Past exam of the probability, statistics and options I course of the University of Oxford / 2007 / O (Statistics)

A confidence interval is a pair of statistics $L(X),U(X)$ such that, for each parameter value $\mu$,In repeated sampling, at least the stated fraction of intervals constructed by this rule cover the fixed unknown parameter. It is not a posterior probability statement about the parameter after one interval has been observed. Discrete exact constructions can have coverage greater than the nominal value.

For independent observations from a Poisson distribution, $S={\sum }_i{X}_i$ has a Poisson distribution with mean $\lambda =n\mu$. Invert equal-tail tests of that total mean. At an observed integer $s>0$, the lower endpoint solvesand the upper endpoint solvesThen divide both endpoints by $n$. If $s=0$, use lower endpoint zero. Each inversion allocates at most $\alpha /2$ noncoverage probability to one side, giving at least $1-\alpha$ total coverage.

Here the single total count is 39. The supplied values with the correct adjacent count indices are $F_{27.7}(38)=0.975$ for the lower endpoint and $F_{53.3}(39)=0.025$ for the upper one. ThusThe values 28.6 and 52.2 use the other count index and are not the endpoints of this central interval. This is adjacent count tails in an exact Poisson confidence interval.

The statistical power of a test $\phi$ is $\beta (\theta )={\mathbb{E}}_{\theta }\phi (X)$, its rejection probability at parameter $\theta$. Its size is ${\sup }_{\theta \in H_0}\beta (\theta )$; for a simple null this is just $\beta ({\theta }_0)$.

The Neyman-Pearson lemma states that for testing a simple null probability density function $f_0$ against a simple alternative probability density function $f_1$, a test rejecting where $f_1/f_0>k$, accepting where it is below $k$, and if needed randomizing on equality to attain the desired size is most powerful among all tests of that size. It follows by integrating $(\phi -{\phi }_{\ast })(f_1-k{f}_0)\le 0$ and using the size constraint.

The proposed probability density function is nonnegative, andFor an independent sample, cancellation of the factors $(x_i+1)$ in the likelihood ratio givesFor ${\theta }_1<{\theta }_0$ this is increasing in $S={\sum }_i{X}_i$. The most powerful test therefore has rejection regionThe sum has a continuous positive probability density function on $(0,\infty )$, so no boundary randomization is required. The threshold depends on the null and chosen size, but not on which ${\theta }_1<{\theta }_0$ was specified. The same test is therefore a uniformly most powerful test for the composite alternative $0<\theta <{\theta }_0$.

Minimizing the squared residual norm gives the normal equationsFull column rank makes $X^{T}X$ invertible. With columns corresponding to $1,x,x^2-4$, the design matrix isSymmetry gives zero cross products between the odd column and either even column, and $\sum (x^2-4)=28-28=0$ gives the remaining orthogonality. Their squared norms are seven, 28 and 84, so $X^{T}X=\mathrm{diag}(7,28,84)$. ThereforeSince the error covariance is ${\sigma }^{2}I$, least squares has covariance ${\sigma }^2(X^{T}X{)}^{-1}$. ThusThe estimators have zero covariances. independence would require an additional distributional assumption such as Gaussian errors; the stated mean and variance assumptions alone give only the covariance calculation.