Negative Binomial Distribution - max. likelihood estimator

[dhs316]

From a negative binomial distribution with a random sample size of 4, unknown p and r=3, calculate the value of the maximum likelihood estimator of p. The values of the sample are: 3, 6, 8, 15.

I got an answer of eight, but I'm not sure if my method is correct. Please help me out with all the steps. Thanks.

 

[galactus]

The negative binomial is \(\displaystyle L(p)=\binom{n-1}{r-1}\cdot p^{r}\cdot (1-p)^{n-r}\)

Find the value of p that maximizes L.

The value of p that maximizes L also maximizes ln(L).

\(\displaystyle ln[L(p)]=ln\binom{n-1}{r-1}+r\cdot ln(p)+(n-r)ln(1-p)\)

Differentiate:

\(\displaystyle \frac{d[ln[L(p)]]}{dp}=\frac{r}{p}-\frac{n-r}{1-p}\)

Now, set to 0 and solve for p and we find the max likelihood estimate is \(\displaystyle p=\frac{r}{n}\)

Thus, the estimator is \(\displaystyle {\hat{P}}=\frac{R}{n}\)

 

[dhs316]

Thanks for the quick reply. I arrived to a similar derivation with slightly different notation, but I was able to work through yours.

So R=3 (given in problem statement). n is simply the average of the four numbers, correct?