Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 13J (Statistical Modelling)

A generalized linear model assumes independent responses from one exponential dispersion family,where ${\sigma }_i^2=a_i{\sigma }^2$, ${\mu }_i=\mathbb{E}[Y_i]=b^{\prime }({\theta }_i)$, and a link function relates the mean to the linear predictor by $g(\mu )=X\beta$. It follows from normalization of the density after shifting the natural parameter thatIndependence therefore gives the joint moment-generating functionwhenever every shifted natural parameter lies in its natural-parameter domain.

Let $m=1/{\sigma }^2$. Within block $i$, the $m$ replicated observations have dispersion $a_i$ and common natural parameter ${\theta }_i$. Evaluating their joint MGF at the blockwise argument ${\sigma }^2{t}_i$ givesMultiplying over blocks reproduces the MGF of $Y$ from part (a). Since moment-generating functions uniquely determine distributions in this family, $\bar{Y}\stackrel{d}{=}Y$.

Writing $B_i$ for block $i$ and using ${\sum }_{j\in B_i}{\tilde{y}}_j={\bar{y}}_i/{\sigma }^2$, the replicated likelihood factors asThe second factor is independent of $\beta$, while the first depends on the data only through $\bar{y}$. The Fisher-Neyman factorization theorem therefore makes $\bar{Y}$ a sufficient statistic for $\beta$ in the replicated experiment.

Letbe the block-saturated mean space, and letbe the fitted GLM mean space. Full column rank and the regularity assumptions give dimensions $n$ and $p$. The factorization in part (ii) makes the nuisance factor cancel from their likelihood ratio, while $\bar{Y}\stackrel{d}{=}Y$ givesAs ${\sigma }^2\to 0$, the replicate count $m=1/{\sigma }^2$ tends to infinity. Wilks theorem applied to these nested regular models makes twice the log-likelihood ratio converge in distribution to a chi-squared variable with the difference in dimensions: