# Skewness for Maximum Likelihood Estimators of the Negative Binomial Distribution

## Abstract

The probability generating function of one version of the negative binomial distribution being (p + 1 - pt){sup -k}, we study elements of the Hessian and in particular Fisher's discovery of a series form for the variance of k, the maximum likelihood estimator, and also for the determinant of the Hessian. There is a link with the Psi function and its derivatives. Basic algebra is excessively complicated and a Maple code implementation is an important task in the solution process. Low order maximum likelihood moments are given and also Fisher's examples relating to data associated with ticks on sheep. Efficiency of moment estimators is mentioned, including the concept of joint efficiency. In an Addendum we give an interesting formula for the difference of two Psi functions.

- Authors:

- ORNL

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- Work for Others (WFO)

- OSTI Identifier:
- 931488

- DOE Contract Number:
- DE-AC05-00OR22725

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Far East Journal of Theoretical Statistics; Journal Volume: 22; Journal Issue: 1

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; MAXIMUM-LIKELIHOOD FIT; FORECASTING; DISTRIBUTION; EFFICIENCY; PROBABILITY; STATISTICS

### Citation Formats

```
Bowman, Kimiko o.
```*Skewness for Maximum Likelihood Estimators of the Negative Binomial Distribution*. United States: N. p., 2007.
Web.

```
Bowman, Kimiko o.
```*Skewness for Maximum Likelihood Estimators of the Negative Binomial Distribution*. United States.

```
Bowman, Kimiko o. Mon .
"Skewness for Maximum Likelihood Estimators of the Negative Binomial Distribution". United States.
doi:.
```

```
@article{osti_931488,
```

title = {Skewness for Maximum Likelihood Estimators of the Negative Binomial Distribution},

author = {Bowman, Kimiko o},

abstractNote = {The probability generating function of one version of the negative binomial distribution being (p + 1 - pt){sup -k}, we study elements of the Hessian and in particular Fisher's discovery of a series form for the variance of k, the maximum likelihood estimator, and also for the determinant of the Hessian. There is a link with the Psi function and its derivatives. Basic algebra is excessively complicated and a Maple code implementation is an important task in the solution process. Low order maximum likelihood moments are given and also Fisher's examples relating to data associated with ticks on sheep. Efficiency of moment estimators is mentioned, including the concept of joint efficiency. In an Addendum we give an interesting formula for the difference of two Psi functions.},

doi = {},

journal = {Far East Journal of Theoretical Statistics},

number = 1,

volume = 22,

place = {United States},

year = {Mon Jan 01 00:00:00 EST 2007},

month = {Mon Jan 01 00:00:00 EST 2007}

}