A new method for generating families of continuous distributions

In this paper, a new method is proposed for generating families of continuous distributions. A random variable $$X$$, “the transformer”, is used to transform another random variable $$T$$, “the transformed”. The resulting family, the $$T$$-$$X$$ family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function of the random variable $$X$$. Many new distributions, which are members of the family, are presented. Several known continuous distributions are found to be special cases of the new distributions.

[1]  C. Chatfield.  Continuous Univariate Distributions, Vol. 1 , 1995 .

[2]  Pandu R. Tadikamalla,et al.  A Probability Distribution and its Uses in Fitting Data , 1979 .

[3]  Carl Lee,et al.  On the Properties of Beta-Gamma Distribution , 2007 .

[4]  F. Famoye,et al.  BETA-NORMAL DISTRIBUTION AND ITS APPLICATIONS , 2002 .

[5]  G. Cordeiro,et al.  The Kumaraswamy Generalized Half-Normal Distribution for Skewed Positive Data , 2012, Journal of Data Science.

[6]  Deo Kumar Srivastava,et al.  The exponentiated Weibull family: a reanalysis of the bus-motor-failure data , 1995 .

[7]  Ayman Alzaatreh,et al.  Gamma-Pareto distribution and its applications , 2012 .

[8]  F. Famoye,et al.  The beta-Pareto distribution , 2008 .

[9]  Marshall Freimer,et al.  a study of the generalized tukey lambda family , 1988 .

[10]  Gauss M. Cordeiro,et al.  The Kumaraswamy generalized gamma distribution with application in survival analysis , 2011 .

[11]  Gauss M. Cordeiro,et al.  A new family of generalized distributions , 2011 .

[12]  M. Steel,et al.  A Constructive Representation of Univariate Skewed Distributions , 2006 .

[13]  I. W. Burr.  Cumulative Frequency Functions , 1942 .

[14]  P. Kumaraswamy.  A generalized probability density function for double-bounded random processes , 1980 .

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

[16]  M. C. Jones.  Families of distributions arising from distributions of order statistics , 2004 .

[17]  Samuel Kotz,et al.  The beta exponential distribution , 2006, Reliab. Eng. Syst. Saf..

[18]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[19]  S. Nadarajah,et al.  The beta Gumbel distribution , 2004 .

[20]  K. Song.  Rényi information, loglikelihood and an intrinsic distribution measure , 2001 .

[21]  James B. McDonald,et al.  Some Generalized Functions for the Size Distribution of Income , 1984 .

[22]  Bruce W. Schmeiser,et al.  An approximate method for generating symmetric random variables , 1972, CACM.

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

[24]  K. Pearson.  Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material , 1895 .

[25]  F. Galton.  Inquiries into Human Faculty and Its Development , 1883 .

[26]  N. L. Johnson,et al.  Systems of frequency curves generated by methods of translation. , 1949, Biometrika.

[27]  J. Moors,et al.  A quantile alternative for kurtosis , 1988 .

[28]  M. C. Jones.  Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages , 2009 .

[29]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[30]  L. Amoroso,et al.  Ricerche intorno alla curva dei redditi , 1925 .

[31]  Tim R. L. Fry.  Univariate and Multivariate Burr Distributions: A Survey , 1989 .

[32]  Saralees Nadarajah,et al.  The Kumaraswamy Weibull distribution with application to failure data , 2010, J. Frankl. Inst..

[33]  K. Pearson.  Contributions to the Mathematical Theory of Evolution , 1894 .

[34]  F. Famoye,et al.  Beta-Normal Distribution: Bimodality Properties and Application , 2004 .