Studying The Exponentiated Frechet
Distribution
Studying The Exponentiated Frechet Distribution
By
Majdah M. Badr
A thesis submitted for the requirements of the degree of Doctor of Philosophy ( Mathematical Specialization
/Mathematical Statistics )
Supervised By
Prof. Shawky Ahmed Ibrahim El-Sayed
Professor of Mathematical Statistics – Department of Statistics - Faculty of Science –
King Abdul Aziz University
FACULTY OF GIRLS COLLEGE OF EDUCATION KING ABDUL AZIZ UNIVERSITY
JEDDAH-SAUDI ARABIA
SAFAR(1431)H – JANUARY( 2010 )G
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( 2 - 1 )
( 3 - 1 ) تبحرًنا ثاءاصحلإا ووسع ...
101
( 3 - 1 - 1 ) تبحرًنا ثاءاصحلإن ةدرفًنا ووسعنا .
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101
( 3 - 1 - 2 ) تبحرًنا ثاءاصحلإن تجودسًنا ووسعنا .
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114
( 3 - 1 - 3 ) تبحرًنا ثاءاصحلإن تيثلاثنا ووسعنا .
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111
( 3 - 1 - 4 ) ا ووسعنا تبحرًنا ثاءاصحلإن تيعابرن
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121
( 3 - 2 ) ءاصحلإا ىهع داًخعلااب يئاصحلإا للادخسلاا يسلآا جشيرف عيزوح ني تبحرًنا ثا
(أ
ًهعًن سيحخي ريغ يطخ ردقي مضف شايقناو عضوًنا ي
...) 126
( 3 - 3 ) رلاا ثاقلاعنا تيدادح
( تيراركخنا )
تبحرًنا ثاءاصحلإا ووسع نيب .
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132
( 4 - 1 ) يسلآا جشيرف عيزوح ني ايهعنا تهجسًنا ىيقنا ووسع .
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165
( 4 - 1 - 1 ) ايهعنا تهجسًنا ىيقهن ةدرفًنا ووسعنا .
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165
( 4 - 1 - 2 ) ايهعنا تهجسًنا ىيقهن تجودسًنا ووسعنا .
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161
( 4 - 1 - 3 ) ايهعنا تهجسًنا ىيقهن تيثلاثنا ووسعنا .
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111
( 4 - 1 - 4 ) ايهعنا تهجسًنا ىيقهن تيعابرنا ووسعنا .
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115
( 4 - 2 ) جسًنا ىيقنا ىهع داًخعلااب يئاصحلإا للادخسلاا يسلآا جشيرف عيزوح ني ايهعنا ته
مضفأ
(
ًهعًن سيحخي ريغ يطخ ردقي شايقناو عضوًنا ي
...) 180
( 4 - 2 - 1 ) شايقنا و عضوًنا يًهعًن سيحخي ريغ يطخ ردقي مضفأ ...
. 180
( 4 - 3 ) تيراركخنا ثاقلاعنا (
تيدادحرلاا )
ىيقنا ىهع داًخعلااب يسلآا جشيرف عيزوح ووسع نيب
ايهعنا تهجسًنا .
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181
( 4 - 3 - 1 ) ووسعنا نيب تيدادحرلاا ثاقلاعنا ايهعنا تهجسًنا ىيقهن ةدرفًنا
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181
( 4 - 3 - 2 ) ايهعنا تهجسًنا ىيقهن تجودسًنا ووسعنا نيب تيدادحرلاا ثاقلاعنا .
...
188
( 4 - 3 - 3 ) ايهعنا تهجسًنا ىيقهن تيثلاثنا ووسعنا نيب تيدادحرلاا ثاقلاعنا .
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190
( 4 - 3 - 4 ) ا ايهعنا تهجسًنا ىيقهن تيعابرنا ووسعنا نيب تيدادحرلاا ثاقلاعن .
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191
( 4 - 4 ) تيراركخنا ثاقلاعنا (
تيدادحرلاا )
نيبناج ني رخبنا وأ عطقنا ىهع دًخعح يخنا ...
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192
( 4 - 4 - 1 ) تيدرفنا ووسعنا تناح يف .
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193
( 4 - 4 - 2 ) تجودسًنا ووسعنا تناح يف .
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195
( 4 - 5 ) ايهعنا تهجسًنا ىيقنا ووسعن ةدنوًنا تنادنا .
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191
( 4 - 5 - 1 ) يقهن ةدرفًنا ووسعهن ةدنوًنا تنادنا ايهعنا تهجسًنا ى
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191
( 4 - 5 - 2 ) ايهعنا تهجسًنا ىيقهن تجودسًنا ووسعن ةدنوًنا تنادنا .
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198
( 4 - 5 - 3 ) ايهعنا تهجسًنا ىيقهن تيثلاثنا ووسعهن ةدنوًنا تنادنا .
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202
( 4 - 5 - 4 ) ايهعنا تهجسًنا ىيقهن تيعابرنا ووسعهن ةدنوًنا تنادنا .
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206
سماخلا بابلا :
يسلأ اتشيرف عيزوتل قيفوتلا ةدوج ثارابتخا ...
212
( 5
عجارملا ...
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234
تيسيلجنلاا تغللاب صلختسملا ...
i
تيسيلجنلاا تغللاب صخلملا ...
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ii
Studying The Exponentaited Frechet Distribution Abstract
Majdah M. Badr
The Exponentaited Frechet distribution is considered to be one of the newest model of lifetime models , So our aim in this dissertation is to give a comprehensive study about the interims of its statistical properties of this model. We also, are interested in studying statistical inferences about its parameter(s) exponentaited frechet distribution by goodness- of- fit tests , Bayesian estimation and non Bayesian estimation.
Studying the characteristics of order statics and record values from Exponentaited Frechet distribution also discussed of inference a round parameters of the distribution depending on the order statistics and the record values. We using mont carlo method to create tables of critical values to goodness- of -fit tests of the modified Kolmogorov-Smirnov(KS), Anderson- Darling (AD) and Cramer-von Mises (CVM) for the Exponentaited Frechet distribution with unknown shape parameter based on complete and type II censored samples.
Where we use the approximation to find Bayesian estimators and then we had numerical comparisons between these estimators by using the Monte Carlo simulation.
Studying The Exponentiated Frechet Distribution Summary
Majdah M. Badr
In this dissertation, we consider the Exponentaited Frechet distribution as a new model of lifetime models. Also, We are interested in studying statistical inferences about its parameter(s). Studying order statics and record values from Exponentaited Frechet distribution and associated inference are also one of our main objectives in this dissertation. We are started this dissertation with a general introduction; giving an overview about lifetime distributions as exponential, Weibull and gamma distributions with a literature review. Also, We are propose an Exponentaited, Exponentaited Weibull (EW) and Exponentaited gamma (EG) distributions as new lifetime
models which were introduced by Gupta et al. (1998).
The dissertation consists of five chapters ordered as follows:
In chapter 1, we have introduced basic concepts and review of some literatures about the subject of our thesis. Then, we have discussed Exponentaited Frechet distribution and its statistical properties. Moments, moment generating function, some measures of tendency and dispersion have been also derived.
Also, We have studied measures of skewness and kurtosis.
Moreover, we are study the failure rate function and mean residual life functions. Finally, we have introduced relation between the Exponentaited Frechet distribution and several other distributions.
In chapter 2 we have study methods of non Bayesian estimation of the two shape parameters have been considered and find fisher information matrix in complete and type II Censored samples, then we are comparison between deferent estimators and chose best between this estimators by use simulation. So, we are study estimators Bayesian absence find
estimator two shape parameters of exponentaited frechet distribution, reliability function, failure rate function, Bayesian interval estimation of the parameters, Bayesian estimation of reliability function, Bayesian interval estimation of reliability function, Bayesian estimation of failure rate function by use two kind lose function: squared error loss and lose by use complete and type II Censored samples and comparison between estimators Bayesian and maximum likelihood to shape parameter, reliability function, failure rate function by
simulation numerical.
In Chapter 3 discussed the order statistics from Exponentaited Frechet distribution. We are derived higher order moments, moment generating function and recurrence relations of order statistics from the Exponentaited Frechet distribution. We are deduction many recurrence relations based on doubly truncated in single and product moments. We are study Inferences based on order statistics have been made.
best liner unbiased estimators for the location and scale parameters EF distribution under type II censored have been obtained. The variances and covariance's of these estimators are also presented. The for the two shape parameters is derived. Estimators based on order statistics for shape parameter are obtained by . Comparisons between the estimators are made based on simulation study.
In chapter 4 we have discussed the upper record values from an Exponentaited Frechet distribution. We are study properties of this distribution based on the upper record values, so we are find the moments around zero and the moment function, inference statistics around parameters to this distribution based on the upper record values. We are find for the location and scale parameters EF distribution. The
calculator variances and covariance's of these estimators of deferent size samples. Also, we have established general recurrence relations between moments to this distribution based on the upper record values. We deduction many recurrence relations based on doubly truncated in single and product moments.
In chapter 5 we are used method mont carlo by tables of critical values of the modified Kolmogorov-Smirnov(KS), Anderson- Darling (AD) and Cramer-von Mises (CVM) goodness- of-fit tests for the Exponentaited Frechet distribution with unknown shape parameter based on complete and type II censored samples are given in this chapter. The powers of these tests are given for a number of alternative distributions.
Finally, we are used approximation to find Bayesian estimator and then we had a numerical comparisons between these estimators by used the monte carlo simulation.
Tables and figures are enclosed in this dissertation. All our numerical results in this dissertation have been computed with
Mathematica 4.0 and Mathematica 5.0.
