Goodness-of-Fit Tests for Stationary Gaussian Processes with Tapered Data
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Goodness-of-Fit Tests for Stationary Gaussian Processes with Tapered Data Mamikon S. Ginovyan1
Received: 30 July 2020 / Accepted: 11 November 2020 © Springer Nature B.V. 2020
Abstract The paper is concerned with the construction of goodness-of-fit tests for testing a hypothesis H0 that the hypothetical spectral density of a stationary Gaussian process X(t) has the specified form, based on the tapered data. We show that in the case where the hypothetical spectral density of X(t) does not depend on unknown parameters (the hypothesis H0 is simple), then the suggested test statistic has a limiting chi-square distribution. In the case where the hypothesis H0 is composite, that is, the hypothetical spectral density of X(t) depends on an unknown parameter, we choose an appropriate estimator for unknown parameter and describe the limiting distribution of the test statistic. This distribution is similar to that of obtained by Chernov and Lehman (Ann. Math. Stat. 25(3):579–586, 1954) in the case of independent observations. Mathematics Subject Classification (2010) 62F03 · 60G10 · 62G05 · 62G20 Keywords Goodness-of-fit test · Tapered data · Stationary processes · Periodogram · Spectral density · Chi-square distribution
1 Introduction Let {X(u), u ∈ U} be a centered real-valued stationary Gaussian process with spectral density f (λ), λ ∈ Λ, and covariance function r(t), t ∈ U. We consider simultaneously the continuous-time (c.t.) case, where U = R := (−∞, ∞), and the discrete-time (d.t.) case, where U = Z := {0, ±1, ±2, . . .}. The domain Λ of the frequency variable λ is Λ = R in the c.t. case, and Λ := [−π.π] in the d.t. case. The research was partially supported by National Science Foundation Grant #DMS-1309009 at Boston University.
B M.S. Ginovyan
[email protected]
1
Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA
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M.S. Ginovyan
The present paper is concerned with the following problem of hypotheses testing. Suppose we observe a finite realization XT of the process X(t): {X(t), t = 1, . . . , T } in the d.t. case, XT := (1.1) {X(t), 0 ≤ t ≤ T } in the c.t. case. Based on the sample XT we want to construct goodness-of-fit tests for testing a hypothesis H0 that the spectral density of the process X(t) has the specified form f (λ). We will distinguish the following two cases. a) The hypothesis H0 is simple, that is, the hypothetical spectral density f (λ) of X(t) does not depend on unknown parameters. b) The hypothesis H0 is composite, that is, the hypothetical spectral density f (λ) of X(t) depends on an unknown p–dimensional vector parameter θ = (θ1 , . . . , θp ) , that is, f (λ) = f (λ, θ ), λ ∈ Λ, θ ∈ Θ ⊂ Rp . The above stated problem for different models has been considered by many authors. For instance, for independent observations, the problem was considered in Chernov and Lehman [4], Chibisov [5], Cramer [6], Dzhaparidze and Nikulin [12], and Kendall and Stuart [23]. For observations generated by discrete-time Gaussian stationary
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