Asymptotic Relative Efficiency in Testing

Making a substantiated choice of the most efficient statistical test of several ones being at the disposal of the statistician is regarded as one of the basic problems of Statistics. This problem became especially important in the middle of XX century when appeared computationally simple but ”inefficient” rank tests. Asymptotic relative efficiency (ARE) is a notion which enables to implement in large samples the quantitative comparison of two different tests used for testing of the same statistical hypothesis. The notion of the asymptotic efficiency of tests is more complicated than that of asymptotic efficiency of estimates. Various approaches to this notion were identified only in late fourties and early fifties, hence, 20–25 years later than in the estimation theory. We proceed now to their description. Let {Tn} and {Vn} be two sequences of statistics based on n observations and assigned for testing the null-hypothesis H against the alternative A. We assume that the alternative is characterized by real parameter θ and for θ = θ0 turns into H. Denote by NT (α, β, θ) the sample size necessary for the sequence {Tn} in order to attain the power β under the level α and the alternative value of parameter θ. The number NV (α, β, θ) is defined in the same way. It is natural to prefer the sequence with smaller N . Therefore the relative efficiency of the sequence {Tn} with respect to the sequence {Vn} is specified as the quantity