Parameterization of the COM-Poisson Distribution through the Mean Using Spectral Algorithms for Solving Nonlinear Equations

Abstract

The Poisson regression is popularly used to model count data. However, real data often do not satisfy the assumption of equality of the mean and variance which is an important property of the Poisson distribution. The Poisson – Gamma (Negative binomial) distribution and the recent Conway-Maxwell-Poisson (COM-Poisson) distributions are some of the proposed models for over- and under-dispersion respectively. Nevertheless, the parameterization of the COM-Poisson distribution still remains a major challenge in practice as the location parameter of the original COM-Poisson distribution rarely represents the mean of the distribution. As a result, this paper proposes a new parameterization of the COM-Poisson distribution via the central location (mean) so that more easily-interpretable models and results can be obtained. The parameterization involves solving nonlinear equations which do not have analytical solutions. The nonlinear equations are solved using the efficient and fast derivative free spectral algorithm. Implementation of the parameterization in R (R Core Team, 2018) is used to present useful numerical results concerning the relationship between the mean of the COM-Poisson distribution and the location parameter in the original COM-Poisson parameterization. The proposed technique is further used to fit COM-Poisson probability models to real life datasets. It was found that obtaining estimates via this parameterization makes the estimation easier and faster compared to directly maximizing the likelihood function of the standard COM-Poisson distribution.

Households’ Consumption Pattern of Snail (Archachatina species) in Oluyole Local Government Area of Oyo State, Nigeria

“…This implies that marital status is a deciding factor to snail meat consumption in some households. This result also agrees with Adeniyi et al (2013) who reported that majority of the respondents were married (66.2 percent). Table 1 further reveals that a large percentage of the respondents (35.0%) are Artisans followed by those that are civil servants (23.3%), Farmers (20.8%), Traders (17.5%) and the unemployed respondents (3.3%).…”

“…Table 4 showed that the major problems/constraints associated with snail meat consumption in the study area were Price of snail, followed by financial constraint, unavailability all year round and cultural belief. This corroborates the report of Adeniyi et al, (2013) that scarcity (unavailability all year round), price and cultural belief were major factors hindering the consumption of food products.…”

“…The coefficient of Price is negatively significant at 5% level, showing that if the price increase the amount spent on snail meat will reduce. This is in agreement with Adeniyi et al, (2013) who showed that the coefficient of Price of snail is negatively significant at 5% level, indicating that an increase in price will lead to decrease in amount spent on snail meat and vice-versa. The coefficient of Household income is positively significant at 1%, indicating that an increase in household income will cause an increase in snail meat consumption.…”

“…This implies that females are decision makers in most cases when it comes to household consumption. This result is in line with Adeniyi et al (2013) who reported that 43.7 percent of the respondents were males while 56.3% were females. The result also indicates that age group between 31 and 40 has the highest representation of 45.0% of the respondent.…”

“…Furthermore, Table 2 reveals that majority of the respondents (85%) consume snail meat because of its taste and 80.8% claimed to consume snail meat because it is easy to prepare. This is in consonance with the work of Adeniyi et al (2013) who reported that nutritional factor and taste played prominent roles in household consumption pattern but influence as a result of easy to prepare negates the findings of Adeniyi et al (2013). A good percentage of the respondents (72.5%) get snail meat from the market followed by 15% that get snail meat from the farm, 7.5% from food restaurants while 5% get their snail meat from the beer parlour.…”