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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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 This article is categorized under: Algorithms and Computational Methods > Random Number Generation Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

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.

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