The multivariate asymmetric slash Laplace distribution and its applications
AbstractWe have introduced a multivariate asymmetric-slash Laplace distribution, a flexible distribution that can take skewness and heavy tails into account. This distribution is useful in simulation studies where it can introduce distributional challenges in order to evaluate a statistical procedure. It is also useful in analyzing data sets that do not follow the normal law. We have used the microarray data set for illustration. The asymmetric slash Laplace distribution provides the possibility of modelling impulsiveness and skewness required for gene expression data. Hence, the probability distribution presented in this paper will be very useful in estimation and detection problems involving gene expression data. The multivariate asymmetric-slash Laplace distribution introduced in this article is clearly an alternative to multivariate skew-slash distributions because it can model skewness, peakedness and heavy tails. One interesting advantage of the multivariate asymmetric slash Laplace distribution is that its moments can be computed analytically by taking advantage of the moments of the multivariate asymmetric Laplace distribution, see the discussion in Section 3. Another attractive feature is that simulations from the multivariate asymmetric-slash Laplace distribution are straightforward from softwares that permit simulations from the multivariate asymmetric Laplace or Laplace distribution. We believe that the new class will be useful for analyzing data sets having skewness and heavy tails. Heavy-tailed distributions are commonly found in complex multi-component systems like ecological systems, biometry, economics, sociology, internet traffic, finance, business etc.
H. AKAIKE, (1973), Information theory and an extension of the maximum likelihood principle, “In breakthrough in Statistics (Kotz and Johnson eds.)”, Vol I Springer Verlag, New-York, pp. 610-624.
O. ARSLAN, (2008), An alternative multivariate skew-slash distribution, “Statistics & Probability Letters”, 78, pp. 2756-2761.
O. ARSLAN, (2009), Maximum likelihood parameter estimation for the multivariate skew-slash distribution, “Statistics & Probability Letters”, 78, pp. 2158-2165.
O. ARSLAN, A. I. GENC, (2009), A generalization of the multivariate slash distribution, “Journal of Statistical Planning and Inference”, 78, pp. 1164-1170.
O. E. BARNDORFF-NIELSEN, (1997), Normal inverse Gaussian distributions distribution and stochastic volatility modelling, “Scandinavian Journal of Statistics”, 24, pp. 1-13.
P. P. BINDU, (2011), A new family of skewed slash distributions generated by the normal kernal, “Statistica”, anno LXXI, n. 3, 2011.
P. P. BINDU, (2012)(a), A new family of skewed slash distributions generated by the Cauchy kernal, “Communications in Statistics-Theory and Methods”, to appear.
P. P. BINDU, (2012)(b), The multivariate skew-slash t and skew-slash Cauchy distributions, “Model Assisted Statistics Applications”, 7, 33-40.
K. BURNHAM, D. ANDERSON, (1998), Model selection and Inference, Springer, New York.
W. S. CLEVELAND, S. J. DELVIN, (1988), Locally Weighted regression: an approach to regression analysis by local fitting, “Journal of the American Statistical Association”, 83(403), pp. 596-610.
S. DHARMADHIKARI, K. JOAG-DEV, (1988), Unimodality, Convexity, and Applications, Academic Press, San Diego.
M. D. ERNST, (1998), A multivariate generalized Laplace distribution, “Computational Statistics”, 13, pp. 227-232.
K. T. FANG, S. KOTZ, K. W. NG, (1990), Symmetric multivariate and Related distributions, Chapman and Hall, London, New-York.
C. F. FERNAN’DEZ, J. OSIEWALSKI, M. F. J. STEEL, (1995), Modeling and inference with v-distributions, “Journal of the American Statistical Association”, 90, pp. 1331-1340.
A. GEN, (2007), A Generalization of the Univariate slash by a Scale-Mixtured Exponential Power Distribution, “Communications in Statistics - Simulation and Computation”, 36(5), 937-947.
H. W. GOME’Z, F. A. QUINTANA AND F. J. TORRES, (2007), A New Family of Slash-Distributions with Elliptical Contours, “Statistics & Probability Letters”, 77 (7), 715-727.
R. A. HARO-LO’PEZ, A. F. M. SMITH, (1999), On robust Bayesian analysis for location and scale parameters, “Journal of Multvariate Analysis”, 70, pp. 30-56.
K. K. JOSE, T. LISHAMOL, (2007), On slash Laplace distributions and applications, ‘‘STARS International Journal (Sciences) ”, 1, pp. 1-10.
K. KAFADAR, (1982), A biweight approach to the one-sample problem, “Journal of the American Statistical Association”, 77, pp. 416-424.
K. KAFADAR, (1988), Slash distribution, “Encyclopedia of Statistical Sciences”, Johnson, N.L., Kotz, S., Read, C., Eds., Wiley, New York: 510-511.
S. KOTZ, K. KOZUBOWSKI, PODGORSKI, (2001), The Laplace Distribution and Generalizations: A revisit with Applications to Communications, Economics, Engineering and Finance, Birkhäuser, Boston.
R DEVELOPMENT CORE TEAM, (2006), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing: Vienna, Austria, http://www.R-project.org/.
F. TAN, H. PENG, (2005), The slash and skew-slash student t distributions, “Pre-print”, available from http://home.olemiss.edu/ mmpeng/sst2.pdf.
J. WANG, M. G. GENTON, (2006), The multivariate skew-slash distribution, “Journal of Statistical Planning and Inference”, 136, pp. 209-220.
Y. H. YANG, S. DUDOIT, P. LUU, D. M. LIN, V. PENG, J. NGAI, T. P. SPEED, (2002), Normalization for cDNA microarray data: a robust composite method addressing single and multiple slide systematic variation, “Nucleic Acids Research”, 30(4), e15.