Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE

Abstract

Abstract. We consider goodness-of-fit tests of the Cauchy distribution based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard Cauchy distribution. For standardization of data Ciirtler and Henze (2000, Annals of the Institute of Statistical Mathematics, 52, 267-286) used the median and the interquartile range. In this paper we use the maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE), which minimizes the weighted integral. We derive an explicit form of the asymptotic covariance function of the characteristic function process with parameters estimated by the MLE or the EISE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distributions of the test statistics are obtained by the residue theorem. A simulation study shows that the proposed tests compare well to tests proposed by Giirtler and Henze and more traditional tests based on the empirical distribution function.

A Comparison of Alternative Approaches to Supremum-Norm Goodness of Fit Tests with Estimated Parameters

“…It should be noted that the body of theory represented in Piterbarg (1996) is very general and applicable to a wide variety of Gaussian processes and fields, and as such may serve as a fruitful point of departure for solutions to more general problems, for example the extension of these techniques to test statistics that converge to Gaussian processes in higher dimensions. Approximation P 2 is also quite flexible -it may be applied to any sup-norm test for which the empirical process has a Gaussian limit, as is for example the case with the empirical characteristic function (Matsui and Takemura, 2005, Theorem 2.1). For goodness of fit tests based on regression residuals, very few modifications must be made -see van der Vaart and Wellner (2007).…”

A comparison of alternative approaches to sup-norm goodness of fit tests with estimated parameters

“…While Subsection 3.2 may appear to be only a serendipitous confluence of results from some quite different theoretical starting points, it should be noted that the body of theory represented in Piterbarg (1996) is very general and applicable to a wide variety of Gaussian processes and fields, and as such may serve as a fruitful point of departure for solutions to more general problems, for example the extension of these techniques to test statistics that converge to Gaussian processes in higher dimensions. On the other hand, approximation P 2 is also very flexible -it may be applied to any sup-norm test for which the empirical process has a Gaussian limit, as is for example the case with the empirical characteristic function (Matsui and Takemura, 2005, Theorem 2.1). For goodness of fit tests based on regression residuals, very few modifications must be made; in the dynamic case, some regularity is required on the sequence of score functions to ensure weak convergence of the process -see Bai (2003).…”

Fourier methods for model selection

“…3 and 4, we have estimated the parameters θ and γ by means of their ISE estimators because, according to the definition of θ * and γ * , the ISE estimators are their natural estimators. Nevertheless, motivated by the fact that in certain settings the calculation of the ISE estimators can be time consuming (see for example Matsui and Takemura 2005), other estimators could be used. In such a case, although the proposed methods could be applied, some asymptotic properties may differ while, as seen in Remarks 1 and 2, others continue to be true, whenever the estimators satisfy certain assumptions.…”