On testing simple hypotheses in finite time with Hellman-Cover automata

Asymptotically
\varepsilon
-optimal automata were developed by Hellman and Cover [4] for testing simple hypotheses concerning the parameter of an independent identically distributed sequence of Bernoulli random variables. These automata permit transitions only between adjacent states and employ artificial randomization only at extreme states. In this paper we study the problem of approximating the optimal Hellman-Cover automaton in fixed-sample-size problems. It is shown that the optimal level of the parameter, which regulates the probability of transitions out of an extreme state, tends to zero at the rate
\ln n/n
in symmetric testing problems where
n
is the sample size. We develop an approximation for the optimal parameter value valid for
n
sufficiently large.