(i) Suppose $Y_{1}, \ldots, Y_{n}$ are independent Poisson variables, and

$$\mathbb{E}\left(Y_{i}\right)=\mu_{i}, \quad \log \mu_{i}=\alpha+\beta t_{i}, \quad \text { for } \quad i=1, \ldots, n,$$

where $\alpha, \beta$ are two unknown parameters, and $t_{1}, \ldots, t_{n}$ are given covariates, each of dimension 1. Find equations for $\hat{\alpha}, \hat{\beta}$, the maximum likelihood estimators of $\alpha, \beta$, and show how an estimate of $\operatorname{var}(\hat{\beta})$ may be derived, quoting any standard theorems you may need.

(ii) By 31 December 2001, the number of new vCJD patients, classified by reported calendar year of onset, were

$$8,10,11,14,17,29,23$$

for the years

$$1994, \ldots, 2000 \text { respectively }$$

Discuss carefully the (slightly edited) $R$ output for these data given below, quoting any standard theorems you may need.

year

year

[1] 1994199519961997199819992000

$>$ tot

[1] $\begin{array}{lllllll}8 & 10 & 11 & 14 & 17 & 29 & 23\end{array}$

first.glm - glm(tot year, family = poisson)

$>\operatorname{summary}$ (first.glm)

Call:

glm(formula $=$ tot year, family $=$ poisson $)$

Coefficients

Estimate Std. Error z value $\operatorname{Pr}(>|z|)$

(Intercept) $-407.8128599 .35366-4.1054 .05 \mathrm{e}-05$

year $\quad 0.20556 \quad 0.04973 \quad 4.1333 .57 e-05$

(Dispersion parameter for poisson family taken to be 1)

Null deviance: $20.7753$ on 6 degrees of freedom

Residual deviance: $2.7931$ on 5 degrees of freedom

Number of Fisher Scoring iterations: 3

Part II 2003