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Abstract In many applied areas like lifetime analysis, nance, insurance and biology, there is a clear need to nd an appropriate distribution that represents the data in the best way. Data modeling is a great challenge. Therefore the distribution theory was widely studied. The generalized distributions (obtained by adding parameters to a well-known distribution) are appeared to provide great exibility to model of various types of data. Moreover it includes a simpler model as a limiting case. The generalized distributions are also useful in survival analysis, where the focus in this case will be on the survival and hazard rate functions while in the data modeling, the focus is on the indices of skewness and kurtosis. Many generalized classes of distributions have been developed and applied to describe various phenomena. In this study we aim to consider dierent classes of distribution functions, each of which includes the normal distribution as a particular member. Many examples of such classes can be found in the literature such as, the generalized normal distributions, which widely adopted in signal processing eld Box and Tiao (1973) , rst discussed its characters, see, also Nadarajah (2005), the kum-normal distribution proposed by Cordeiro and Castro (2012), Azzalini’s skew normal proposed by Azzalini (1985), the beta normal distribution proposed by Eugene et al.(2002), g and h distribution, which has some facility generating asymmetric data values and was suggested by Tukey (1977) and discussed later by Hoaglin and Peters (1979) and Hoaglin (1983) , SS-normal family, which was suggested by Barakat (2015) as a family that contains the reverse of every df that belongs to it and the normal full families suggested by Barakat and Khaled (2017) that contain all the possible types of cdf’s (nine types). The rst aim of this work is to carry out a comparison of most capable families of distributions for modeling asymmetry. Kum-normal, stable-symmetric normal family and two of the full families were chosen, where the quality of the t, exibility and amount of asymmetry parameters were factors used for comparison. The second aim of this work is to introduce a new method to add two shape parameters to any baseline bivariate cumulative distribution function (cdf) to get a more exible family of bivariate df’s. This method is applied to yield a new two-parameter extension of the bivariate standard normal distribution, denoted by BSSN. The statistical properties of the BSSN family are studied. Moreover, via a mixture of the BSSN family and the standard bivariate logistic cdf, we get a more capable family, denoted by FBSSN. 2 Finally, we compare the families BSSN and FBSSN with some competitors important generalized families of bivariate df’s via real data examples. Chapter 1: It includes general review of four generalized distributions (skew- normal, Kumaraswamy-normal (Kum-normal), ss normal and FN-normal). The nor- mal (Gaussian) distribution is considered as the baseline distribution, where the four studied generalized distributions turn into the normal distribution as special case. In addition, it includes some needed statistical methods of estimation and other methods for tting. Finally, we introduce a brief description of the real data, which we use in this. Chapter 2: In this chapter, we introduce overview of the R Project for statistical computing and some other packages that we will use. We brie y review the eight packages: maxLik, fBasics, fGarch, ghyp, sn, VGAM, fMultivar and moments. Chapter 3: In this chapter we present a comparison of most capable fami- lies of distributions for modeling asymmetry. Kum-normal, stable-symmetric normal family and two of the full families were chosen, where the quality of the t, exi- bility and amount of asymmetry parameters were factors used for comparison. The objective of the study of this chapter is to generate data with increasing levels of asymmetry and choose the best t. The distributions were also compared in mod- eling two data sets of pollution of the drinking water in El-Sharkia governorate in Egypt. Most of this study is concerned with distribution theory, exploring the prop- erties of some new recent families of distributions and, where appropriate, extolling their virtues; relatively much of the material of this chapter is devoted to practical application. Finally, the material of this chapter is published in Barakat et al.(2018) Chapter 4: This chapter are introduced a new method to add two shape parameters to any baseline bivariate cdf to get a more exible family of bivariate cdf’s. Through the additional parameters we can fully control the type of the resulting family. This method is applied to yield a new two-parameter extension of the bivariate standard normal distribution, denoted by BSSN. The statistical properties of the BSSN family are studied. Moreover, via a mixture of the BSSN family and the standard bivariate logistic cdf, we get a more capable family, denoted by FBSSN. Theoretically, each of the marginals of the FBSSN contains all the possible types of cdf’s and possesses very wide range of the indices of skewness and kurtosis. Finally, we compare the families BSSN and FBSSN with some competitors important generalized 3 families of bivariate cdf’s via real data examples. Finally, the material of this chapter is accepted for publication in Barakat et al. (2018) |