A new method for generating families of continuous distributions
10 Apr 2013-Vol. 71, Iss: 1, pp 63-79
TL;DR: In this article, a new method is proposed for generating families of continuous distributions, where a random variable is used to transform another random variable and the resulting family, the $$T$$¯¯ -=-=-=-=-=-=-=-=-=-=-=-=- family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function.
Abstract: 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.
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TL;DR: In this article, a new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility, namely one-parameter exponential distribution, for data analysis purposes.
Abstract: A new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility. A special case has been considered in detail, namely one-parameter exponential distribution. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, mode, moment-generating function, mean residual lifetime, stochastic orders, order statistics, and expression of the entropies, are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non linear equations only. Further, we consider an extension of the two-parameter exponential distribution also, mainly for data analysis purposes. Two datasets have been analyzed to show how the proposed models work in practice.
211 citations
Cites methods from "A new method for generating familie..."
...Alzaatreh et al. (2013) introduced a new method for generating families of continuous distributions called T-X family by replacing the beta PDF with a PDF, r(t ), of a continuous random variable and applying a function W (F(x)) that satisfies some specific conditions....
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TL;DR: In this paper, a new family of distributions called exponentiated Weibull-exponential (exponential) distribution is defined and studied, and some properties including distribution shapes, limit behavior, hazard function, Shannon entropy, moments, skewness and kurtosis are discussed.
Abstract: In this paper, a new family of distributions called exponentiated $T$-$X$ distribution is defined. Some of its properties and special cases are discussed. A member of the family, namely, the three-parameter exponentiated Weibull-exponential distribution is defined and studied. Some of its properties including distribution shapes, limit behavior, hazard function, Shannon entropy, moments, skewness and kurtosis are discussed. The flexibility of the exponentiated Weibull-exponential distribution is assessed by applying it to three real data sets and comparing it with other distributions. The exponentiated Weibull-exponential distribution is found to adequately fit left-skewed and right-skewed data sets.
145 citations
TL;DR: In this paper, the authors introduce a new family of continuous distributions generated from a logistic random variable called the logistic-X family, which can be expressed as a linear combination of exponentiated densities based on the same baseline distribution.
Abstract: The logistic distribution has a prominent role in the theory and practice of statistics. We introduce a new family of continuous distributions generated from a logistic random variable called the logistic-X family. Its density function can be symmetrical, left-skewed, right-skewed, and reversed-J shaped, and can have increasing, decreasing, bathtub, and upside-down bathtub hazard rates shaped. Further, it can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon entropy, and order statistics. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. We also investigate the properties of one special model, the logistic-Frechet distribution, and illustrate its importance by means of two applications to real data sets.
129 citations
Cites background or methods from "A new method for generating familie..."
...…(i) W [G(x)] ∈ [a, b] (ii) W [G(x)] is differentiable and monotonically non decreasing, and (iii) W [G(x)] → a as x → −∞ andW [G(x)] → b as x → ∞ (1) Recently, Alzaatreh et al. (2013) defined the T-X family of distributions by F(x) = ∫ W [G(x)] a r(t ) dt (2) whereW [G(x)] satisfies the…...
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...Alzaatreh et al. (2013) obtained the Shannon’s entropy of the T-X family, whereW [G(x)] = −log[1 − G(x)]....
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...One can use the same technique as in Alzaatreh et al. (2013) to derive the Shannon’s entropy of the LX family when W [G(x)] = log{−log[1 − G(x)]} as ηx = E [ log { g [ G−1 ( 1 − e−eT )]}] − E(eT ) + μT + ηT (27) whereμT and ηT are themean and Shannon’s entropy for the random variable T,…...
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...…and Montazeri, 2012), log-gamma-G (Amini et al., 2012), logistic-G (Torabi andMontazeri, 2014), exponentiated generalized-G (Cordeiro et al., 2013), transformed-transformer (T-X; Alzaatreh et al., 2013), exponentiated T-X (exp T-X; Alzaghal et al., 2013), and Weibull-G (Bourguignon et al., 2014)....
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TL;DR: A new family of continuous distributions called the odd generalized exponential family, whose hazard rate could be increasing, decreasing, J, reversed-J, bathtub and upside-down bathtub is proposed, which includes as a special case the widely known exponentiated-Weibull distribution.
Abstract: We propose a new family of continuous distributions called the odd generalized exponential family, whose hazard rate could be increasing, decreasing, J, reversed-J, bathtub and upside-down bathtub. It includes as a special case the widely known exponentiated-Weibull distribution. We present and discuss three special models in the family. Its density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon and Renyi entropies and order statistics. For the first time, we obtain the generating function of the Frechet distribution. Two useful characterizations of the family are also proposed. The parameters of the new family are estimated by the method of maximum likelihood. Its usefulness is illustrated by means of two real lifetime data sets.
AMS Subject Classification Primary 60E05; secondary 62N05; 62F10
125 citations
Cites background from "A new method for generating familie..."
...…and Montazari (2014), odd exponentiated generalized (odd exp-G) by Cordeiro et al. (2013), transformed-transformer (T-X) (Weibull-X and gamma-X) by Alzaatreh et al. (2013), exponentiated T-X by Alzaghal et al. (2013), odd Weibull-G by Bourguignon et al. (2014), exponentiated half-logistic by…...
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TL;DR: In this paper, five general methods of combination and their variations are discussed: (1) method of generating skew distributions, (2) method adding parameters (e.g., exponentiation), (3) beta generated method, (4) transformed-transformer method, and (5) composite method.
Abstract: There has been a renewed interest in developing more flexible statistical distributions in recent decades. A major milestone in the methods for generating statistical distributions is undoubtedly the system of differential equation approach. There is a recent renewed interest in generating skewed distributions. Generally speaking, the methods developed prior to 1980s may be summarized into three categories: (1) method of differential equation, (2) method of transformation, and (3) quantile method. Techniques developed since 1980s may be categorized as ‘methods of combination’ for the reason that these methods attempt to combine existing distributions into new distributions or adding new parameters to an existing distribution. This article discusses five general methods of combination and their variations. These five are (1) method of generating skew distributions, (2) method of adding parameters (e.g., exponentiation), (3) beta generated method, (4) transformed-transformer method, and (5) composite method. WIREs Comput Stat 2013, 5:219–238. doi: 10.1002/wics.1255
Conflict of interest: The authors have declared no conflicts of interest for this article.
123 citations
References
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TL;DR: This final installment of the paper considers the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now.
Abstract: In this final installment of the paper we consider the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now. To a considerable extent the continuous case can be obtained through a limiting process from the discrete case by dividing the continuum of messages and signals into a large but finite number of small regions and calculating the various parameters involved on a discrete basis. As the size of the regions is decreased these parameters in general approach as limits the proper values for the continuous case. There are, however, a few new effects that appear and also a general change of emphasis in the direction of specialization of the general results to particular cases.
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TL;DR: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
Abstract: Continuous Distributions (General) Normal Distributions Lognormal Distributions Inverse Gaussian (Wald) Distributions Cauchy Distribution Gamma Distributions Chi-Square Distributions Including Chi and Rayleigh Exponential Distributions Pareto Distributions Weibull Distributions Abbreviations Indexes
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TL;DR: In this paper, a nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard.
Abstract: On introduit une nouvelle classe de fonctions de densite dependant du parametre de forme λ, telles que λ=0 corresponde a la densite normale standard
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"A new method for generating familie..." refers background in this paper
...Azzalini [4] introduced the skew normal family of distributions....
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2,085 citations
"A new method for generating familie..." refers methods in this paper
...Johnson [18] proposed a system for generating distributions using normalization transformation with the general form...
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TL;DR: The first systematic, scientific treatise on individual differences in psychological phenomena Francis Galton's work marked the beginning of the scientific study of imagery and the association of ideas Here for the first time he sketched out the essentials of a theory of mental testing as discussed by the authors.
Abstract: The first systematic, scientific treatise on individual differences in psychological phenomena Francis Galton's work marked the beginning of the scientific study of imagery and the association of ideas Here for the first time he sketched out the essentials of a theory of mental testing
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