Tests for Comparing Time‐Invariant and Time‐Varying Spectra Based on the Pearson Statistic

Abstract

Two tests are proposed in this paper for comparing spectra of two univariate time series. One is a Pearson‐like statistic based only on periodograms of the compared time series and applicable for testing the equality of two time‐invariant spectra of two independent or dependent time series, with an asymptotic chi‐squared distribution under the null hypothesis. The other is based on the maximum of the Pearson‐like statistics. Not only does this test, again, depend only on periodograms but also approximately equals the maximum of a chi‐squared distribution of the same degrees of freedom under the null. It can be used to test the equality of spectra of two locally stationary time series regardless of whether they are dependent or independent. Multiple simulation examples show that both statistics achieve good performance. The proposed approach is illustrated by an application to longitudinal vibration data from a container ship.

Tests for comparing time‐invariant and time‐varying spectra based on the Anderson–Darling statistic

“…We base our discussion on a multivariate extension (cf. Zhang & Tu, 2018) of the model of Dette et al (2011b). Let {x t,T , t = 1, … , T} (T ∈ N) be a bivariate locally stationary time series, where each observation x t,T exhibits a linear representation of the form…”

“…If two spectral densities are continuous in (0, 𝜋) and L is taken greater, the periodograms I T,11 ( l−1∕2 L 𝜋) and I T,22 ( l L 𝜋) are expected to estimate the respective spectral densities f 11 (⋅) and f 22 (⋅) at the same frequency point. By lemma 2.1 of Zhang and Tu (2018), under the null (5), the periodogram-ratios…”

“…With arguments similar to the proof of theorem 3.1 of Zhang and Tu (2018), we obtain the asymptotic properties of (24) under the null below.…”

“…We conclude from Table 2 as follows. First, as T increases, the performance of test Zhang and Tu (2018). Models F-I are also employed in the simulation studies of Zhang and Tu (2018), with identical settings.…”

“…sampled from F(2, 2), Zhang and Tu (2018) proposed a Pearson-like statistic to test the equality of two time-invariant spectra, where F(2, 2) denotes an F distribution with 2 and 2 degrees of freedom. Using blocking, they further extended the approach to the more general setting of comparing two time-varying spectra of locally stationary time series.…”