New closed-form efficient estimators for the negative binomial distribution

Abstract

The negative binomial (NB) distribution is of interest in various application studies. New closed-form efficient estimators are proposed for the two NB parameters, based on closed-form $\sqrt{n}$-consistent estimators. The asymptotic efficiency and normality of the new closed-form efficient estimators are guaranteed by the theorem applied to derive the new estimators. Since the new closed-form efficient estimators have the same asymptotic distribution as the maximum likelihood estimators (MLEs), these are denoted as MLE-CEs. Simulation studies suggest that the MLE-CE of dispersion parameter r performs better than its MLE and the method of moments estimator (MME) for some parameter ranges. The MLE-CE of the probability parameter p exhibits the best performance for relatively large p values, where the positive-definite expected Fisher information matrix exists. MLE performs better than MME in this parameter space. The MLE-CE is over 200 times faster than the MLE, especially for large sample sizes, which is good for the big data era. Considering the estimated accuracy and computing time, MLE-CE is recommended for small r values and large p values, whereas MME is recommended for other conditions.

Appendices

Appendix A

The assumptions for Theorem 1:

(A1):

The distributions of the observations are distinct;

(A2):

The distributions have common support;

(A3):

The observations are $\textbf{X}=\left( X_1,\ldots , X_n\right)$, where the $X_i$s are i.i.d. with probability density or probability mass $f(x|{\varvec{\theta }})$ which is assumed to be continuous or discrete at x;

(A4):

There exists an open subset $\omega$ of $\Omega$ containing the true parameter point ${\varvec{\theta }}^0$ such that for almost all x, the density $f(x|{\varvec{\theta }})$ admits all third derivatives ${\partial ^3\over \partial \theta _j \partial \theta _k\partial \theta _l}f(x|{\varvec{\theta }})$ for all ${\varvec{\theta }}$ in $\omega$;

(A5):

The first and second logarithmic derivatives of f satisfy the equations

$$\begin{aligned} E_{\varvec{\theta }}\left[ {\partial \over \partial \theta _j} \log f( X |{\varvec{\theta }})\right] =0\text { for }j=1,\ldots ,s, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {I}}_{jk}({\varvec{\theta }})= & {} E_{\varvec{\theta }} \left[ {\partial \over \partial \theta _j}\log f( X |{\varvec{\theta }}) \cdot {\partial \over \partial \theta _k}\log f( X |{\varvec{\theta }})\right] \\= & {} E_{\varvec{\theta }}\left[ -{\partial ^2\over \partial \theta _j\partial \theta _k} \log f( X |{\varvec{\theta }})\right] ; \end{aligned}$$

(A6):

Assume that the ${\mathcal {I}}_{jk}({\varvec{\theta }})$ are finite and that the matrix ${\mathcal {I}}$ is positive definite for all ${\varvec{\theta }}$ in $\omega$, and hence that the score statistics ${\partial \over \partial \theta _j}\log f(x|{\varvec{\theta }})$ for $j=1,\ldots ,s$, are affinely independent with probability 1;

(A7):

Suppose that there exist functions $M_{jkl}(x)$ such that

$$\begin{aligned} \left| {\partial ^3\over \partial \theta _j\partial \theta _k\partial \theta _l} f(x|{\varvec{\theta }})\right| \le M_{jkl}(x) \text { for all } {\varvec{\theta }}\in \omega , \end{aligned}$$

where $m_{jkl}(x)=E_{{\varvec{\theta }}^0}\left[ M_{jkl}(X) \right] <\infty$ for all j, k, l.

Appendix B

Elements of the new closed-form efficient estimators of parameters ${\varvec{\theta }}_*=(r,\mu )$ in the Sect. 4:

$$\begin{aligned} l_{*1}'= & {} {\partial l(\varvec{\theta }_*)\over \partial r}=\sum \limits _{i=1}^n\varphi (x_i+r)-n\varphi (r)-{1\over \mu +r}\sum \limits _{i=1}^nx_i+n\log {r\over \mu +r}+ {n\mu \over \mu +r},\\ l_{*2}'= & {} {\partial l(\varvec{\theta }_*)\over \partial \mu }=\left( {1\over \mu }-{1\over \mu +r}\right) \sum \limits _{i=1}^nx_i-{nr\over \mu +r},\\ l_{*11}''= & {} {\partial ^2 l(\varvec{\theta }_*)\over \partial r^2}=\sum \limits _{i=1}^n\varphi _1(x_i+r)-n\varphi _1(r)+{1\over (\mu +r)^2}\left( \sum \limits _{i=1}^nx_i-n\mu \right) +{n\over r}-{n\over \mu +r}, \\ l_{*12}''= & {} {\partial ^2 l(\varvec{\theta }_*)\over \partial r\partial \mu }=l_{*21}''={\partial ^2 l(\varvec{\theta }_*)\over \partial \mu \partial r}={1\over (\mu +r)^2}\left( \sum \limits _{i=1}^nx_i-n\mu \right) ,\\ l_{*22}''= & {} {\partial ^2 l(\varvec{\theta }_*)\over \partial \mu ^2}=\left( -{1\over \mu ^2}+{1\over (\mu +r)^2}\right) \sum \limits _{i=1}^nx_i+{nr\over (\mu +r)^2}. \end{aligned}$$