[Nonparametric change of trend Detection]

by

Acknowledgement

I would take this opportunity to thank my research supervisor, family and friends for their support and guidance without which this research would not have been possible.

DECLARATION

I, [type your full first names and surname here], declare that the contents of this dissertation/thesis represent my own unaided work, and that the dissertation/thesis has not previously been submitted for academic examination towards any qualification. Furthermore, it represents my own opinions and not necessarily those of the University.

Signed __________________ Date _________________

Abstract

The dissertation considers construction of confidence intervals for a cumulative distribution function F(z) and its inverse, F-1(u), at some fixed points z and u on the basis of an i.i.d. sample X = {xi}i=1, where the sample size is relatively small. The sample is modeled as having the flexible Generalized Gamma distribution with all three parameters being unknown. This approach can be viewed as an alternative to nonparametric techniques which do not specify distribution of X and lead to less efficient procedures. The confidence intervals are constructed by objective Bayesian methods and use the Jeffreys non-informative prior. Performance of the resulting confidence intervals is studied via Monte Carlo simulations and compared to the performance of nonparametric confidence intervals based on binomial proportion. In addition, techniques for change point detection are analyzed and further evaluated via Monte Carlo simulations. The effect of a change point on the interval estimators is studied both analytically and via Monte Carlo simulations.

Table of Contents

CHAPTER 2: INTRODUCTION………………………………………………………..………………………..7

Problem Statement12

Research Questions13

General Hypothesis14

Hypothesis 114

Hypothesis 214

Hypothesis 315

Hypothesis 415

Research Purpose15

Research Objective16

Relevance of this Study16

CHAPTER 2: LITERATURE REVIEW18

Technical Approach18

The Generalized Gamma Distribution19

Bayesian Analysis23

Bayes Estimation23

Bayes Hypothesis Testing24

Bayesian Credible Sets26

The Jeffreys Prior27

Change Point Analysis29

CHAPTER 3: TECHNICAL RESULTS31

Derivation of the Jeffreys Prior32

The Posterior Distribution of a, b, and ?34

The Posterior Distribution of F(z) and F-1(u)36

Confidence Intervals38

Change Point Detection and Location39

CHAPTER 4: SIMULATIONS41

Random Sample Generation42

Confidence Intervals without Change Points42

Robustness of Technique46

Change Point Detection and Location Techniques50

Confidence Intervals with Change Points55

Change-Point Analysis in Statistical Process Control57

Previous Work on Change-Point Analysis in SPC60

Change-Point Estimator for Variation63

Conover.s Squared Ranks for Variances63

Change-Point Estimator for Variation63

Fit of the Prediction Models65

Summary68

Further Research71

CHAPTER 5: CONCLUSIONS74

Chapter 1: Introduction

The understanding of all interactions and non linear behaviors inside a system is nearly impossible, and the presence of trends makes the situation even worst. Only a non deterministic approach can help decision makers to overcome such situation. Under the label of Statistical Process Control (SPC), many procedures and tools have been developed to assist managers to handle the resulting uncertainty, and a known approach has been the use of control charts. They are useful tools extensively applied to establish if a process is in statistical control, and if not, assist in the detection of special causes of variation, which can then be assessed to bring the process back to control. Knowing the moment of a change helps in the search for assignable causes of the process variation. However, when sustained changes are involved in time series, control charts are not capable of identifying the initial moment of the change. The study of the problem to estimate ...