الفهرس 
chapter 1Only 14 pages are availabe for public view
 |  | from | 123 |  |  |
| | | from | 123 | | |
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 di erent 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)