Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / O / 2 (O2)

With immediate replacement and uniform mixing, each capture is a Bernoulli trial with tagged probability $p=k/N$. Counting failures before the $r$th tagged capture gives the stated negative binomial distribution. Independent runs on the successive days give the sample model.

Writing $S={\sum }_i{x}_i$, the log-likelihood is $\ell (p)=nr\log p+S\log (1-p)+\text{constant}$. Its derivative vanishes atThe second derivative is negative. If $S=0$, this formula gives the boundary maximum $p=1$; on the strict physical parameter space $p<1$, it is a supremum rather than an attained interior maximum.

Using the supplied negative-binomial mean, $\mathbb{E}S=nr(1-p)/p$. Thus total Fisher information for the sample is

For an interior true parameter and large $n$, standard asymptotic normality of a maximum likelihood estimator givesEstimate its standard error by $s_e=\hat{p}\sqrt{(1-\hat{p})/(nr)}$. A two-sided approximate Wald confidence interval is $(L,U)=(\hat{p}-z_{1-\alpha /2}{s}_e,\hat{p}+z_{1-\alpha /2}{s}_e)$, intersected with the parameter range $(0,1)$. The normal quantile is chosen so a standard normal lies between its positive and negative values with probability $1-\alpha$.

The relation $N=k/p$ is strictly decreasing. Thus an interval $0<L<U<1$ transforms by reversing endpoints:Keep integer population sizes satisfying $N>k$. If the clipped lower confidence endpoint for $p$ is zero, the transformed upper endpoint for $N$ is infinite. This monotone transformation preserves the approximate coverage probability supplied by the interval for $p$.