Empirical estimation of d-risks at distinguishing one-sided hypotheses

Abstract

This paper deals with problem of distinguishing between the two hypotheses H₀: θ ≤ 0, H₁: θ > 0 based on a fixed volume sample with a normal distribution N(θ, 1), θ ∈ R. It is considered by suppose that the true value θ is a realization of a random value ϑ with some unknown a priori density g(θ). An empirical estimate g(θ) based on the estimate of archive data of prior distribution characteristic function is suggested for the d-risk of the optimal criteria (conditional probability of justice of hypothesis H_j in condition that it is rejected, j = 0, 1). Consistency of empirical estimators of d-risks and appropriate critical values are studied. The rate of convergence is discovered from obtained estimates.