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.
The space of nonstrictly stationary processes is vast and, therefore, in this manuscript we focus the evaluation of the power of the PHR test on a few alternative hypotheses. We do not study PHR's power against the alternative of a process with periodicities in the mean and random variation in the periodic structure, as in @cite_0 . In the space of tests for strict stationarity in the time domain, @cite_14 proposed a test for strict stationarity and compared its power to the KPSS test of level stationarity. Similarly to @cite_14 we compare the power of the PHR test to the power of the KPSS test. We calculate the power for the alternative hypotheses of unit root, varying variance, and their combination. Our PHR test excels in detecting stationarity violations due to second moment.
@cite_0 @cite_14
171502498 2112502349 2299462150 1499910526
We introduce a test for strict stationarity based on the fluctuations of the quantiles of the data, and we show that this test has power against the alternative hypothesis of unconditional heteroskedasticity while other tests for first order (level) stationarity as the KPSS test (, 1992) and, its robust version, the IKPSS test (de , 2007) have low power against this alternative of time-varying variance. Moreover, our test has power against the alternative hypothesis of time-varying kurtosis, while the test for second order (covariance) stationarity introduced by Xiao and Lima (2007) has power close to size against this alternative. We consider the problem of the reconstruction of a continuous-time function f(x) spl isin H from the samples of the responses of m linear shift-invariant systems sampled at 1 m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H=L sub 2 ) is considerably larger than the subspace of band-limited functions. Second, we use a more general specification of the reconstruction subspace V( spl psi ), so that the output of the system can take the form of a band-limited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximation f spl isin V( spl psi ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V( spl psi ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filter bank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the band-limited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines. Stationarity is a cornerstone property that facilitates the analysis and processing of random signals in the time domain. Although time-varying signals are abundant in nature, in many practical scenarios, the information of interest resides in more irregular graph domains. This lack of regularity hampers the generalization of the classical notion of stationarity to graph signals. This paper proposes a definition of weak stationarity for random graph signals that takes into account the structure of the graph where the random process takes place, while inheriting many of the meaningful properties of the classical time domain definition. Provided that the topology of the graph can be described by a normal matrix, stationary graph processes can be modeled as the output of a linear graph filter applied to a white input. This is shown equivalent to requiring the correlation matrix to be diagonalized by the graph Fourier transform; a fact that is leveraged to define a notion of power spectral density (PSD). Properties of the graph PSD are analyzed and a number of methods for its estimation are proposed. This includes generalizations of nonparametric approaches such as periodograms, window-based average periodograms, and filter banks, as well as parametric approaches, using moving-average, autoregressive, and ARMA processes. Graph stationarity and graph PSD estimation are investigated numerically for synthetic and real-world graph signals. We discuss a one-sample location test that can be used in the case of highdimensional data. For high-dimensional data, the power of Hotelling’s test decreases when the dimension is close to the sample size. To address this loss of power, some non-exact approaches were proposed, e.g., Dempster (1958, 1960), Bai and Saranadasa (1996) and Srivastava and Du (2006). In this paper, we focus on Hotelling’s test and Dempster’s test. The comparative merits and demerits of these two tests vary according to the local parameters. In particular, we consider the situation where it is difficult to determine which test should be used, that is, where the two tests are asymptotically equivalent in terms of local power. We propose a new statistic based on the weighted averaging of Hotelling’s T 2 statistic and Dempster’s statistic that can be applied in such a situation. Our weight is determined on the basis of the maximum local asymptotic power on a restricted parameter space that induces local asymptotic equivalence between Hotelling’s test and Dempster’s test. In addition, some good asymptotic properties with respect to the local power are shown. Numerical results show that our test is more stable than Hotelling’s T 2 statistic and Dempster’s statistic in most parameter settings.