Empirical Likelihood Ratio-Based Goodness-of-Fit Test for the Laplace Distribution
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Empirical Likelihood Ratio-Based Goodness-of-Fit Test for the Laplace Distribution Hadi Alizadeh Noughabi1
Received: 7 July 2015 / Revised: 22 March 2016 / Accepted: 24 October 2016 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2016
Abstract The Laplace distribution can be compared against the normal distribution. The Laplace distribution has an unusual, symmetric shape with a sharp peak and tails that are longer than the tails of a normal distribution. It has recently become quite popular in modeling financial variables (Brownian Laplace motion) like stock returns because of the greater tails. The Laplace distribution is very extensively reviewed in the monograph (Kotz et al. in the laplace distribution and generalizations—a revisit with applications to communications, economics, engineering, and finance. Birkhauser, Boston, 2001). In this article, we propose a density-based empirical likelihood ratio (DBELR) goodness-of-fit test statistic for the Laplace distribution. The test statistic is constructed based on the approach proposed by Vexler and Gurevich (Comput Stat Data Anal 54:531–545, 2010). In order to compute the test statistic, parameters of the Laplace distribution are estimated by the maximum likelihood method. Critical values and power values of the proposed test are obtained by Monte Carlo simulations. Also, power comparisons of the proposed test with some known competing tests are carried out. Finally, two illustrative examples are presented and analyzed. Keywords Likelihood ratio · Laplace distribution · Density-based empirical likelihood ratio · Goodness-of-fit test · Monte Carlo simulation · Power study Mathematics Subject Classification 62G10 · 62G20
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Hadi Alizadeh Noughabi [email protected] Department of Statistics, University of Birjand, Birjand, Iran
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H. Alizadeh Noughabi
1 Introduction The Laplace distribution can be compared against the normal distribution. This distribution has an unusual, symmetric shape with a sharp peak and tails that are longer than the tails of a normal distribution. Other properties of this distribution can be found in Balakrishnan and Nevzorov [3]. The Laplace distribution has been used in various areas of applied research including finance, engineering, astronomy, and environmental sciences (see, for example, [4,7, 9–11,13] and references therein). Therefore, it is necessary to ascertain model validity of Laplace model to the observations by assessing goodness-of-fit tests. In the statistical literature, goodness-of-fit tests based on the empirical distribution function (EDF) are well known and widely used in practice and statistical software. The known EDF tests are Cramer–von Mises (CH), Kolmogorov–Smirnov (D), Kuiper (V ), Watson (U), and Anderson–Darling (AD). For more details about these tests, one can see D’Agostino and Stephens [5]. Most of goodness-of-fit tests for the Laplace distribution are based on EDF. According to Neyman–Pearson lemma, the maximum likelihood
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