Method of Moments: Negative Binomial Distribution

Given a collection of data that may fit the negative binomial distribution, we would like to estimate the parameters which best fit the data. We illustrate the method of moments approach on this webpage. Click here to see another approach, using the maximum likelihood method.

Parameter estimates

As shown in Negative Binomial Distribution, we can estimate the sample mean and variance for the negative binomial distribution by the population mean and variance, as follows:

Moments

Solving for k and p, we get

Method of moments

We also see that

p estimation

Log-likelihood function

The likelihood function for the negative binomial distribution is

Likelihood function

It follows that the log-likelihood function is

Log-likelihood function

LL part 2

LL part 3

Example

Example 1: Use the method of moments to find the parameters of the negative binomial distribution that best fits the data in Figure 1.

Negative Binomial Distribution example

Figure 1 – Method of Moments example

We see from column H that p = .622789 and k = 10.40152. If we require that k be an integer, then we see that k = 11 is better than k = 10 (i.e. higher MLE). Actually, we see from column L that k = 13 and p = .673575 achieve a higher value for LL.

Note that the formula in cell H9 is

=SUMPRODUCT(GAMMALN($B3:$E7+H8))-SUMPRODUCT(LN(FACT($B3:$E7)))-H3*GAMMALN(H8)+H3*H8*LN(H7)+LN(1-H7)*SUM($B3:$E7)

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

Reference

Wikipedia (2023) Negative binomial distribution
https://en.wikipedia.org/wiki/Negative_binomial_distribution

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