Multinomial models for contingency tables

In Example 3.9 when $p_{i j}=p_{i+} p_{+j}$, determine the variances of the two unbiased estimators $\delta_{0}=n_{i j} / n$ and $\delta_{1}=n_{i+} n_{+j} / n^{2}$ of $p_{i j}$, and show directly that $\operatorname {var}\left(\delta_{0}\right)>\operatorname {var}\left(\delta_{1}\right)$ for all $n>1$.

In Example 3.9, show that independence of $A$ and $B$ implies that $(n_{1+}, \ldots , n_{I^{+}})$ and $\left(n_{+1}, \ldots , n_{+J}\right)$ are independent with multinomial distributions as stated.

Verify (3.20).

In Example 3.10, show that the estimator $\delta_{1}$ of $p_{i j k}$ is unbiased for the model (3.20).

In Example 6.1, show that $\gamma_{i j}=0$ for all $i, j$ is equivalent to $p_{i j}=p_{i+} p_{+j}$.

In Example 6.2, show that the conditional independence of $A, B$ given $C$ is equivalent to $\alpha_{i j k}^{A B C}=\alpha_{i j}^{A B}=0$ for all $i, j$, and $k$.

In Example 6.1, show that the conditional distribution of the vectors $(n_{i 1}, \ldots , n_{i J})$ given the values of $n_{i^{+}}(i=1, \ldots , I)$ is that of $I$ independent vectors with multinomial distribution $M\left(p_{1 \mid i}, \ldots , p_{J \mid i} ; n_{i+}\right)$ where $p_{j \mid i}=p_{i j} / p_{i+}$.

Show that the distribution of the preceding problem also arises in Example 6.1 when the $n$ subjects, rather than being drawn from the population at large, are randomly drawn: $n_{1+}$ from Category $A_{1}, \ldots , n_{I^{+}}$from Category $A_{I}$.

An application of log linear models in genetics is through the Hardy-Weinberg model of mating. If a parent population contains alleles $A, a$ with frequencies $p$ and $1-p$, then standard random mating assumptions will result in offspring with genotypes $A A$, $A a$, and $a a$ with frequencies $\theta_{1}=p^{2}, \theta_{2}=2 p(1-p)$, and $\theta_{3}=(1-p)^{2}$.

A city has been divided into $I$ major districts and the $i$ th district into $J_{i}$ subdistricts, all of which have populations of roughly equal size. From the police records for a given year, a random sample of $n$ robberies is obtained. Write the joint multinomial distribution of the numbers $n_{i j}$ of robberies in subdistrict $(i, j)$ for this nested two-way layout as $e^{\Sigma \Sigma n_{i j} \xi_{i j}}$ with $\xi_{i j}=\mu +\alpha_{i}+\beta_{i j}$ where $\Sigma_{i} \alpha_{i}=\Sigma_{j} \beta_{i j}=0$, and show that the assumption $\beta_{i j}=0$ for all $i, j$ is equivalent to the assumption that $p_{i j}=p_{i+} / J_{i}$ for all $i, j$.

Instead of a sample of fixed size $n$ in the preceding problem, suppose the observations consist of all robberies taking place within a given time period, so that $n$ is the value taken on by a random variable $N$. Suppose that $N$ has a Poisson distribution with unknown expectation $\lambda$ and that the conditional distribution of the $n_{i j}$ given $N=n$ is the distribution assumed for the $n_{i j}$ in the preceding problem. Find the UMVU estimator of $\lambda p_{i j}$ and show that no unbiased estimator $p_{i j}$ exists.