Empirical Likelihood Ratio-Based Goodness-of-Fit Test for the Laplace Distribution

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

References

[1] Alizadeh Noughabi H. Empirical likelihood ratio-based goodness-of-fit test for the logistic distribution. J. Appl. Stat.. 2015, 42 1973-1983

[2] Bain LJ, Englehardt M. Interval estimation for the two parameter double exponential distribution. Technometrics. 1973, 15 875-887

[3] Balakrishnan N, Nevzorov VB. A Primer on Statistical Distributions. 2003 Blackwell: Wiley

[4] Chen G, Gott JR, Ratra B. Non-Gaussian error distribution of hubble constant measurements. Publ. Astron. Soc. Pac.. 2003, 115 1269-1279

[5] D’Agostino RB, Stephens MA. Goodness-of-Fit Techniques. 1986 New York: Marcel Dekker

[6] Duncan AJ. Quality Control and Industrial Statistics. 1974 Homewood, IL: Irwin

[7] Granger WJ, Ding Z. Some properties of absolute return. Ann. D’econ. Stat.. 1995, 40 67-91

[8] Gurevich G, Vexler A. A Two-sample empirical likelihood ratio test based on samples entropy. Stat. Comput.. 2011, 21 657-670

[9] Kotz S, Kozubowski TJ, Podgorski K. The Laplace Distribution and Generalizations—A Revisit with Applications to Communications, Economics, Engineering, and Finance. 2001 Boston: Birkhauser

[10] Linden M. A model for stock return distribution. Int. J. Finance Econ.. 2001, 6 159-169

[11] Mittnik S, Paolella MS, Rachev ST. Unconditional and conditional distributional models for the Nikkei Index. Asia Pac. Financ. Mark.. 1998, 5 99-128

[12] Puig P, Stephens MA. Tests of fit for the Laplace distribution with applications. Technometrics. 2000, 42 417-424

[13] Sabarinath A, Anilkumar AK. Modeling of sunspot numbers by a modified binary mixture of laplace distribution functions. Sol. Phys.. 2008, 250 183-197

[14] Safavinejad M, Jomhoori S, Alizadeh Noughabi H. A density-based empirical likelihood ratio goodness-of-fit test for the Rayleigh distribution and power comparison. J. Stat. Comput. Simul.. 2014, 85 3322-3334

[15] Shan G, Vexler A, Wilding GE, Hutson AD. Simple and exact empirical likelihood ratio tests for normality based on moment relations. Commun. Stat. Simul. Comput.. 2011, 40 141-158

[16] Shepherd, L.A., Tsai, W.-M., Vexler, A., Miecznikowski, J.C.: dbEmpLikeNorm: test for joint assessment of normality. R package. http://cran.r-project.org/web/packages/dbEmpLikeNorm/index.html (2013)

[17] Tanajian, H., Vexler. A., Hutson, A.D.: Novel and efficient density based empirical likelihood procedures for symmetry and K-sample comparisons: STATA package. http://sphhp.buffalo.edu/biostatistics/research-and-acilities/software/stata.html (2013)

[18] Vexler A, Gurevich G. Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy. Comput. Stat. Data Anal.. 2010, 54 531-545

[19] Vexler A, Gurevich G. A note on optimality of hypothesis testing. Math. Eng. Sci. Aerosp.. 2011, 2 243-250

[20] Vexler A, Yu J. Two-sample density-based empirical likelihood tests for incomplete data in application to a pneumonia study. Biom. J.. 2011, 53 628-651

[21] Vexler A, Kim YM, Yu J, Lazar NA, Hutson AD. Computing critical values of exact tests by incorporating Monte Carlo simulations combined with statistical tables. Scand. J. Stat.. 2014

[22] Vexler A, Liu S, Schisterman EF. Nonparametric likelihood inference based on cost-effectively-sampled-data. J. Appl. Stat.. 2011, 38 769-783

[23] Vexler A, Shan G, Kim S, Tsai WM, Tian L, Hutson AD. An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions. J. Stat. Plan. Inference. 2011, 141 6 2128-2140

[24] Vexler A, Tsai W-M, Malinovsky Y. Estimation and testing based on data subject to measurement errors: from parametric to non-parametric likelihood methods. Stat. Med.. 2012, 31 2498-2512

[25] Vexler A, Tsai W-M, Gurevich G, Yu J. Two-sample density-based empirical likelihood ratio tests based on paired data, with application to a treatment study of Attention-Deficit/Hyperactivity Disorder and Severe Mood Dysregulation. Stat. Med.. 2012, 31 1821-1837

[26] Vexler A, Yu J, Hutson AD. Likelihood testing populations modeled by autoregressive process subject to the limit of detection in applications to longitudinal biomedical data. J. Appl. Stat.. 2011, 38 1333-1346

[27] Yu J, Vexler A, Kim S, Hutson AD. Two-sample empirical likelihood ratio tests for medians in application to biomarker evaluations. Can. J. Stat.. 2011, 39 671-689