Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series

Traditionally the most important problem of mathematical statistics dealing with random stationary processes X_t, t = …,-1,0,1, … is the problem of estimating the second order characteristics, namely the covariance function or its Fourier transform — the spectral density f = f(λ) (under the assumption that the spectral density exists). For this reason, a vast amount of periodical and monographic literature is devoted to the nonparametric statistical problem of estimating the function β(τ) and especially that of f(λ) (see, for example, the books [4,21,22,26,56,77,137,139,140,]). However, the empirical value f _n ^* of the spectral density f obtained by applying a certain statistical procedure to the observed values of the variables X₁, … , X_n, usually depends in a complicated manner on the cyclic frequency λ. This fact often presents difficulties in applying the obtained estimate f _n ^* of the function f to the solution of specific problems related to the process X_t.

Let X_t, t = …, -1,0,1, … be a Gaussian stationary process with zero expected value E(X_t) = 0, finite variance D(X_t) = \(E(X_t^2) < \infty\), and absolutely continuous spectral function where f = f(λ) is the spectral density of the process X_t.¹

Let X_t, t = …, -1,0,1, … be a Gaussian process with zero expectation and spectral density f depending on an unknown vector-valued parameter θ so that f = f_θ, θ ∈ θ, where θ is a subset in R_p. Assume furthermore, that it is required to estimate the value of the unknown parameter θ based on a sequence of observations from the random process X_t, for t = 1, …, n.

The examples considered in Sections 4 and 5 of the preceding Chapter indicate that the asymptotic m.l. estimators \(\mathop \theta \limits^ \sim \) of the parameters θ appearing in the expression for spectral density f_θ of a Gaussian random process X_t, t = …,-1,0,1, … while they are simpler than the exact m.l. estimators \(\mathop \theta \limits^\_ \), they are nevertheless most often roots of rather complex nonlinear equations so that their determination also requires a substantial amount of time and effort. Only the problem of estimating the parameters ⌊₁, …, ⌊_q and σ² in the autoregressive process with spectral density (II.4.3) was an exception. In Subsection 4.1 of the preceding Chapter it was shown that for this problem the asymptotic m.l. estimators \(\mathop L\limits^ \sim \)₁, …, \(\mathop L\limits^ \sim \)_q are roots of a simple system of linear equations (II.4.6) with respect to the variables ⌊₁ …, ⌊_q and that the estimator \(\mathop \sigma \limits^ \sim \)² of the parameter σ² is given by a relatively simple formula (II.4.7).

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

In this chapter the problem of testing the hypothesis H₀ concerning the form of spectral density f of a linear process X_t of the form (II.6.1) is considered. Unlike in the preceding chapter more general assumptions on the nature of the process X_t are imposed. Namely, it is assumed that the coefficients g₁, g₂, … and the sequence of identically distributed random variables ε_t, t = …,-1,0,1, …, satisfy the following conditions which are more stringent than those in Chapter II, Subsection 6.1: for some \( \delta > 0,{\text{ }}\sum _{{j = 1}}^{\infty }{{j}^{{1/2 + \delta }}}\left| {{{g}_{j}}} \right| < \infty \) and for some r > 4, E(∣ε_t∣^r) < ∞.