T-normal family of distributions: a new approach to generalize the normal distribution

Abstract

The idea of generating skewed distributions from normal has been of great interest among researchers for decades. This paper proposes four families of generalized normal distributions using the T-X framework. These four families of distributions are named as T-normal families arising from the quantile functions of (i) standard exponential, (ii) standard log-logistic, (iii) standard logistic and (iv) standard extreme value distributions. Some general properties including moments, mean deviations and Shannon entropy of the T-normal family are studied. Four new generalized normal distributions are developed using the T-normal method. Some properties of these four generalized normal distributions are studied in detail. The shapes of the proposed T-normal distributions can be symmetric, skewed to the right, skewed to the left, or bimodal. Two data sets, one skewed unimodal and the other bimodal, are fitted by using the generalized T-normal distributions.

Family of generalized gamma distributions: Properties and applications

“…The logarithm of the egg diameters of the asteroids data has a bimodal shape. We fit the logarithm of the egg diameters of the asteroids data and compared it with the beta-normal distribution (Famoye et al, 2004) and logistic-normal{logistic} distribution (Alzaatreh et al, 2014b). The results of the maximum likelihood estimates, the log-likelihood value, the AIC, the K-S test statistic, and the p-value for the K-S statistic for the fitted distributions are reported in Table 6.…”

“…The T -R{Y } framework defined in Aljarrah et al (2014) (see also Alzaatreh et al, 2014b) is briefly described in the following. Let T , R and Y be random variables with CDF…”

A Generalization of the Weibull Distribution with Applications

“…In this example we consider the percentage of body fat (%Bfat) variable for the 202 athletes. The LWD, the beta-normal distribution (BND) defined by Eugene et al (2002), the logistic-normal{logistic} distribution (LND) defined by Alzaatreh et al (2014b), and the Weibull distribution (WD) are applied to fit the data set. Table 7 contains the estimates, standard errors of the estimates, log-likelihood values, AIC, K-S test statistic, and the corresponding p-values.…”

“…in the T-X family in (1) and called it the T-R{Y} family. Following the notation proposed by Alzaatreh, Famoye, and Lee (2014b), the CDF of the T-R{Y} family, as defined by Aljarrah et al (2014), is given by …”

Some Generalized Families of Weibull Distribution: Properties and Applications

“…W in (1.4) as a quantile function of an existing random variable Y. A unified definition of this family was given in Alzaatreh et al (2014b) and named as T-R{Y}. Let T, R and Y be random variables with CDF ( ) ( ) 5) and the PDF corresponding to (1.5) is given by Vol.…”