We are concerned here with establishing the consistency and
asymptotic normality for the maximum likelihood estimator of a “merit
vector” $(u_0,\dots,u_t)$, representing the merits of $t +1$ teams
(players, treatments, objects), under the Bradley–Terry model, as $t \to
\infty$. This situation contrasts with the well-known Neyman–Scott
problem under which the number of parameters grows with $t$ (the amount of
sampling), and for which the maximum likelihood estimator fails even to attain
consistency. A key feature of our proof is the use of an effective
approximation to the inverse of the Fisher information matrix. Specifically,
under the Bradley–Terry model, when teams $i$ and $j$ with respective
merits $u_i$ and $u_j$ play each other, the probability that team $i$ prevails
is assumed to be $u_i/(u_i + u_j)$. Suppose each pair of teams play each other
exactly $n$ times for some fixed $n$. The objective is to estimate the merits,
$u_i$’s, based on the outcomes of the $nt(t +1)/2$ games. Clearly, the
model depends on the $u_i$’s only through their ratios. Under some
condition on the growth rate of the largest ratio $u_i/u_j (0 \leq i, j \leq
t)$ as $t \to \infty$, the maximum likelihood estimator of
$(u_1/u_0,\dots,u_t/u_0)$ is shown to be consistent and asymptotically normal.
Some simulation results are provided.