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Skewness and elongation are two factors that directly determine the shape of a probability distribution. Thus, to obtain a flexible distribution it is always desirable that the parameters of the distribution directly determine the skewness and elongation. To meet this purpose, Tukey (1977) introduced a family of distributions called g-and-h family (gh family) based on a transformation of the standard normal variable where g and h determine the skewness and the elongation, respectively. The gh family of distributions was extensively studied by Hoaglin (1985) and Martinez and Iglewicz (1984). For its flexibility in shape He and Raghunathan (2006) have used this distribution for multiple imputations. Because of the complex nature of this family of distributions, it is not possible to have an explicit mathematical form of the density function and the estimates of the parameters g and h fully depend on extensive numerical computations.In this study, we have developed algorithms to numerically compute the density functions. We present algorithms to obtain the estimates of g and h using method of moments, quantile method and maximum likelihood method. We analyze the performance of each method and compare them using simulation technique. Finally, we study some special cases of gh family and their properties. / Department of Mathematical Sciences

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The family of univariate skew-normal probability distributions, an extension of symmetric normal distribution to a general case of asymmetry, was originally proposed by Azzalani [1]. Since its introduction, very limited research has been conducted in this area. An extension of the univariate skew-normal distribution to the multivariate case was considered by Azzalani and Dalla Valle [4]. Its application in statistics was recently considered by Azzalani and Capitanio [3]. As a general result, Azzalani (1985) [See [1]] showed that, any symmetric distribution can be viewed as a member of a more general class of skewed distributions.In this study we establish some properties of general family of skewed distributions. Examples of general family of asymmetric distributions is presented in a way to show their differences from the corresponding symmetric distributions. The skew-logistic distribution and its properties are considered in great details. / Department of Mathematical Sciences

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