Abstract:
The purpose of extreme value frequency analysis is to analyze past records of
extremes to estimate future occurrence probabilities, nature, intensity and frequency.
It is only possible if most suitable probability distribution is employed with proper
estimation method. Many probability distributions and parameter estimation methods
have been proposed in last couple of decade, but the quest of best fit has always been
of concern. In the continuity of this dimension, the fundamental aim of this
dissertation is to model the extreme events by proper probability distributions using
the most suitable method of estimation. This objective is achieved by reviewing and
employing the concept of L- and TL-moments and quadratic rank transmutation map.
The L- and TL-moments of some specific distributions are derived, and parameter
estimation is approached through the method of L- and TL-moments. In this study
three transmuted and two double-bounded transmuted distributions are developed and
proposed with their properties and applications. Moreover, the generalized
relationships are also established to obtain the properties of the transmuted
distributions using their parent distribution.
In the first part of the dissertation, it is observed that the Singh Maddala, Dagum, and
generalized Power function distribution are suitable candidates for extreme value
frequency analysis, as these densities are heavy-tailed in their range. In literature, the
theory of L- and TL-moments is considered best and extensively used for such
analysis. Therefore, the L- and TL-moments are derived, and the parameters of these
densities are estimated by employing the method of L- and TL-moments. These
estimation methods are compared with the method of maximum likelihood estimation
and method of moments using some real extreme events data sets. Simulation studies
have also been carried out for the same purpose. In these studies, superiority of the
method of L- and TL-moments has been justified.
In the second part of the dissertation, three heavy-tailed, flexible and versatile
distributions are introduced using the quadratic rank transmutation map to model the
extreme value data. The proposed distributions are the transmuted Singh Maddala,
transmuted Dagum and transmuted New distribution. The mathematical properties
viiiand reliability behaviors are derived for each of the proposed transmuted distribution.
The densities of order statistics, generalized TL-moments, and its special cases are
also studied. Parameters are estimated using the method of maximum likelihood
estimation. The appropriateness of the transmuted distributions for modeling extreme
value data is illustrated using some real data sets. The empirical results indicated that
the proposed transmuted distributions perform better as compared to the parent
distributions.
In literature, continuous double-bounded data is fairly popular. However, it is quite
unrealistic to analyze such kind of data using normal theory models. This type of data
is also targeted, and two new double-bounded distributions have been introduced, in
the third part of the dissertation. These developed distributions termed as transmuted
Kumaraswamy and transmuted Power function distribution. The most common
mathematical properties are derived, and it has been observed that the hazard rate
function have either increasing or bathtub shaped for these distributions. The method
of maximum likelihood estimation is employed for the parameter estimation and the
construction of the confidence intervals. The application and potential of these
distributions are investigated using real data sets. Comparatively, proposed double
bounded transmuted distributions performed better than their parent distributions in
real applications.
Finally, it has already been proved that transmuted distributions are better than their
parent distributions. But directly dealing with the transmuted density is complicated
and exhaustive especially for order statistics analysis. To make it simple, the
relationships between transmuted and parent distributions are established for the
single and product moments of order statistics. In addition, the generalized TL-
moments of the transmuted distribution and its special cases are derived using single
moments of the parent distribution. The established relationships are used for
parameter estimation, and a simulation study is also carried out to investigate the
behavior of the estimators. Moreover, the transmuted and parent distributions
relationships are illustrated through two well-known distributions and two real data
sets. Furthermore, it can be claimed on the base of established results; now it is quite
convenient to find the moments of order statistics, parameter estimates and especially
generalized TL-moments for transmuted distributions.