Past exam of the options I course of the University of Oxford / 2009 / O / O2 / i

Solution

Let

S = \sum_{i} x_{i}

and

\overset{x}{ˉ} = S / n

. Terms independent of

p

can be dropped from the log-likelihood, leaving

ℓ (p) = n r lo g p + S lo g (1 - p) + C, ℓ^{'} (p) = \frac{n r}{p} - \frac{S}{1 - p}, ℓ^{''} (p) = - \frac{n r}{p ^{2}} - \frac{S}{( 1 - p ) ^{2}} .

For

S > 0

p = \frac{n r}{n r + S} = \frac{r}{r + x ˉ} .

S = 0

, the likelihood increases up to

p = 1

: this is the boundary estimate on the closed parameter range, or a supremum if the model insists on

p < 1

Using the given mean, the sample Fisher information is

I_{n} (p) = - E ℓ^{''} (p) = \frac{n r}{p ^{2}} + \frac{n r ( 1 - p ) / p}{( 1 - p ) ^{2}} = \frac{n r}{p ^{2} ( 1 - p )} .

I_{1} (p) = r / [p^{2} (1 - p)]

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