# Testing Hypotheses on Spectrum Parameters of a Gaussian Time Series

• K. Dzhaparidze
Chapter
Part of the Springer Series in Statistics book series (SSS)

## 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 Pnθ, θ ∈ θ for some choice of random vector Δn,θ = Δn, θ(x), xXn, and the nonrandom matrices Γθ satisfy the conditions (D1)–(D4) for τn = $$\mathop n\limits^\_$$ of asymptotic differentiability as well as the condition (D5) which assures the asymptotic normality of the vector Δn,θ (cf. the Introduction, page 21 and Section 1 of Chapter III). Assume for definiteness that the set θ ∈ Rp 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 Φn = Φn(x) defined on the sample space xXn. Any measurable function taking on values 0 ⩽ Φn ⩽ 1 may serve as a test function which determines the probability Φn(x) that hypothesis H0 will be rejected when x is observed.

## Keywords

Random Vector Critical Region Consistent Estimator Spectrum Parameter Noncentrality Parameter
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