الفهرس 
A Detailed Study of The Logistic DistributionOnly 14 pages are availabe for public view
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Abstract
The simplicity of the logistic distribution and its importance as a
growth curve has made it one of the many important statistical
distributions. When would we use a logistic distribution? The compendium
says that is often used instead ”as an approximation to other symmetrical
distributions due to the mathematical tractability of its cumulative
distribution function (cdf).” In a more simple way, the logistic gives a nice
looking S-shaped curve with a relatively simple mathematical formula. The
S-shaped curve is used in the so-called logistic regression model, which uses
input variables to make predictions about likelihood of certain outcomes.
The S-shaped curve of the logistic cdf is thought to be a substantively useful
description of how the probability of an ”event” or other outcome rises as a
function of some input variables.
The logistic distribution has also attracted interesting applications in
the modeling of the dependence of chronic obstructive respiratory disease
prevalence on smoking and age, degrees of pneumoconiosis in coal miners,
geological issues, hemolytic uremic syndrome data for children,
physiochemical phenomenon, psychological issues, survival time of
diagnosed leukemia patients, and weight gain data.
In statistics, the logistic distribution function plays a leading role in
the methodology of logistic regression, where it makes an important
contribution to the literature on classification. The logistic distribution
function has also appeared in many guises in neural network research. In
early work, in which continuous time formalisms tended to dominate, it was
justified via its being the solution to a particular differential equation. In
later work, with the emphasis on discrete time, it was generally used more
heuristically as one of the many possible smooth, monotonic functions that
map real values into a bounded interval. More recently however, with the
increasing focus on learning, the probabilistic properties of the logistic
function have begun to be emphasized. This emphasis has led to better
learning methods and has helped to strengthen the links between neural
networks and statistics.
2
Logistic distribution functions are good models of biological
population growth in species which have grown so large that they are near
to saturating their ecosystems, or of the spread of information within
societies. They are also common in marketing where they chart the sales of
new products over time, in a different context, they can also describe
demand curves.
In order to improve the fit of the logistic model for bioassay and
quintal response data, many generalized types of the logistic distribution
have been proposed recently. These generalized distributions (indexed by
one or more shape parameters) are developed to extend the scope of the
logistic model to asymmetric probability curves and to improve the fit in the
non-central probability regions.
In this thesis, we discuss two generalizations of the logistic
distribution by introducing two extra shape parameters referred to as the
beta generalized logistic (BGL) distribution and the other is referred to as
the gamma generalized logistic (GGL) distribution. The role of the additional
parameters is to introduce skewness and to vary tail weights and provide
greater flexibility in the shape of the generalized distribution and
consequently in modeling observed data. It may be mentioned that
although several skewed distribution function exist on the positive real axis,
not many skewed distributions are available on the whole real line, which
are easy to use for data analysis purpose.
The thesis is organized as follows: In Chapter 1, some definions,
properties and characterizations of the family of logistic distributions are
recalled. Chapter 2 presents various results on the Beta Generalized Logisc
distribution (BGL). The Gamma Generalized Logistic distribution is studied in
Chapter 3. followed by an application to real data with a comparison
between both generalized distributions. All the results given in chapter 2
and chapter 3 are published in ” Journal of the Egypan Mathemacal
Society ”.