Some Comments about the p-Generalized Negative Binomial (NBp) Model

Abstract

This paper describes various selected properties and features of negative binomial (NB) random variables, with special reference to NB2 (i.e., p = 2), and some generalizations to NBp (i.e., p ≥ 2), specifications. It presents new results (e.g., the NBp moment-generating function) with regard to the relationship between a sample mean and its accompanying variance, as well as spatial statistical/econometric numerical and empirical examples, whose parameter estimators are maximum likelihood or method of moment ones. Finally, it highlights the Moran eigenvector spatial filtering methodology within the context of generalized linear modeling, demonstrating it in terms of spatial negative binomial regression. Its overall conclusion is a bolstering of important findings the literature already reports with a newly recognized empirical example of an NB3 phenomenon.

1. Introduction

Many contemporary random variables (RVs) measure counts (with a zero to infinity support), implying that they are Poisson variates, but without equi-dispersion (i.e., a violation of the μ = σ² property, most often such that μ < σ²), further implying that their best distributional description is a Poisson variable with a gamma-distributed mean (i.e., a gamma–Poisson mixture), which translates nicely into a negative binomial (NB) RV. However, a standard NB does not always successfully account for this mean–variance discrepancy in such circumstances. Cameron and Trivedi [1] introduce a generalized version of the NB probability model now known as the NBp model, where p is a positive integer, whose goal is to better capture under/over-dispersion; this variate differs from the negative binomial process (NBP) model (e.g., [2]). The literature contains little discussion about this former generalized model, with the exception of Cameron and Trivedi’s initial treatment of it and papers by Greene [3] and by Di et al. [4]. Of note is that an R package (version ≥ 3.00) exists for implementing this model [4]. The purpose of this paper is to address this gap in the literature by (1) clarifying/correcting several misleading technical details in part of Greene’s paper; (2) presenting new results; (3) illuminating selected small and large sample properties on NB models; and (4) presenting new empirical spatial statistics/econometrics instances involving spatial autocorrelation (SA).

A parametric mixture conceptualization—analogous to a Bayesian conjugate prior formulation—is the context of this discussion: the NB specification describes a count’s variable Y that is Poisson distributed, with its mean μ being gamma distributed rather than constant. Accordingly, the gamma RV has parameters αμ^2–p and μ^p–1/α, where μ denotes its mean and 1/α denotes its dispersion parameter, yielding a global NBp model with parameters r = αμ^2–p and probability p = 1/(1 + μ^p–1/α) for some non-negative integer p > 0; p = 1, 2 are standard NB specifications, with the former being the Poisson RV and the latter being the popular NB2 variate.

2. Materials and Methods: Clarifying Greene’s Presentation

Confusing equations appearing in Greene [1] include (2-13) and (2-14), particularly when one contrasts them with his appendix equation (A-2) with regard to the p exponent. In addition, this latter equation is missing a subscript i on the λ variable appearing in the numerator of its third term on its right-hand side. Discussion pertaining to equations (2-4) and (2-5) would be clearer if the posited gamma distribution is stated in its standard symbology as Γ(θ, 1/θ). Greene’s equations (2-13)—in which he incorrectly states q = 1/1(1 + θ) rather than q = 1/(1 + 1/θ)—and (2-14)—in which he states s = λ/(λ + θλ^2–p) rather than 1/(1 + μ^p–2/α)—yield, for independent and identically distributed (iid) NBps, the following probability mass function:

$$\frac{\mathsf{\Gamma }(\mathrm{y}+\alpha {\mu }^{3-\mathrm{p}})}{\mathsf{\Gamma }(\mathrm{y}+1)\mathsf{\Gamma }(\alpha {\mu }^{3-\mathrm{p}})}{\left(\frac{1}{1+{\mu }^{\mathrm{p}-2}/\alpha }\right)}^{{\alpha \mu }^{3-\mathrm{p}}}{\left(\frac{{\mu }^{\mathrm{p}-2}/\alpha }{1+{\mu }^{\mathrm{p}-2}/\alpha }\right)}^{\mathrm{y}},\mathrm{y}=0,1,2,\dots ,$$

(1)

where, adopting more typical literature-appearing notation (particularly for gamma–Poisson specifications) in this paper, μ (denoting the mean) is equivalent to λ and α is the same as θ in Greene’s paper, the subscripts i are not used in this version because it represents a global equation, and the ratio of gamma functions term also can be rewritten employing combinatorial binomial coefficients. The difference is that Greene’s equation (2-14) has an exponent of (2–p) rather than (p–2). Equation (1) renders the revised results of

E(Y) = μ and E[(Y − μ)²] = μ(1 + μ^p–1/α).

(2)

Greene’s equation (2-14) specification renders

$$\frac{\mathsf{\Gamma }(\mathrm{y}+\alpha {\mu }^{2-\mathrm{p}})}{\mathsf{\Gamma }(\mathrm{y}+1)\mathsf{\Gamma }(\alpha {\mu }^{2-\mathrm{p}})}{\left(\frac{1}{1+{\alpha \mu }^{1-\mathrm{p}}}\right)}^{{\alpha \mu }^{2-\mathrm{p}}}{\left(\frac{\alpha {\mu }^{1-\mathrm{p}}}{1+{\alpha \mu }^{1-\mathrm{p}}}\right)}^{\mathrm{y}},\mathrm{y}=0,1,2,\dots ,$$

which is not his equation (2-15), even for the specific standard case of p = 2. His probability mass function should be Equation (1), rather than

$$\frac{\mathsf{\Gamma }(\mathrm{y}+\alpha {\mu }^{2-\mathrm{p}})}{\mathsf{\Gamma }(\mathrm{y}+1)\mathsf{\Gamma }(\alpha {\mu }^{2-\mathrm{p}})}{\left(\frac{1}{1+{\alpha \mu }^{1-\mathrm{p}}}\right)}^{{\alpha \mu }^{2-\mathrm{p}}}{\left(\frac{\alpha {\mu }^{1-\mathrm{p}}}{1+{\alpha \mu }^{1-\mathrm{p}}}\right)}^{\mathrm{y}},\mathrm{y}=0,1,2,\dots ,$$

which is neither the result for a Poisson parametric mixture probability defined by Prob(y) = e^−μμ^y/(y!) with μ ~ Γ(αμ^2–p, μ^p–1/α), nor the outcome from Greene’s replacing α with αμ^2–p in this mixture, which renders E(Y) = μ and E[(Y − μ)²] = μ(1 + μ^p–3/α).

3. Results: Articulating NBp Variates

Both mathematical statistics analytical results and simulation experimental output help establish an understanding of a generalization of NB1 and NB2 to NBp. This section furnishes some of each.

3.1. The NBp Moment-Generating Function and Asymptotic Normality

Particularly because the method of moments is a useful estimation technique for parameter α [5], knowing the moments of the NBp specification is desirable. The absence of closed-form maximum likelihood estimators (MLEs, e.g., see [6,7,8,9,10]) makes this ensuing invention even more appealing.

3.1.1. The NBp Moment-Generating Function (mgf)

The NBp mgf is

$${\left[\frac{1}{1+({\mu }^{\mathrm{p}-1}/\alpha )(1-{\mathrm{e}}^{\mathrm{t}})}\right]}^{\alpha {\mu }^{2-\mathrm{p}}},$$

with the first moment about the origin being

$${\left.\frac{\partial }{\partial \mathrm{t}}{\left[\frac{\alpha {\mu }^{2-p}}{\alpha {\mu }^{2-p}+\mu (1-e^t)}\right]}^{\alpha {\mu }^{2-p}}\right|}_{t=0}=\mu ,$$

and, hence, the moments about the mean mgf being

$${\mathrm{e}}^{-\mu \mathrm{t}}{\left[\frac{\alpha {\mu }^{2-p}}{\alpha {\mu }^{2-p}+\mu (1-e^t)}\right]}^{\alpha {\mu }^{2-p}}.$$

(3)

The second moment about the mean is μ(1 + μ^p–1/α).

3.1.2. Asymptotic Normality and an NBp RV

Matching moments is one way to establish equivalency between distributions [11]. Skewness and kurtosis involve the third and fourth moments about the mean, which are derivable from Equation (3), such that for an NBp RV

$$\mathrm{skewness}=\frac{1+2{\mu }^{\mathrm{p}-1}/\alpha }{\sqrt{\mu }\sqrt{1+{\mu }^{\mathrm{p}-1}/\alpha }},\mathrm{and}\mathrm{kurtosis}=3+\frac{6{\mu }^{\mathrm{p}-2}}{\alpha }+\frac{1}{\mu (1+{\mu }^{\mathrm{p}-1}/\alpha )}.$$

The Poisson skewness term, $1/\sqrt{\mu }$, is visible in this preceding first expression. Here, skewness does not necessarily converge on zero, and kurtosis does not necessarily converge on 3, unless the over-dispersion parameter, 1/α, as well as p exhibit certain behaviors and/or properties. In other words, asymptotical normality for the NBp distribution is not obvious. Bagui and Mehra [12] offer ways to prove this asymptotic feature, one of which involves its mgf.

Meanwhile, a classical theorem states that if random samples are drawn from the same normal distribution (i.e., iid), then their sample mean and variance statistics are independent [because s² is a function of (y_i − $\stackrel{\_}{y}$); Figure 1a]. Figure 1 portrays scatterplots of relationships between sample s² and s² from simulation experiment results for 10,000 replications of samples of size n = 100 (a well-known appropriate size to ensure that the classical Central Limit Theorem is at work). For this experiment, the linear correlation between $\stackrel{\_}{y}$ and s² for a standard normal RV (i.e., iid) is –0.014 (Figure 1a), whereas that for an NBp RV with p = 2, μ = 1000, and α = 1000 (i.e., iid) is 0.027 (Figure 1b). An extension of this conventional, well-known theorem is as follows:

Theorem 1.

The linear correlation between $\stackrel{\_}{y}$ and s² for a mixture of independent and non-identically distributed (i~id) normal RVs is zero.

Proof of Theorem 1.

For a normal RV, μ can be any real value, and σ² can be any positive real number. Because these two sets of values can be paired in any of the infinite number of possible ways that exist, each value of μ is paired with relatively large, intermediate, and small values of σ², and each value of σ² is paired with relatively large, intermediate, and small values of μ. Consequently, the linear correlation between these two variates is zero. □

This result (see Figure 1c; r = 0.001) is in contrast to that for an NBp RV for which μ appears in its variance formula, introducing linear correlation between the two (Figure 1d; r = 0.905). The formal postulation extension here is as follows:

Theorem 2.

The linear correlation between $\stackrel{\_}{y}$ and s² for a mixture of i~id NBp RVs is $\frac{\sqrt{3(2\mathrm{p}+1)}}{p+2}>0$, p ≥ 1.

Proof of Theorem 2.

The linear product moment correlation coefficient is

$${\mathsf{\rho }}_{\mathrm{c}}=\frac{\left\{{\int }_0^{\mathrm{c}}\left[\mu -\left({\int }_0^{\mathrm{c}}\mu \mathrm{d}\mu \right)/\mathrm{c}\right]\times \left[\mu (1+{\mu }^{\mathrm{p}-1}/\alpha )-\left({\int }_0^{\mathrm{c}}\mu (1+{\mu }^{\mathrm{p}-1}/\alpha )\mathrm{d}\mu \right)/\mathrm{c}\right]\mathrm{d}\mu \right\}}{\sqrt{{\int }_0^{\mathrm{c}}{\left[\mu -\left({\int }_0^{\mathrm{c}}\mu \mathrm{d}\mu \right)/\mathrm{c}\right]}^2\mathrm{d}\mu }\times \sqrt{{\int }_0^{\mathrm{c}}{\left[\mu (1+{\mu }^{\mathrm{p}-1}/\alpha )-\left({\int }_0^{\mathrm{c}}\mu (1+{\mu }^{\mathrm{p}-1}/\alpha )\mathrm{d}\mu \right)/\mathrm{c}\right]}^2\mathrm{d}\mu }}.$$

$\underset{\mathrm{c}\to \mathrm{\infty }}{\lim }{\mathsf{\rho }}_{\mathrm{c}}=\frac{\sqrt{3(2\mathrm{p}+1)}}{\mathrm{p}+2}$ because p > 0. □

• This new theorem indicates that if p = 1 (i.e., a Poisson RV), then ρ = 1, and for the standard NB2 case of p = 2, ρ ≈ 0.968. The parameter p needs to be 20 before this linear correlation is approximately 0.5. The dispersion parameter α does not affect this linear correlation. This new theorem indicates that for the aforementioned NBp, the sample mean and variance, $\stackrel{\_}{y}$ and s², are correlated (Figure 1d). In other words, the linear correlation between $\stackrel{\_}{y}$ and s² for a mixture of normal distributions is zero, whereas for a mixture of NBp distributions, it is asymptotically zero as p → ∞. Although this linear correlation remains zero when normal distributions are pooled, it can be as large as 1 for pooled NBp distributions.

Of note is that Theorem 1 holds even for a normal distribution restricted in such a way that σ² = μ(1 + μ^p–1/α), with suitable restrictions on both p and μ (e.g., p must be even and the μ domain interval [–1, 0] must be removed from the range of σ²). Linear correlation occurs for this particular normal distribution case by, for example, imposing a constraint such as μ > 0, which then also allows p to be odd and linear correlation results to be described by Theorem 2.

3.2. MLE and Method of Moments Estimation (MME) of the Dispersion Parameter α

Various estimation techniques exist, including MLE, MME (e.g., see [6,13]), and least squares (see [14,15,16]). Frequently, these techniques yield exactly the same estimators. The estimation of the dispersion parameter 1/α of an NBp often is by MLE and/or MME. A μ closer to zero and a small sample size can create a situation in which an MLE and/or MME is inaccurate and/or unstable. MME is the simpler of the two techniques for estimating 1/α [5]. An MLE is unable to produce a solution that is an estimator in closed form for this NBp parameter (see Appendix A).

3.2.1. Estimation of Parameter Exponent p

Rather than setting p = 2, Equation (1) alludes to p as a parameter, albeit one restricted to the positive integers (i.e., discrete, not continuous). As a third parameter, its MME estimation requires, for example, adding the third moment about the mean, m₃, to the set of estimation equations. Accordingly,

$$\widehat{\mu }={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{y}}_{\mathrm{i}}/\mathrm{n}=\stackrel{\_}{\mathrm{y}},$$

$$\widehat{\alpha }=\widehat{\mu }\frac{{\widehat{\mu }}^{\mathrm{p}}}{\widehat{\mu }+{\sum }_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{y}}_{\mathrm{i}}-\widehat{\mu }{)}^2/(\mathrm{n}-1)},\mathrm{and}$$

$$\frac{1+2{\widehat{\mu }}^{\mathrm{p}-1}/\widehat{\alpha }}{\sqrt{\widehat{\mu }}\sqrt{1+{\widehat{\mu }}^{\mathrm{p}-1}/\widehat{\alpha }}}=\frac{{\sum }_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{y}}_{\mathrm{i}}-\widehat{\mu }{)}^3}{\mathrm{n}-1}.$$

Solving this triplet of equations for p yields an equation that does not contain p, nor does the one incorporating the fourth moment. Thus, MME cannot render an estimate of p. Meanwhile, an MLE, which involves the difference calculus here because p is an integer, fails to produce a closed-form solution.

The implication of this outcome is that generalizing the standard NB2 variance term from μ(1 + μ/α) to the generalized NBp term μ(1 + μ^p–1/α) is similar to quasi-likelihood estimation with regard to parameter p. Denote $\frac{1}{\widehat{\alpha }}$ with ${\widehat{\alpha }}^{-1}$; if ${\widehat{\alpha }}^{-1}$ seems too large, then a k^th multiple of $\widehat{\mu }$ (i.e., ${\widehat{\mu }}^{\mathrm{k}}$), where k is a positive integer, can be factored from it, effectively setting p = 2 + k. This factorization is nonconstant across the values of μ_i, with subscript i denoting an individual observation, when the mean varies from observation to observation (this is the situation Greene [1] presents).

3.2.2. A Simple n = 3 Numerical Example

Consider a sample of size n = 3, with the number of counts per observation, Y, being as follows: y_i = {2, 5, 8}. Then

$$\stackrel{\_}{y}=(2+5+8)/3=5\mathrm{and}{s}^2=18/(3-1)=9=5[1+5/(25/4\left)\right].$$

The MLEs are

$$\widehat{\mu }=5\mathrm{and}{\widehat{\alpha }}^{-1}\approx 0.0476.$$

The MMEs are

$$\widehat{\mu }=5\mathrm{and}{\widehat{\alpha }}^{-1}=0.1600.$$

Table 1. Summary Monte Carlo simulation experiment results for NB2[5, 5(1 + 5/0.0476)] and NB2[5, 5(1 + 5/0.1600)].

Parameter	Estimate	Standard Deviation	Range	% 0 s	% < 0
using MLEs as the population parameters
μ	4.984	1.410	0.7–11.7
1/αMLE	0.073	0.231	0.0–7.2	71.1
1/αMME	0.113	0.213	0.0–2.7	5.4	52.2
using MMEs as the population parameters
μ	5.003	1.714	0.3–14.3
1/αMLE	0.137	0.315	0.0–7.7	58.4
1/αMME	0.190	0.295	0.0–2.8	1.3	40.6

summarizes results from a Monte Carlo simulation experiment with 10,000 replications (to invoke the Law of Large Numbers), for NB2[5, 5(1 + 5/0.0476)] and NB2[5, 5(1 + 5/0.1600)]. Figure 2 portrays the collective MLE and MME estimates for these simulation results.

As expected, both estimation techniques yield a sound estimate of the population mean, μ. The techniques furnish different estimates of the dispersion parameter, although both are within the 95% confidence interval of the other’s estimate. The MLEs are zero more often than MMEs are. Most of these zero MMEs actually are negative estimates that have been replaced by zero via truncation (these zero values were employed when calculating the summary statistics appearing in Table 1). These values should be considered infeasible because α is the shape parameter of a gamma RV, and as such must be positive. Figure 2 furnishes a visual box-and-whisker plot—the upper and lower edges of a box denote the first and third quartiles, the internal box line denotes the median, and the whisker tips denote 1.5 times the inter-quartile range (i.e., the third minus the first quartile values)—comparison between the two estimators (i.e., the MLEs and MMEs), highlighting that the MLEs have more outliers (denoted by asterisks). Excessive zeroes and frequency of outlier estimates for MLEs, and negative estimates for MMEs, are among the reasons these estimators are considered inaccurate and/or unstable. One general implication of these results is that neither estimator is particularly good for small samples, but that MME may well be preferable. This small sample case illustrates that an MLE and MME can yield different estimates (also see [17]).

3.2.3. Moran Eigenvector Spatial Filtering: A Brief Overview

The next section presents a simple example of interest, especially to criminologists, ecologists, entrepreneurs and marketers, environmental scientists, epidemiologists, geospatial analysts, sociologists, urban planners and urbanists, and urban and regional economists, among others, namely, population density, perhaps one of the most commonly utilized covariates in the social and behavioral sciences; it serves as a preface to Section 4. Georeferenced data require spatial statistics/econometrics methodology in order to properly account for latent SA in those data. SA is the correlation among observations that arises because of their relative proximity in geographic space, because either they are affected by some common factor or they directly interact with each other over geographic space. It is analogous to, but more complex than, serial correlation in time series data (e.g., see [18]).

Moran eigenvector spatial filtering (MESF) is a relatively novel methodology to handle SA contained in data [19,20,21]. MESF uses a set of synthetic proxy variables, which are extracted as eigenvectors from a doubly centered n-by-n spatial weights matrix (SWM), say C, that often is binary (0–1) and that ties n geographic objects together in space (indicating which are pairwise directly correlated by an entry of one), and then adds these vectors as control variables to a regression model specification; SWMs are analogous to undirected planar graph adjacency matrices. These control variables identify and isolate stochastic spatial dependencies among a set of georeferenced observations, filtering these dependencies out of a model’s residuals and adding them to the model’s mean response, thus allowing regression model building to proceed with georeferenced observations that mimic being independent.

The Moran Coefficient (MC; [22]) SA index, one of several such indices that exist—e.g., the Geary Ratio (GR; [23])—furnishes the basis for MESF; its numerator contains the doubly centered SWM exploited by MESF. This index can be written in matrix form as follows:

(n/1^T C1)Y^T(I − 11^T/n)C(I − 11^T/n)Y/Y^T(I − 11^T/n)Y,

(4)

where n is the number of, for example, polygons in a geographic information system shapefile; Y is an n-by-1 vector of attribute values; I is an n-by-n identity matrix; 1 is an n-by-1 vector of ones; superscript T denotes matrix transpose and, here, C is a binary 0–1 SWM for which 1 denotes that the row and column areal units (i.e., locations, e.g., shapefile polygons) share a non-zero length common boundary, and 0 denotes that they do not (i.e., the rook adjacency definition, phraseology based upon chess game pieces and their moves). The eigenfunctions (i.e., the paired eigenvalues and eigenvectors) are extracted from the following matrix, which is doubly centered because matrix C is pre- and post-multiplied by the projection matrix (I − 11^T/n), appearing in the numerator of expression (4) as follows:

(I − 11^T/n)C(I − 11^T/n).

(5)

When multiplied by (n/1^TC1), an eigenvalue of matrix (5) is converted to the MC measuring the SA in its associated eigenvector map pattern of real number elements [24,25]. The sign of an eigenvalue indicates the nature of SA represented by its corresponding eigenvector, whereas its magnitude indicates the degree of SA. The extreme eigenvalues of matrix expression (4) determine the limits of MC, which are not necessarily ±1 [26].

Extracting the eigenfunctions from adjusted SWMs constitutes a spectral decomposition of those matrices. The extracted eigenvectors with associated eigenvalues relatively far from zero, and hence representing other-than-negligible SA, may be viewed as portraying global, regional, or local components of SA because of the particular map patterns they depict when visualized [27,28]. In other words, SA manifests itself in terms of similar (positive SA; PSA) or dissimilar (negative SA; NSA) values of variable Y clustering on a map: hot spots (i.e., similar relatively high values clustering), cold spots (i.e., similar relatively low values clustering), or contrasts (i.e., relatively dissimilar values clustering).

3.2.4. An Empirical 2010 Puerto Rico Population Density Toy Illustration

Puerto Rico is partitioned into five agricultural administrative regions, as a scheme applied to it for a century or more, each focusing on one of the island’s major urban areas (Figure 3). The 1899 geographic distribution of population density for this spatial resolution displays excessive extra-Poisson variation (deviance = 1723, which is far greater than its expected value of 1) and moderate PSA. The table in Figure 3b reports the adjusted SWM eigenvector representing this PSA [MC = 0.221, with its maximum (MC_max) being 1.094, and GR = 0.474]. This scenario acknowledges a well-recognized source of over-dispersion, namely correlation among observations (i.e., dependent data, autocorrelation), which is frequent in spatial statistics; for example, SA accounts for roughly half of any detected over-dispersion (e.g., extra-Poisson variation).

An NB2 description of this geographic distribution of population density yields the global MLE of $\widehat{\mu }$ = e^5.6361 ≈ 280.4 people per square mile, with a dispersion parameter MLE of ${\widehat{\alpha }}^{-1}$ ≈ 0.0010 (i.e., nearly a Poisson RV). The eigenvector spatial filter (ESF) specification, a linear combination of a subset of the eigenvectors extracted from the adjusted SWM, describes a spatially varying mean population density by region [i.e., 283.6 (SJ), 278.6 (A), 295.7 (M), 271.6 (P), and 272.4 (C)] about a global mean of 280.3 and reduces the over-dispersion parameter to 0.0006 (even more closely resembling a Poisson RV).

The global MME of $\widehat{\mu }$ also is e^5.6361 ≈ 280.4. However, the dispersion parameter MME of ${\widehat{\alpha }}^{-1}$ is –0.0001, which would result in its being rounded to 0. Finally, the spatially varying mean population density by region is as follows: 280.1 (SJ), 280.6 (A), 280.8 (M), 280.4 (P), and 280.0 (C). These outcomes seem inferior to their preceding MLE counterparts.

4. Discussion: Two MESF Empirical Examples

The preceding examples in Section 3.2.2 and Section 3.2.4 are only for exemplification purposes. This section presents two empirical examples whose results furnish informative substantive implications.

4.1. An Empirical 2010 Puerto Rico Urban Population Density Case Study

Figure 4a portrays the 2010 geographic distribution of urban population density (people per square mile) across the main island of Puerto Rico. The map pattern characterizing this geographic distribution has an MC of 0.487 and a GR of 0.541, implying moderate PSA. From the MLEs for an NB2 model specification, the global density is e^7.0549 ≈ 1158.5, with a dispersion parameter of 0.6707 (i.e., sizeable extra-Poisson variation). Historically, the population has been concentrated on the coastal lowlands of the island, the preferred locations of the initial Spanish settlers because of convenience and accessibility, furnishing a rationale for including mean elevation as a covariate in an NB2 regression. Primarily because of the way urban areas expand (e.g., the establishment of infrastructure at the rural–urban fringe), this variable should contain PSA. In this case, executing NB2 stepwise regression estimation results in the selection of five PSA and no NSA eigenvectors from the candidate set extracted from the adjusted SWM for which MC_j/MC_max ≥ 0.25, with subscript j denoting the jth eigenvalue.

Table 2. Summary of model estimation results.

Model Specification	NB2 Urban Population Density		Rural Population Density
			NB2		NB3
	${\widehat{\mathit{\alpha }}}^{-1}$	Pseudo–R2	${\widehat{\mathit{\alpha }}}^{-1}$	Pseudo–R2	${\widehat{\mathit{\alpha }}}^{-1}$	Pseudo–R2
Intercept only	0.6707	0	1.8760	0	1.8285	0
Single covariate	0.4848	0.18	1.3964	0.48	1.1847	0.48
Covariate + ESF	0.2021	0.79	1.0399	0.52	1.0065	0.65

NOTE: pseudo—R² (analogous to Efron’s) denotes the R² from a bivariate linear regression of the observed on the NBp-predicted population density (e.g. [29,30,31,32]).

summarizes the MLE results. It reveals that SA accounts for roughly 60% of the over-dispersion in these data, and the single covariate coupled with the constructed ESF accounts for more than three-fourths of the spatial variation in urban population density across the main island of Puerto Rico. With a final ${\widehat{\alpha }}^{-1}$ < 0.2, a dramatic shrinkage from nearly 0.7, p = 2 seems adequate for the NB model specification.

4.2. An Empirical 2010 Puerto Rico Rural Population Density Illustration

Figure 4b portrays the 2010 geographic distribution of rural population density across the main island of Puerto Rico. The map pattern characterizing this geographic distribution has an MC of 0.496 and a GR of 0.511, implying moderate PSA. From the MLEs for an NB2 model specification, the global density is e^4.1298 ≈ 62.2, with a dispersion parameter of 1.8760. Rural and urban land are mutually exclusive, furnishing a rationale for including the density of rural land as a covariate. Because rural land tends to concentrate outside of urban areas, suggesting it contains PSA, and primarily because the expansion of urban land means the contraction of rural land, particularly along the rural–urban fringe, with this spatial competition suggesting it contains NSA, this variable should contain a PSA-NSA mixture. In this case, the stepwise NB2 estimation results in the selection of five eigenvectors, four with PSA and one with NSA (i.e., a mixture). Table 2 summarizes the MLE results. It reveals that SA accounts for roughly 26% of the over-dispersion in these data, and the single covariate coupled with the constructed ESF accounts for more than half of the spatial variation in rural population density across the main island of Puerto Rico. With a final ${\widehat{\alpha }}^{-1}$ > 1, p = 2 may seem inadequate for the NB model specification. Setting p = 3—with estimation executed via SAS PROC NLMIXED (see Appendix B), in which the NB3 probability density log-likelihood function was written expressing the modified NB as a general distribution [see https://stats.oarc.ucla.edu/sas/faq/how-can-i-compute-negative-binomial-models-with-random-intercepts-and-slopes-using-nlmixed/ (last accessed on 20 March 2024)], alters ${\widehat{\alpha }}^{-1}$, which remains greater than one, while increasing the pseudo-R² values when an ESF is present; these latter values decrease when p = 4, implying that p = 3 is the adequate value.

4.3. An Empirical 2010 Puerto Rico Rural Population Den

The two population density geographic distributions have markedly different island-wide averages and extra-Poisson variation. Although they have very similar SA index values, their ESFs indicate that they have very different latent map patterns (e.g., one is characterized by PSA, whereas the other is characterized by a mixture of PSA and NSA). A population classified as urban is not rural, and vice versa, which implies a negative relationship between these two geographic distributions. The bivariate Poisson RV correlation between them is –0.62 (see [33]). Finally, urban population density seems to be well described by an NB2 model, whereas rural population density may benefit from being described by an NB3 model.

5. Concluding Comments and Implications

This paper contributes to the academic literature in a number of different ways. First, it clarifies/corrects several inaccurate technical details in Greene’s earlier paper. Second, it presents new results that include the mgf for the NBp model, two new theorems, and original empirical spatial statistical/econometric applications. Third, it exemplifies selected small and large sample properties on NBp models. Finally, yet again, it demonstrates the power of MESF.

One prominent implication is that quantitative analysts, and especially applied spatial statisticians, studying Poisson and/or NB variables need to expand their toolbox to include NBp specifications for which p > 2. Another is the need for a wider appreciation and adoption of MESF-type methodology for correlated data, including for traditional dependent datasets, time series, and contemporary social network projects. A third repercussion of this paper is a recognition that the NBp random variable needs a Wikipedia page paralleling the existing one for the NB (https://en.wikipedia.org/wiki/Negative_binomial_distribution; last accessed on 15 May 2024).

Perhaps the most serious NBp limitation is its lack of closed-form MLEs. Although this variable drawback is not new to statistics, it does constitute an impediment to how the utilization of a variable flourishes. This certainly is a theme that should motivate subsequent research about the NBp. Another of its limitations is its newness, which means establishing strengths and weaknesses of employing it vis à vis alternative RVs is yet to be determined.