Testing Hypotheses on Spectrum Parameters of a Gaussian Time Series

Abstract

Following the general ideas of LeCam [80–82] (cf. also [110]) we shall consider a sequence of experiments

$${E_n} = \{ {X_n},{\mathfrak{A}_n},{P_{n,}}_\theta ,\theta \in \theta \} ,\,n = 1,2,...,$$

where the family of distributions P_{n θ}, θ ∈ θ for some choice of random vector Δ_n,θ = Δ_{n, θ}(x), x ∈ X_n, and the nonrandom matrices Γ_θ satisfy the conditions (D1)–(D4) for τ_n = √$\mathop n\limits^\_$ of asymptotic differentiability as well as the condition (D5) which assures the asymptotic normality of the vector Δ_n,θ (cf. the Introduction, page 21 and Section 1 of Chapter III). Assume for definiteness that the set θ ∈ R_p of possible values of the vector-valued parameter θ contains the origin and consider the problem of testing the hypothesis H₀ that the parameter θ takes on the value 0. A test for this hypothesis is given by a sequence of test functions Φ_n = Φn(x) defined on the sample space x ∈ X_n. Any measurable function taking on values 0 ⩽ Φ_n ⩽ 1 may serve as a test function which determines the probability Φ_n(x) that hypothesis H₀ will be rejected when x is observed.