 'rregm'
AIC(object, ..., k=2)
## S3 method for class 'rregm'
BIC(object, ...)
## S3 method for class 'rregm'
coef(object, ...)
## S3 method for class 'rregm'
logLik(object, ...)
## S3 method for class 'rregm'
print(x, digits = max(3L, getOption("digits") - 3L), ...)
## S3 method for class 'rregm'
summary(object, ...)

Arguments

Details

Supported regression models are: - reparametrized beta - reparametrized gamma - reparametrized beta prime - reparametrized inverse gamma

Value

A complete summary for the coefficients extracted from a "rregm" object.

Author(s)

Diego Gallardo and Marcelo Bourguignon.

References

Gallardo and Bourguignon (2022).

Examples

set.seed(2100)
n=100; x1=rnorm(max(n)) ##drawing covariates, the same for mu and sigma
mu=exp(0.5-0.4*x1); sigma=exp(-0.1+0.05*x1)
y=rRGA(n, mu, sigma, param="MD") ## model parameterized in the median
data=list(y=y, x1=x1)
aux.RGA=fit.RGA(y~x1, sigma.formula=~x1, data=data, param="MD")
summary(aux.RGA)
qqnorm(res(aux.RGA, type="mod.pearson"))