Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 13J (Statistical Modelling)

No intercept should be included unless the ordering of the two teams has an external meaning such as home advantage. Ifthen reversing the teams gives ${\eta }_{k\ell ,ij}=-{\eta }_{ij,k\ell }$, so the logistic model automatically givesAn intercept would not change sign when the teams are swapped and would violate this required symmetry. Thus an intercept does not make sense in the stated model.

Whenever players 1 and 2 occur, the linear predictor contains their parameters only throughFor any constants $a,b$, replacingleaves every fitted probability and hence the likelihood function unchanged. The parameterization is therefore nonidentifiable: at most ${\gamma }_{12}$ can be estimated uniquely. Hence there is no unique maximum-likelihood estimate of the three separate parameters.

Let $x$ be the parameter-contrast vector for the specified game: it has entries $+1$ for ${\beta }_i,{\beta }_j,{\beta }_{\{i,j\}}$, entries $-1$ for ${\beta }_k,{\beta }_{\ell },{\beta }_{\{k,\ell \}}$, and zero elsewhere. PutUnder the stated regularity assumptions, asymptotic normality of a maximum likelihood estimator giveswhere $i_N(\beta )/N\to I(\beta )$. Since $g^{\prime }(\eta )=p(1-p)$, the delta method givesEquivalently, for large $N$,with $p=e^{x^T\beta }/(1+e^{x^T\beta })$.