The energy goodness-of-fit test and E-M type estimator for asymmetric Laplace distributions

Recently the asymmetric Laplace distribution and its extensions have gained attention in the statistical literature. This may be due to its relatively simple form and its ability to model skewness and outliers. For these reasons, the asymmetric Laplace distribution is a reasonable candidate model for certain data that arise in finance, biology, engineering, and other disciplines. For a practitioner that wishes to use this distribution, it is very important to check the validity of the model before making inferences that depend on the model. These types of questions are traditionally addressed by goodness-of-fit tests in the statistical literature. In this dissertation, a new goodness-of-fit test is proposed based on energy statistics, a widely applicable class of statistics for which one application is goodness-of-fit testing. The energy goodness-of-fit test has a number of desirable properties. It is consistent against general alternatives. If the null hypothesis is true, the distribution of the test statistic converges in distribution to an infinite, weighted sum of Chi-square random variables. In addition, we find through simulation that the energy test is among the most powerful tests for the asymmetric Laplace distribution in the scenarios considered. In studying this statistic, we found that the current methods for parameter estimation of this distribution were lacking, and proposed a new method to calculate the maximum likelihood estimates of the multivariate asymmetric Laplace distribution through the expectation-maximization (E-M) algorithm. Our proposed E-M algorithm has a fast computational formula and often yields parameter estimates with a smaller mean squared error than other estimators