$$\alpha $$α Logarithmic Transformed Family of Distributions with Application

In this paper, a new three-parameter distribution, called $$\alpha $$α logarithmic transformed generalized exponential distribution ($$\alpha LTGE$$αLTGE) is proposed. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, moment generating function, mean deviation about the mean and median, mean residual life, Bonferroni curve, Lorenz curve, Gini index, Rényi entropy, stochastic ordering and order statistics are derived. It appears to be a distribution capable of allowing monotonically increasing, decreasing, bathtub and upside-down bathtub shaped hazard rates depending on its parameters. The maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non-linear equations only. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance covariance matrix. Finally, two empirical applications of the new model to real data are presented for illustrative purposes.

[1]  P. Phillips,et al.  Lorenz Curve and Gini coefficient: novel tools for analysing seasonal variation of environmental radon gas. , 2009, Journal of environmental management.

[2]  Dorel Aiordachioaie,et al.  Signal segmentation in time-frequency plane using Rényi entropy - Application in seismic signal processing , 2013, 2013 Conference on Control and Fault-Tolerant Systems (SysTol).

[3]  Gauss M. Cordeiro,et al.  The beta generalized exponential distribution , 2008, 0809.1889.

[4]  Faton Merovci,et al.  TRANSMUTED EXPONENTIATED EXPONENTIAL DISTRIBUTION , 2013 .

[5]  Jerald F. Lawless,et al.  Statistical Models and Methods for Lifetime Data. , 1983 .

[6]  J. Kenney,et al.  Mathematics of statistics , 1940 .

[7]  M. H. Tahir,et al.  The odd generalized exponential family of distributions with applications , 2015 .

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

[9]  Ayman Alzaatreh,et al.  Methods for generating families of univariate continuous distributions in the recent decades , 2013 .

[10]  M. Gail.  Applying the Lorenz curve to disease risk to optimize health benefits under cost constraints. , 2009, Statistics and its interface.

[11]  M. P. Tarvainen,et al.  Renyi entropy in identification of cardiac autonomic neuropathy in diabetes , 2012, 2012 Computing in Cardiology.

[12]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[13]  Mohamed Elsayed Ahmed Mead,et al.  A NEW GENERALIZATION OF BURR XII DISTRIBUTION , 2014 .

[14]  Umesh Singh,et al.  A New Method of Proposing Distribution and Its Application to Real Data , 2016 .

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

[16]  M. Nassar,et al.  THE BETA GENERALIZED PARETO DISTRIBUTION , 2011 .

[17]  Debasis Kundu,et al.  Generalized exponential distribution: Existing results and some recent developments , 2007 .

[18]  Konstantinos Adamidis,et al.  A Family of Lifetime Distributions , 2012 .

[19]  D. Kundu,et al.  Marshall-Olkin generalized exponential distribution , 2015 .

[20]  Ayman Alzaatreh,et al.  A new method for generating families of continuous distributions , 2013 .

[21]  A. Radice.  Use of the Lorenz Curve to Quantify Statistical Nonuniformity of Sediment Transport Rate , 2009 .

[22]  Alexandre B. Simas,et al.  Some Results for Beta Fréchet Distribution , 2008, 0809.1873.

[23]  Malwane M. A. Ananda,et al.  Modeling actuarial data with a composite lognormal-Pareto model , 2005 .

[24]  M. O. Lorenz,et al.  Methods of Measuring the Concentration of Wealth , 1905, Publications of the American Statistical Association.

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

[26]  S. Nadarajah,et al.  Extreme Value Distributions: Theory and Applications , 2000 .

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

[28]  Debasis Kundu,et al.  A new method for generating distributions with an application to exponential distribution , 2017 .

[29]  G. S. Mudholkar,et al.  Exponentiated Weibull family for analyzing bathtub failure-rate data , 1993 .

[30]  S. Kotz,et al.  Statistical Size Distributions in Economics and Actuarial Sciences , 2003 .

[31]  G. Casella,et al.  Statistical Inference , 2003, Encyclopedia of Social Network Analysis and Mining.

[32]  Tamás Linder,et al.  High-Resolution Scalar Quantization With Rényi Entropy Constraint , 2010, IEEE Transactions on Information Theory.

[33]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[34]  Larry Wasserman,et al.  Forest Density Estimation , 2010, J. Mach. Learn. Res..

[35]  L Egghe,et al.  Development of hierarchy theory for digraphs using concentration theory based on a new type of Lorenz curve , 2002 .

[36]  M. H. Tahir,et al.  The Gumbel-Lomax Distribution: Properties and Applications , 2016, J. Stat. Theory Appl..

[37]  A. Maldonado,et al.  Depression and cognition: new insights from the Lorenz curve and the Gini index , 2007 .

[38]  Boualem Boashash,et al.  Estimating the number of components of a multicomponent nonstationary signal using the short-term time-frequency Rényi entropy , 2011, EURASIP J. Adv. Signal Process..

[39]  I. Olkin,et al.  A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families , 1997 .

[40]  Sanku Dey,et al.  A New Extension of Generalized Exponential Distribution with Application to Ozone Data , 2017 .

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