| Type: | Package |
| Title: | Maximum Likelihood Estimation of the Generalized Poisson Distribution |
| Version: | 1.1 |
| Date: | 2023-10-23 |
| Author: | Michail Tsagris [aut, cre] |
| Maintainer: | Michail Tsagris <mtsagris@uoc.gr> |
| Depends: | R (≥ 4.0) |
| Imports: | Rfast, RNGforGPD, stats |
| Description: | Functions to estimate the parameters of the generalized Poisson distribution with or without covariates using maximum likelihood. The references include Nikoloulopoulos A.K. & Karlis D. (2008). "On modeling count data: a comparison of some well-known discrete distributions". Journal of Statistical Computation and Simulation, 78(3): 437–457, <doi:10.1080/10629360601010760> and Consul P.C. & Famoye F. (1992). "Generalized Poisson regression model". Communications in Statistics - Theory and Methods, 21(1): 89–109, <doi:10.1080/03610929208830766>. |
| License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
| NeedsCompilation: | no |
| Packaged: | 2023-10-23 14:07:06 UTC; mtsag |
| Repository: | CRAN |
| Date/Publication: | 2023-10-23 15:50:16 UTC |
Maximum Likelihood Estimation of the Generalized Poisson Distribution.
Description
Maximum likelihood estimation of the generalized Poisson distribution. Regression modelling is also supported.
Details
| Package: | gp |
| Type: | Package |
| Version: | 1.1 |
| Date: | 2023-10-23 |
| License: | GPL-2 |
Maintainers
Michail Tsagris mtsagris@uoc.gr.
Author(s)
Michail Tsagris mtsagris@uoc.gr.
References
Nikoloulopoulos A.K. & Karlis D. (2008). On modeling count data: a comparison of some well-known discrete distributions. Journal of Statistical Computation and Simulation, 78(3): 437–457. Consul P.C. & Famoye F. (1992). Generalized poisson regression model. Communications in Statistics - Theory and Methods, 21(1): 89–109.
Cumulative probability mass of the generalized Poisson distribution
Description
Cumulative probability mass of the generalized Poisson distribution.
Usage
pgp(y, theta, lambda)
Arguments
y |
A vector with non negative integer values. |
theta |
The value of the |
lambda |
The value of the |
Details
The cumulative probability mass of the generealized Poisson distribution is computed.
Value
A vector with the probabilities.
Author(s)
Michail Tsagris.
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
References
Nikoloulopoulos A.K. & Karlis D. (2008). On modeling count data: a comparison of some well-known discrete distributions. Journal of Statistical Computation and Simulation, 78(3): 437–457.
Demirtas H. (2017). On accurate and precise generation of generalized Poisson variates. Communications in Statistics - Simulation and Computation, 46(1): 489–499.
See Also
Examples
y <- rgp(1000, 10, 0.5, method = "Inversion")
a <- gp.mle(y)
pgp(y[1:10], a[1], a[2])
Density computation of the generalized Poisson distribution
Description
Density computation of the generalized Poisson distribution.
Usage
dgp(y, theta, lambda, logged = TRUE)
Arguments
y |
A vector with non negative integer values. |
theta |
The value of the |
lambda |
The value of the |
logged |
Should the logarithm of the density values be computed? The default value is TRUE. |
Details
The density of the generealized Poisson distribution is computed.
Value
A vector with the logged density values.
Author(s)
Michail Tsagris.
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
References
Nikoloulopoulos A.K. & Karlis D. (2008). On modeling count data: a comparison of some well-known discrete distributions. Journal of Statistical Computation and Simulation, 78(3): 437–457.
Demirtas H. (2017). On accurate and precise generation of generalized Poisson variates. Communications in Statistics - Simulation and Computation, 46(1): 489–499.
See Also
Examples
y <- rgp(1000, 10, 0.5, method = "Inversion")
a <- gp.mle(y)
f <- dgp(y, a[1], a[2])
sum(f)
Maximum likelihood estimation of the generalized Poisson distribution
Description
Maximum likelihood estimation of the generalized Poisson distribution.
Usage
gp.mle(y)
Arguments
y |
A vector with non negative integer values. |
Details
The probability density function of the generalized Poisson distribution is the following (Nikoloulopoulos & Karlis, 2008):
P(Y=y|\theta, \lambda)=\theta(\theta+\lambda y)^{y-1}\frac{e^{-\theta-\lambda y}}{y!},
\ \ y=0,1... \ \ \theta >0, \ \ 0 \leq \lambda \leq 1.
To ensure that \theta is positive we use the "log" link and for \lambda to lie within 0 and 1
we use the "logit" link within the optim function.
Value
A vector with three numbers, the \theta and \lambda parameters and the value of the log-likelihood.
Author(s)
Michail Tsagris.
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
References
Nikoloulopoulos A.K. & Karlis D. (2008). On modeling count data: a comparison of some well-known discrete distributions. Journal of Statistical Computation and Simulation, 78(3): 437–457.
See Also
Examples
y <- rgp(1000, 10, 0.5, method = "Inversion")
gp.mle(y)
Generalized Poisson regression
Description
Generalized Poisson regression.
Usage
gp.reg(y, x, tol = 1e-7)
gp.reg2(y, x, tol = 1e-7)
Arguments
y |
The response variable, a vector with non negative integer values. |
x |
A data.frame or a matrix with the independent variables. |
tol |
The tolerance value to terminate the optimization. |
Details
The loglikelihood of the generalised Poisson distribution when covariates are present is the following (Consul & Famoye, 1992):
\ell(\beta, \phi)=\sum_{i=1}^n\log(\mu_i) + \sum_{i=1}^n(y_i-1)\log{[\mu_i+(\phi-1)y_i]}-
\log{\phi}\sum_{i=1}^ny_i-\frac{1}{\phi}\sum_{i=1}^n[\mu_i+(\phi-1)y_i]-\sum_{i=1}^n\log{(y_i)},
where \mu_i=e^{\sum_{j=0}^kX_{ij}\beta_j}, n denotes the sample size, k is the number of \beta coefficients, and \phi > 0.
Breslow (1984) suggested the (moment) estimation of a dispersion parameter by equating the chi-square statistic to its degrees of freedom. For the generalised Poisson regression model, this leads to
\sum_{i=1}^n\frac{(y_i-\mu_i)^2}{\mu_i\phi^2}=n-k and we solve this for \phi.
According to Consul and Famoye (1992) we begin by fitting a Poisson regression model and obtain initial values for \beta_s and \phi. If \hat{\phi} \approx 1, it implies that the Poisson regression
model is appropriate and no further estimation needs to be done. However, if \hat{\phi} \neq 1, this is used to obtain new values of the estimated \beta_s by maximizing the log-likelihood. This process is iterated until we obtain a stable solution.
The function as seen below returns the log-likelihood of the initial Poisson regression as well. This is useful if one wants to test, via the log-likelihood ratio test as 1 degree of freedom, if the generalized Poisson regression is to be preferred over the Poisson regression.
gp.reg() estimates the beta coefficients using Newton-Raphson, whereas gp.reg2() uses the optim function. For some reason these two do not always agree. One might yield higher log-likelihood than the other and this is why I offer both ways.
Value
A list including:
pois.loglik |
The initial Poisson regression log-likelihood. |
gp.loglik |
The generalized Poisson regression log-likelihood. |
be |
The estimated |
phi |
The estimated |
Author(s)
Michail Tsagris.
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
References
Consul P.C. & Famoye F. (1992). Generalized poisson regression model. Communications in Statistics - Theory and Methods, 21(1): 89–109.
Breslow N. E. (1984). Extra-Poisson variation in log-linear models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 33(1): 38–44.
See Also
Examples
n <- 500
x <- matrix (rnorm(n * 2), nrow = n, ncol = 2)
be <- c(1, 1)
mi <- x[, 1] * be[1] + x[, 2] * be[2] + 1
mi <- exp(mi)
y <- numeric(n)
for (i in 1:n) y[i] <- rgp(2, mi[i], 0.5, method = "Inversion")[1]
gp.reg(y, x)
gp.reg2(y, x)
Random values simulation from the generalized Poisson distribution
Description
Random values simulation from the generalized Poisson distribution.
Usage
rgp(n, theta, lambda, method)
Arguments
n |
The number of random values to generate. |
theta |
The value of the |
lambda |
The value of the |
method |
The simulation method to use. The available options are: "Inversion", "Branching", "Normal-Approximation", "Build-Up" and "Chop-Down". |
Details
The values \theta and \lambda affect the method of simulation to use.
See Li et al. (2020) for more information.
Value
A vector with values from the generalized Poisson distribution.
Author(s)
Michail Tsagris.
R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.
References
Li H., Demirtas H. & Chen R. (2020). RNGforGPD: An R Package for Generation of Univariate and Multivariate Generalized Poisson Data. R JOURNAL, 12(2): 173–188.
Demirtas H. (2017). On accurate and precise generation of generalized Poisson variates. Communications in Statistics - Simulation and Computation, 46(1): 489–499.
See Also
Examples
y <- rgp(1000, 10, 0.5, method = "Inversion")
gp.mle(y)