The standardized spectral density completely describes serial dependence of a Gaussian process. For non-Gaussian processes, however, it may become an inappropriate analytic tool, because it misses the nonlinear processes with zero autocorrelation. By generalizing the concept of the standardized spectral density, I propose a new spectral tool suitable for both linear and nonlinear time series analysis. The generalized spectral density is indexed by frequency and a pair of auxiliary parameters. It is well defined for both continuous and discrete random variables, and requires no moment condition. Introduction of the auxiliary parameters renders the spectrum able to capture all pairwise dependencies, including those with zero autocorrelation. The standardized spectral density can be derived by properly differentiating the generalized spectral density with respect to the auxiliary parameters at the origin. The consistency of a class of Parzen's kernel-type estimators for the generalized spectral dens...

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Hypothesis Testing in Time Series via the Empirical Characteristic Function: A Generalized Spectral Density Approach

The empirical characteristic function (CF) is the Fourier transform of the sample distribution function. The values of its real and imaginary parts at some real number t are merely sample means of cosine and sine functions of the data, the observations being multiplied by t. Given independent samples from two populations, we develop a test for the two-sample problem which is based on a quadratic form in differences between the respective components of the empirical CFs of the two samples. The power of the CF test compares favorably with that of competing omnibus tests when the data are continuous. In the discrete case the CF procedure is also applicable and quite successful; and in this application it appears to have no competitors.

An omnibus test for the two-sample problem using the empirical characteristic function

Summary A new and consistent rank test for bivariate dependence is developed. Let Xi' and Y' denote the (approximate) normal scores associated with the iid vectors (X, Yi), i = 1 ... , n. Then the proposed test statistic may be obtained by removing the first Haijek projection from the quantity Sn-2 I'- XXk' I Y- YkJ. Empirical characteristic function considerations are used in our development and some related graphical methods are proposed. Some difficulties that arise in extensions to dimension k > 2 are noted. A small simulation study provides evidence of the effectiveness of the new procedure.

https://www.stat.cmu.edu/~ryantibs/journalclub/feuerverger_1993.pdf

A Consistent Test for Bivariate Dependence

A wide selection of classical and recent tests for exponentiality are discussed and compared. The classical procedures include the statistics of Kolmogorov-Smirnov and Cramer-von Mises, a statistic based on spacings, and a method involving the score function. Among the most recent approaches emphasized are methods based on the empirical Laplace transform and the empirical characteristic function, a method based on entropy as well as tests of the Kolmogorov-Smirnov and Cramer-von Mises type that utilize a characterization of exponentiality via the mean residual life function. We also propose a new goodness-of-fit test utilizing a novel characterization of the exponential distribution through its characteristic function. The finite-sample performance of the tests is investigated in an extensive simulation study.

/pdf/recent-and-classical-tests-for-exponentiality-a-partial-42tctkb5u3.pdf

Recent and classical tests for exponentiality: a partial review with comparisons

Is it possible to define a coefficient of correlation which is (a) as simple as the classical coefficients like Pearson's correlation or Spearman's correlation, and yet (b) consistently estimates some simple and interpretable measure of the degree of dependence between the variables, which is 0 if and only if the variables are independent and 1 if and only if one is a measurable function of the other, and (c) has a simple asymptotic theory under the hypothesis of independence, like the classical coefficients? This article answers this question in the affirmative, by producing such a coefficient. No assumptions are needed on the distributions of the variables. There are several coefficients in the literature that converge to 0 if and only if the variables are independent, but none that satisfy any of the other properties mentioned above.

Sándor Csörgő

Papers

Testing for independence by the empirical characteristic function

Kernel-transformed empirical processes