A generalization of Tukey’s g-h family of distributions

Abstract

A new class of distribution function based on the symmetric densities is introduced, these transformations also produce nonnormal distributions and its pdf and cd f can be expressed in parametric form. This class of distributions depend on the two parameters, namely g and h which controls the skewness and the elongation of the tails, respectively. This class of skewed distributions is a generalization of Tukey's g − h family of distributions.In this paper, we calculate a closed form expression for the density and distribution of the Tukey's g − h family of generalized distributions, which allows us to easily compute probabilities, moments and related measures.

Introducción a la teoría estadística del riesgo

“…En la tabla 1.4 se muestran los valores de g y h que aproximan un conjunto seleccionado de distribuciones bien conocidas. En [115], se establecen los momentos ordinarios de orden n de la familia de distribuciones g − h de Tukey para h < 1 n , como sigue:…”

Compounding of distributions: a survey and new generalized classes

“…Some wellknown generalized (or G-) classes are: Marshall-Olkin-G (Marshall and Olkin 1997), exponentiated-G (Gupta et al 1998), beta-G (Eugene et al 2002), Kumaraswamy-G (Cordeiro and de-Castro 2011), McDonald-G (Alexander et al 2012), ZBgamma-G (Zografos and Balakrishnan 2009;Amini et al 2014), RBgamma-G (Ristić and Balakrishanan 2012;Amini et al 2014), odd-gamma-G (Torabi and Montazari 2012), Kummer-beta-G (Pescim et al 2012), beta extended Weibull-G (Cordeiro et al 2012b), odd exponentiated generalized-G , truncated exponential-G (Barreto-Souza and Simas 2013), logistic-G (Torabi and Montazari 2014), gamma extended Weibull-G (Nascimento et al 2014), odd Weibull-G (Bourguignon et al 2014a), exponentiated-half-logistic-G ), Libby-Novick beta-G Ristić et al 2015), Lomax-G (Cordeiro et al 2014d), Harris-G (Batsidis and Lemonte 2015;Pinho et al 2015), modified beta-G (Nadarajah et al 2014b), odd generalized-exponential-G , Kumaraswamy odd log-logistic-G (Alizadeh et al 2015b), beta odd log-logistic-G , KumaraswamyMarshall-Olkin-G (Alizadeh et al 2015c), beta -Marshall-Olkin-G (Alizadeh et al 2015a), Weibull-G (Tahir et al 2016b), exponentiated-Kumaraswamy-G (da- Silva et al 2016), ZBgamma-odd-loglogistic-G ) and Tukey's g-and h-G (Jiménez et al 2015). For more details on some well-established G-classes, the reader is referred to Tahir and Nadarajah (2015).…”

A Mixture of Generalized Tukey’sgDistributions

“…The following properties for pdf, cdf, and quantile functions of Tukey's generalized distribution were established by Jiménez et al [18] in terms of the pdf and cdf of as follows:…”

“…The first expression of (4) allows us to obtain the following pdf associated with Tukey's distribution. Table 1 shows the parameters of the pdf of that we obtain using a selected set of well known symmetrical distributions (from Jiménez et al [18]).…”

“…Jiménez and Arunachalam [17] provided explicit expressions for VaR and CVaR calculations using the family of Tukey's -ℎ distributions. Currently Jiménez et al [18] studied generalization of Tukey's -ℎ family of distributions, when the standard normal random variable is replaced by a continuous random variable with mean 0 and variance 1.…”