Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series pp 236-272 | Cite as

# Testing Hypotheses on Spectrum Parameters of a Gaussian Time Series

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## Abstract

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

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

_{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*H*_{0}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*H*_{0}will be rejected when*x*is observed.## Keywords

Random Vector Critical Region Consistent Estimator Spectrum Parameter Noncentrality Parameter
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag New York Inc. 1986