MATH235 Week 10 - Workshop problems

During World War II, German tanks had sequential serial numbers, i.e. they were numbered $1,2,3,\mathrm{\dots }$. Some of these numbers became known to Allied forces when German tanks were captured. An important question for the Allied forces was to determine the total number of tanks in the German fleet. Suppose the Germans had $N$ tanks in their fleet, with serial numbers $1,2,\mathrm{\dots },N$. So the parameter of interest here is $N$.

1. (a)

   Suppose that the Allied forces have captured $k$ tanks, and that the largest tank serial number of the captured tanks is $m$. Show that the probability of the largest serial number amongst the captured tanks being $m$, given that the Germans have $N$ tanks in total, is given by

   $$\frac{\left(\genfrac{}{}{0pt}{}{m-1}{k-1}\right)}{\left(\genfrac{}{}{0pt}{}{N}{k}\right)}.$$

   Hence write down the likelihood function of $N$.

2. (b)

   Suppose that the Allied forces have actually captured $6$ tanks and the largest observed serial number is $38$. Plot the likelihood function of $N$ over a reasonable range. Find the MLE of $N$, the number of tanks in the German fleet.

3. (c)

   Is the MLE a sensible estimate of $N$?

There are two kinds of stroke: ischemic and hemorrhagic. It is known that 13% of strokes are hemorrhagic. A clinical trial is to be conducted in a hospital on patients suffering a hemorrhagic stroke, and $n=10$ patients are needed for this trial. When stroke patients first arrive in hospital, it is typically unknown whether the stroke is ischemic or hemorrhagic. Patients are quickly tested to determine this, and entered into the trial if their stroke is hemorrhagic. Let $N$ be the number of stroke patients tested to achieve $n=10$ hemorrhagic patients. Write down the likelihood of $N$. Calculate its MLE.
HINT: The negative binomial distribution counts the number of failures, $s$, until $r$ successes are achieved. With failure probability $ϵ$, its pmf is given by

1. (a)

   Write down the definition of deviance.

2. (b)

   By considering a second order Taylor expansion of $l(\theta )$ about $\widehat{\theta }$, show that a quadratic approximation to the deviance is given by

   $$\tilde{D}(\theta )=-l^{\prime \prime }(\widehat{\theta }){(\theta -\widehat{\theta })}^2.$$

3. (c)

   Write down the asymptotic distribution of the maximum likelihood estimator.

4. (d)

   Show that a 95% confidence interval based on the quadratic approximation to the deviance is identical to a 95% confidence interval based on the asymptotic distribution of the MLE.