Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / O

Up to a parameter-independent multinomial factor, the likelihood function is $L={\prod }_{i,j}{\theta }_{ij}^{n_{ij}}$. A Lagrange multiplier for $\sum {\theta }_{ij}=1$ gives $n_{ij}/{\theta }_{ij}=\lambda$, whence $\lambda =n$ andSince all counts are positive, the log-likelihood function is strictly concave on the positive simplex and tends to minus infinity at its boundary, proving this is the unique maximum.

Under independence of random variables, ${\theta }_{ij}={\varphi }_i{\rho }_j$, so the log-likelihood function separates as ${\sum }_i{n}_{i\cdot }\log {\varphi }_i+{\sum }_j{n}_{\cdot j}\log {\rho }_j$. Applying the same constrained maximization twice gives ${\hat{\varphi }}_i=n_{i\cdot }/n$, ${\hat{\rho }}_j=n_{\cdot j}/n$, andTwice the difference between the unrestricted and null maximized log-likelihood functions isthe required likelihood-ratio test statistic. Under the null hypothesis, its asymptotic chi-squared distribution has $(r-1)(c-1)$ statistical degrees of freedom, the parameter-count difference $(rc-1)-(r+c-2)$. Here that is four. Since $25.192>q_4=18.467$, reject independence at the 0.1 percent significance level. For four statistical degrees of freedom, the asymptotic upper-tail probability is $e^{-25.192/2}(1+25.192/2)\approx 4.60\times 10^{-5}$. This establishes an association in these data; it is not a causal conclusion.

Write $s=e^{-\lambda t}$ for individual survival probability. Conditional on a clump of size $n$, its death probability is $(1-s{)}^n$. Averaging with the geometric distribution givesThusHere $0\le \theta <1$ and $t>0$; different clumps and experimental repetitions are understood independent for the sampling model.

The live-clump counts have independent binomial distributions, all with the same $p$. Up to a constant, their combined likelihood function is $p^R(1-p{)}^{N-R}$, maximized at $\hat{p}=R/N$ when $0<R<N$. Solving $p=s/(1-\theta +\theta s)$ for $s$ gives $s=p(1-\theta )/(1-\theta p)$. ThereforeMonotonicity of $p$ in $\lambda$ makes this the maximum-likelihood estimator. At the sampling boundaries, $R=N$ gives $\hat{\lambda }=0$, while $R=0$ gives only a supremum as $\lambda \to \infty$, with no finite maximizer. These are the limiting interpretations of the formula.