On the asymptotic efficiency of tests and estimates

Abstract

Let x₁, x₂, ... be a sequence of independent and identically distributed observations with distributions determined by a real valued parameter θ. For each n=1, 2, ..., let T_n = T_n (x₁, x₂... x_n) be a statistic such that the sequence T_n is a consistent estimate of θ. It is shown, under weak regularity conditions on the sample space of a single observation, that the asymptotic effective standard deviation of T_n cannot be less than [nI(θ)]^{½}. The asymptotic effective standard deviation of T_n is defined, roughly speaking, as the solution τ of the equation P(|T_n-θ|≥ ε|θ)=P(|N|≥ ε/τ) when n is large and ε is a small positive number, where N denotes a standard normal variable. It is also shown, under stronger regularity conditions, that the asymptotic effective standard deviation of the maximum likelihood estimate of θ is [nI(θ)]^-{½}. These conclusions concerning estimates are derived from certain conclusions concerning the relative efficiency of alternative statistical tests based on large samples.