## Doctoral Theses

### Slippage and Change Point Problem with Directional Data.

5-28-2001

5-28-2002

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Applied Statistics Unit (ASU-Kolkata)

#### Supervisor

Sengupta, Ashis (ASU-Kolkata; ISI)

#### Abstract (Summary of the Work)

In this thesis we develop statistical methods for dealing with two problems namely (1) THE SLIPPAGE PROBLEM and (2) THE CHANGE POINT PROB- LEM in the set-up of directional data. In this chapter, we provide an intro- duction to these problems and discuss the importance of the present work. The slippage problem is basically a problem to detect whether any unspecified observation in a given random sample comes from a distribution different from that for all the other remaining observations. This can also be viewed as a problem of outlier detection or that of spuriosity. This problem assumes great importance in many practical situations where directional data are encountered, e.g. in applications to meteorological data, wind directions, movements of icebergs, propagation of cracks, biological and periodic phenomena, quality assurance and productivity measures, etc. However, little seems to be known regarding its theoretical foundations in the context of directional data under a parametric model, say e.g. aslip in terms of the mean direction of the circular normal distribution (see however, Col- let, 1980, Bagchi and Guttman, 1988, 1990 and Upton, 1993). The circular normal distribution with mean direction u and concentration parameter K, denoted by CN(4, K), is one of the most popular distributions for modeling circular data. Some useful facts regarding this distribution can be found in Appendix-I. A survey of the work done on this problem can be found in Barnett and Lewis (1994).In CHAPTER 2, we consider the problem of testing Ho : 0,, j = 1,., k are identically distributed as CN(Ho, k) against H; : O,...,e,-1,0+1...,O are identically distributed as CN(Ha, K) and O, is distributed as CN(P1, K), 1sis k, > Po, P1. Po and K are all known, using a decision theo- retic route. We derive the Bayes test with respect to the prior distribution invariant with respect to permutations of H1,..., H. We also study the per- formance of the Bayes test when the null hypothesis is true and also when one of the alternative hypothesis is true.In CHAPTER 3, we consider the problem of testing Ho against H; : There exists i, i unknown, such that e, is distributed as CN(P1,K) and 01, 02,...,e-1,e41,...,e, are distributed as CN(4o, K), O,...,e, are all independent. We derive the LRT and study its performance using simu- lations. We illustrate the use of this test by analysing two well known data sets. We introduce the notion of a LOCALLY MoST POWERFUL TYPE TEST (LMPTT) and derive it for this problem. We also indicate a Multiple Testing approach for the outlier problem, which can be easily generalized to situa- tions where the underlying distribution is not circular normal.In CHAPTER 4, we provide a simulation based comparison of the various procedures for outlier detection, namely, the L-statistic(Collet, 1980), M- statistic(Mardia, 1975), the LRT, the LMP and the Bayes-rule with different values of p. It is found that the LMPTT performis best when outliers of small magnitude are sought to be detected, LRT performs best when outliers of moderate magnitude are sought to be detected and the Bayes-Test performs best when outliers of large magnitude are sought to be detected.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842963

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