This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs present in the principal domain of the cumulant spectrum. The test uses a frame (window) averaged sample estimate of the second order cumulant spectrum to build a test statistic that has an asymptotic complex standard normal distribution. We derive the test statistic, study the size and power properties of the test, and demonstrate its implementation with intraday stock market return data. The test has conservative size properties and good power to detect varying variance and unit root in the presence of varying variance.
Estimation and inference about the change in the mean of a stationary random process was amply examined in literature. Many articles studies the estimation and inference about the change in the mean of a random process. @cite_17 proposed a likelihood ratio based procedure to detect a change in the mean while assuming constant variance. In a different approach, @cite_11 developed CUSUM and EWMA charts that detect changes in probability distributions via their scale parameter. Also, tests of a unit root with or without drift against an alternative hypothesis of trend stationarity, can be found, for example, in @cite_8 and @cite_2 .
@cite_8 @cite_11 @cite_2 @cite_17
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Abstract Let n observations Y 1, Y 2, ···, Y n be generated by the model Y t = pY t−1 + e t , where Y 0 is a fixed constant and e t t-1 n is a sequence of independent normal random variables with mean 0 and variance σ2. Properties of the regression estimator of p are obtained under the assumption that p = ±1. Representations for the limit distributions of the estimator of p and of the regression t test are derived. The estimator of p and the regression t test furnish methods of testing the hypothesis that p = 1. The authors illustrate the development and implementation of a proposed CUSUM chart and evaluate its properties through a simulation study. Abstract A possible alternative to the hypothesis that the sequence X 1, X 2, …, Xn are iid N(ξ, σ2) random variables is that at some unknown instant the expectation ξ shifts. The likelihood ratio test for the alternative of a location shift is studied and its distribution under the null hypothesis found. Tables of standard fractiles are given, along with asymptotic results.