I= THE NON-LINEAR FUNCTIONS OF ORDER STATISTICS AND THEIR PROPERTIES IN SELECTED PROBABILITY MODELS
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Title of Thesis
THE NON-LINEAR FUNCTIONS OF ORDER STATISTICS AND THEIR PROPERTIES IN SELECTED PROBABILITY MODELS

Author(s)
EHTESHAM HUSSAIN
Institute/University/Department Details
Department of Statistics/ University of Karachi
Session
2006
Subject
Statistics
Number of Pages
113
Keywords (Extracted from title, table of contents and abstract of thesis)
order statistics, non-linear functions, probability models, reliability static, probabilistic model, lindley distribution, stress-strength model, maximum likelihood estimator

Abstract
In this research we study the reliability (reliability static) parameterized by stress-strength model-reliability. The stress-strength model, reliability for a components (system is given by) R=P(Y

In the present study we proposed "Lindley distribution" for stress-strength model, reliability. This distribution so far has not been studied for reliability assessment. The Lindley distribution is a positively skewed distribution with scale parameter e. First some properties of Lindley distribution are reviewed and estimation problem of e is considered. Then after reliability features of single component, parallel case for identical components and series case for two identical components are considered. The maximum likelihood estimators (MLEs) are obtained for these three situations. Unfortunately the formulas of MLEs rapidly become too complicated for useful result to be deduced. Nowadays numerically intensive methods are commonly used to tackle non-standard problems such as present one. We numerically investigated the properties of estimators based on parametric bootstrap (PB) methods. The comparison between true values and estimated values is made. Estimator performances Bias, Mean Squire Error (mse) Efron percentile confidence interval and its centers are assessed. Relevant computations and simulation is done by witting programs in MATHEMATICA. Our simulation studies showed performance of estimators for different combinations of parameters and sample sizes. For all sample sizes considered in this study, center of Efron percentile confidence intervals, are closed to true values of parameters of interest. The accuracy of parametric bootstrap method increases as the sample size increase.

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991.27 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
229.41 KB
2 1 Introduction 01
119.93 KB
  1.1 Reliability 01
  1.2 Statistical Concepts of Reliability 01
  1.3 Reliability In Variant With Time (Static) R 03
  1.4 Stress-Strength Model 05
  1. 5 Schematic Representation of Probability of Unreliability 06
  1.6 Applications 10
  1.7 The system Reliability 11
  1.8 Use Of Order Statistics 15
3 2 Review of Literature 9
102.53 KB
4 3 The Lindley Distribution 30
157.76 KB
  3.1 Introduction 30
  3.2 Estimation o f Parameter 36
  3.3 Sampling Properties o f MLE 37
  3.4 Simulation And Bootstrap Studies 42
  3.5 Simulation Results 46
5 4 Stress-Strength Model Reliability For a Component 48
153.72 KB
  4.1 Introduction 48
  4.2 The Stress-Strength Model For One Component P[Y 49
  4.3 Maximum Likelihood Estimator Of R 52
  4.4 Simulation And Bootstrap Studies 57
  4.5 Simulation Results 64
6 5 The Reliability Function Of Two-Identical Components System Connected In Parallel 66
140.41 KB
  5.1 Introduction 66
  5.2 Reliability R 2p =P[Y 66
  5.3 Reliability Expression 68
  5.4 The Point Estimatio n Of R 2p 70
  5.5 Sampling Properties Of R 2p 71
  5.6 Simulation And Bootstrap Studies 72
  5.7 Simulation Results 79
7 6 The Reliability Function Of Two-Identical Components System Connected In Series 81
268.55 KB
  6.1 Introduction 81
  6.2 Reliability R 2 s = P[Y < min(X 1, X 2 )] 82
  6.3 Reliability Expression 83
  6.4 The Point Estimator of R 2S 85
  6.5 Sampling Properties Of R 2S 86
  6.6 Simulation And Bootstrap Studies Of R 2S 87
  6.7 Simulation Results 94
  6.8 Appendices
  6.9 Selected Bibliography