A regression model for overdispersed data without too many zeros

Abstract

A regression model for overdispersed count data based on the complex biparametric Pearson (CBP) distribution is developed. It is compared with the generalized Poisson regression model, the negative binomial regression model and the zero inflated Poisson regression model, which are based on the generalized Poisson (CBP), negative binomial (NB) and zero inflated Poisson (ZIP) distributions, respectively. It is shown that the CBP distribution is more adequate than the GP, NB and ZIP distributions when the overdispersion is not related to a higher frequency of 0, but to other low values greater than 0, so it may be appropriate for overdispersed cases in which there are external reasons that raise the number of low values different from 0. Firstly, we study the shape and the parameters of the CBP distribution and we compare it with the Poisson, GP, NB and ZIP distributions by means of the probability of 0, the skewness and curtosis coefficients and the Kullback–Leibler divergence. Furthermore, we present an application example where the aforementioned performance is shown by the number of public educational facilities by municipality in Andalusia (Spain). Secondly, we describe two regression models based on the CBP distribution and the estimation method for their parameters. Thirdly, we carry out a simulation study that reveals the performance of the regression models proposed. Finally, one application in the field of sport illustrates that these models can provide more accurate fits than those provided by other usual regression models for count data.

Appendix

It is easy to prove that in the limit case ($\mu =\sigma ^2$) the quotient given in (10) is equal to 1. In general, if we solve the equation $Q_1=1$, we have

$$\begin{aligned} 2\sigma ^4+2\mu (\mu -2)\sigma ^2-2\mu ^2(\mu -1)=0 \end{aligned}$$

(15)

whose solutions are $\sigma ^2=-\mu ^2+\mu$ (which is imposible since $\sigma ^2>\mu$) or $\sigma ^2=\mu$ (as we already knew). Given that the expression (15) is a parabola with positive coefficient of the greatest order, then $Q_1>1$ when $\sigma ^2>\mu$.

In relation to (11), the quotient

$$\begin{aligned} \frac{g_1^{CBP}}{g_1^{GP}}=\frac{(4\mu +1)AI-3\mu }{(\mu +2-AI)(3AI-2\sqrt{AI})} \end{aligned}$$

is greater than 1 if and only if

$$\begin{aligned}&(4\mu +1)AI-3\mu >(\mu +2-AI)(3AI-2\sqrt{AI})\nonumber \\&\quad \Leftrightarrow \mu AI-5AI-3\mu +3AI^2+2\mu \sqrt{AI}+4\sqrt{AI}-2AI\sqrt{AI}>0. \end{aligned}$$

(16)

Applying that $AI>1$, (16) is greater than $3AI^2-2AI\sqrt{AI}-5AI+4\sqrt{AI}$, which is a polinomial of degree 4 in $\sqrt{AI}$. This polynomial is always greater than 0 when $\sqrt{AI}>1$ since the polynomial $3x^3-2x^2-5x+4$ has the form of Fig. 6.

Fig. 6

Graph of the polynomial $3x^3-2x^2-5x+4$