RelDists, DiscreteDists and RealDists

New distributions to complement GAMLSS regression models

Olga Usuga Manco, Freddy HernΓ‘ndez Barajas

Universidad de Antioquia-Colombia , Universidad Nacional de Colombia-Colombia

September 18, 2024

Slides available at

RelDists, DiscreteDists and RealDists

How these packages fit into R and the statistical literature?

Figure 1: Packages

Linear Models

\[\begin{align} Y_i &\sim N(\mu_i, \sigma^2), \\ \mu_i &= \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \cdots + \beta_p X_{pi}, \\ \sigma^2 &= \text{constant}. \end{align}\]

Generalized linear models

\[\begin{align} Y_i &\sim D(\mu_i, \phi), \\ g(\mu_i) &= \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \cdots + \beta_p X_{pi}, \\ \phi &= \text{constant}. \end{align}\]

The distribution \(D\) can be: Normal, Binomial, Negative Binomial, Poisson, Gamma and Inverse gaussian.

GAMLSS: Generalized Additive Models for Location, Scale and Shape

\[\begin{align} Y_i &\sim D(\mu_i, \sigma_i, \nu_i, \tau_i), \\ g_1(\mu_i) &= \beta_{10} + \beta_{11} X_{1i} + \beta_{12} X_{2i} + \cdots + \beta_{1p} X_{pi}, \\ g_2(\sigma_i) &= \beta_{20} + \beta_{21} X_{1i} + \beta_{22} X_{2i} + \cdots + \beta_{2p} X_{pi}, \\ g_3(\nu_i) &= \beta_{30} + \beta_{31} X_{1i} + \beta_{32} X_{2i} + \cdots + \beta_{3p} X_{pi}, \\ g_4(\tau_i) &= \beta_{40} + \beta_{41} X_{1i} + \beta_{42} X_{2i} + \cdots + \beta_{4p} X_{pi}. \end{align}\]

The distribution \(D\) can be: Normal, Binomial, Negative Binomial, Poisson, Gamma and Inverse gaussian, Weibull, beta, ZIP, ….

GAMLSS: Generalized Additive Models for Location, Scale and Shape

More details about gamlss in https://www.gamlss.com.

RelDists

Distributions

Thirty distributions were implemented in the RelDists package. Some of these distributions are:

  1. Additive Weibull AddW.
  2. Beta Generalized Exponentiated BGE.
  3. Cosine Sine Exponential CS2e.
  4. Extended Exponential Geometric EEG.
  5. Flexible Weibull Extension FWE.
  6. …

The complete list can be found at this link.

Functions

Each distribution has the following functions:

dXXX()  # Density
pXXX()  # P(X<=x)
qXXX()  # Quantile
rXXX()  # Random number generator
hXXX()  # Hazard 
 XXX()  # Family

XXX: It refers to the short name of the distribution.

Flexible Weibull Extension (FWE)

This distribution was proposed by Bebbington (2007) and its density function is given by the following expression:

\[ f(y; \mu, \sigma) = \left( \mu+ \frac{\sigma}{y^2} \right) e^{\mu y - \sigma / y} \exp \left( -e^{\mu y - \sigma / y} \right), \]

with \(\mu > 0\), \(\sigma > 0\), \(y>0\).

Flexible Weibull Extension (FWE)

The hazard function of the FWE distribution has great flexibility and it is given by:

\[h(y) = \left( \mu+ \frac\sigma{y^2} \right) e^{\mu y - \sigma / y}.\]

Probability density function

Hazard function

Example 1

n <- 100
mu <- 0.75
sigma <- 1.3

library(RelDists)
set.seed(123)
y <- rFWE(n=n, mu, sigma)

library(gamlss)
mod <- gamlss(y~1, sigma.fo=~1, family="FWE", control=gamlss.control(trace=FALSE))

exp(coef(mod, what="mu"))
(Intercept) 
  0.7776793 
exp(coef(mod, what="sigma"))
(Intercept) 
   1.331533

The estimates are close to the real values πŸ™.

Example 2

n <- 200
library(RelDists)
set.seed(123)
{
  x1 <- runif(n)
  x2 <- runif(n)
  mu <- exp(1.21 - 3 * x1)
  sigma <- exp(1.26 - 2 * x2)
  y <- rFWE(n=n, mu, sigma)
}

library(gamlss)
mod <- gamlss(y~x1, sigma.fo=~x2, family=FWE, control=gamlss.control(trace=FALSE))

coef(mod, what="mu")
(Intercept)          x1 
   1.131260   -2.831916 
coef(mod, what="sigma")
(Intercept)          x2 
   1.305884   -2.063528

DiscreteDists

Distributions

Nine distributions were implemented in the DiscreteDists package. Some of these distributions are:

  1. Discrete Burr Hatke DBH.
  2. Discrete generalized exponential DGEII.
  3. Discrete Inverted Kumaraswamy DIKUM.
  4. Discrete Lindley DLD.
  5. Discrete Generalized Exponential II DGEII.
  6. …

The complete list can be found at this link.

Functions

Each distribution has the following functions:

dXXX()  # Density
pXXX()  # P(X<=x)
qXXX()  # Quantile
rXXX()  # Random number generator
 XXX()  # Family

XXX: It refers to the short name of the distribution.

Discrete Generalized Exponential Distribution

The Discrete Generalized Exponential Distribution (DGEII) distribution with parameters \(\mu\) and \(\sigma\) has a support \(0, 1, 2, ...\) and mass function given by

\[ 𝑓(π‘₯|πœ‡,𝜎)=(1βˆ’πœ‡π‘₯+1)πœŽβˆ’(1βˆ’πœ‡π‘₯)𝜎, \]

with \(0<\mu<1\) and \(\sigma>0\). If \(\sigma=1\), the DGEII distribution reduces to the geometric distribution with success probability \(1βˆ’\mu\).

Mass probability function

Example

n <- 100
mu <- 0.75
sigma <- 0.5

library(DiscreteDists)
set.seed(123)
y <- rDGEII(n = n, mu, sigma)

library(gamlss)
mod <- gamlss(y~1, family=DGEII, control=gamlss.control(n.cyc=500, trace=FALSE))

inv_logit <- function(x) 1/(1 + exp(-x))
inv_logit(coef(mod, what="mu"))
(Intercept) 
  0.7472655 
exp(coef(mod, what="sigma"))
(Intercept) 
  0.4681355

RealDists

Distributions

One distribution was implemented in the RealDists package:

  1. Generalised exponential-Gaussian GEG.

The package can be found at this link.

Functions

The distribution has the following functions:

dXXX()  # Density
pXXX()  # P(X<=x)
qXXX()  # Quantile
rXXX()  # Random number generator
 XXX()  # Family

XXX: It refers to the short name of the distribution.

Generalised exponential-Gaussian

The Generalised exponential-Gaussian with parameters \(\mu, \sigma, \nu\) and \(\tau\) has density given by

\[ 𝑓(π‘₯|πœ‡,𝜎,𝜈,𝜏)=\frac{𝜏}{𝜈}\exp(𝑀)Ξ¦\left(π‘§βˆ’\frac{𝜎}{𝜈}\right)\left[Ξ¦(𝑧)βˆ’\exp(𝑀)Ξ¦\left(π‘§βˆ’\frac{𝜎}{𝜈}\right)\right]^{πœβˆ’1}, \]

for \(βˆ’βˆž<π‘₯<∞\). With \(𝑀=\frac{πœ‡βˆ’π‘₯}{𝜈}+\frac{\sigma^2}{2\nu^2}\) and \(𝑧=\frac{π‘₯βˆ’πœ‡}{𝜎}\) and \(Ξ¦\) is the cumulative function for the standard normal distribution.

Density probability function

Example

n <- 500
mu <- -5 ; sigma <- 4 ; nu <- 2.5 ; tau <- 1

library(RealDists)
set.seed(123)
y <- rGEG(n=n, mu, sigma, nu, tau)

library(gamlss)
mod <- gamlss(y ~ 1, family=GEG, control=gamlss.control(n.cyc=1000, trace=FALSE))

coef(mod, what="mu")
(Intercept) 
  -5.918508 
exp(coef(mod, what="sigma"))
(Intercept) 
   3.725721 
exp(coef(mod, what="nu"))
(Intercept) 
   2.978824 
exp(coef(mod, what="tau"))
(Intercept) 
   1.116511

Final comments

These new R packages allow:

  1. Use the functions dXXX(), pXXX(), qXXX(), and rXXX().

  2. Estimate distribution parameters and regression model parameters can be estimated.

  3. Integrate the new distributions into their statistical analyses.

Future work

If you want to be part of the team that develops these packages, please write to us at: fhernanb@unal.edu.co and olga.usuga@udea.edu.co.

Conference

Conference

COBIPE - 1ΒΊ ColΓ³quio Binacional Brasil-ColΓ΄mbia de Probabilidade e EstatΓ­stica