The energy goodness-of-fit test for univariate stable distribution

The family of stable distributions is an important class of four-parameter continuous distributions. It has appealing properties such as being the only possible limit distribution of a suitably normalized sum of independent and identically distributed random variables. Therefore, it has wide applications in modeling distributions with heavy tails, such as the return of financial assets. However, there are also critics against using stable distribution in modeling financial assets such as stock and futures. It is very important to check the validity of the model assumption before making inferences based on the model.;Previous work has been done in the goodness-of-fit test for several special cases including normal distributions, Cauchy distributions and more generally, symmetric stable distributions. Classical goodness-of-fit methods such as the Kolmogorov-Smirnov test and the Anderson-Darling test are not able to handle the stable distributions directly because of the lack of closed-form probability density functions (PDF) and cumulative distribution functions (CDF). Since stable distributions can be fully characterized by their characteristic functions, goodness-of-fit tests based on the empirical characteristic function (ECF) have also been studied in recent years.;In this dissertation, a new goodness-of-fit test is proposed for general stable distributions based on the energy statistic, which is invariant under rigid motions. The test statistic is essentially a weighted L 2-norm of the distance between the empirical characteristic function and the hypothetical characteristic function of the null distribution, and it can also be expressed as a V-statistic with degenerate kernel. By asymptotic theory of degenerate kernel V-statistics, the test statistic converges in distribution to an infinite sum of weighted chi 2 random variables if the null hypothesis of stability is true. It can be proved that the test is consistent against a large class of alternatives. A relatively simple computation formula is derived for the test statistic, which involves numerical integration in general. Bootstrap method and critical values based on the asymptotic distribution of the test statistic can be applied to implement the test.;The dissertation is organized as follows. In Chapter 1, the class of stable distributions and its properties are reviewed. In Chapter 2, existing methods of goodness-of-fit test for stable distributions will be discussed. In Chapter 3, theoretical properties of the test statistic, including the definition, computation issues and asymptotic results, are developed. In Chapter 4, simulation studies are presented to illustrate the empirical type I error and power of testing stable distributions against alternative distributions including stable distributions with different parameters and other interesting light-tailed and heavy-tailed distributions. Simulation results show that our test is sensitive in detecting the difference either in the center or extreme values in the tail. In Chapter 5, some basic work has been finished to study the asymptotic distribution of the energy statistic for testing Cauchy when parameters are estimated by maximum likelihood estimators (MLE).