Analyzing skewed data by power normal model

Abstract The skew normal distribution, proposed by Azzalini (1985, A class of distributions which include the normal. Scand J Stat 12:171–178), can be a suitable model for the analysis of data exhibiting a unimodal density having some skewness present, a structure often occurring in data analysis. It has been observed that there are some practical problems in estimating the skewness parameter for small to moderate sample sizes. In this paper we point out those problems and propose another skewed model which we call “Power normal model”. The basic structural properties of the proposed model including its reliability properties are presented. The closeness of skew normal and power normal distributions is investigated. It is shown that the proposed model has some nice properties which make it feasible to study the estimation and testing of the skewness parameter. This can be achieved by transforming the data to the exponential model which has been studied extensively in the literature.

[1]  Erich L. Lehmann,et al.  The Power of Rank Tests , 1953 .

[2]  Moshe Shaked,et al.  Preservation of some likelihood ratio stochastic orders by order statistics , 2001 .

[3]  R. Beaver,et al.  The nontruncated marginal of a truncated bivariate normal distribution , 1993 .

[4]  Debasis Kundu,et al.  Discriminating between Weibull and generalized exponential distributions , 2003, Comput. Stat. Data Anal..

[5]  N. Sartori.  Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions , 2006 .

[6]  Marc G. Genton,et al.  Skew-elliptical distributions and their applications : a journey beyond normality , 2004 .

[7]  Harshinder Singh,et al.  The Reversed Hazard Rate Function , 1998, Probability in the Engineering and Informational Sciences.

[8]  Pushpa L. Gupta,et al.  Modeling failure time data by lehman alternatives , 1998 .

[9]  B. Liseo,et al.  A note on reference priors for the scalar skew-normal distribution , 2006 .

[10]  Anna Clara Monti.  A note on the estimation of the skew normal and the skew exponential power distributions , 2003 .

[11]  D. Kundu,et al.  Closeness of Gamma and Generalized Exponential Distribution , 2003 .

[12]  D. Kundu,et al.  EXPONENTIATED EXPONENTIAL FAMILY: AN ALTERNATIVE TO GAMMA AND WEIBULL DISTRIBUTIONS , 2001 .

[13]  A. Azzalini.  A class of distributions which includes the normal ones , 1985 .

[14]  Arturo J. Fernández,et al.  Computing maximum likelihood estimates from type II doubly censored exponential data , 2002 .

[15]  Debasis Kundu,et al.  Generalized exponential distribution: different method of estimations , 2001 .

[16]  A. Azzalini,et al.  Statistical applications of the multivariate skew normal distribution , 2009, 0911.2093.

[17]  B. Liseo.  La classe delle densità normali sghembe: aspetti inferenziali da un punto di vista Bayesiano , 1990 .

[18]  S. Kirmani,et al.  On order relations between reliability measures , 1987 .

[19]  D. Kundu,et al.  Theory & Methods: Generalized exponential distributions , 1999 .

[20]  R. Beaver,et al.  Skewed multivariate models related to hidden truncation and/or selective reporting , 2002 .

[21]  Ramesh C. Gupta,et al.  RELIABILITY STUDIES OF THE SKEW-NORMAL DISTRIBUTION AND ITS APPLICATION TO A STRENGTH-STRESS MODEL , 2001 .

[22]  A. Pewsey.  Problems of inference for Azzalini's skewnormal distribution , 2000 .

[23]  Ramesh C. Gupta,et al.  Generalized skew normal model , 2004 .