## Doctoral Theses

### On Logistic and Some New Discrimination Rules:Charecterizations,Inference and Application.

9-28-1999

9-28-2000

#### 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)

Introduction and Summary Consider the problem of classification of an observation into one of two specified populations. Fisher's classification rale, just as several other rules commonly used in practice, depends only on the ratio of the individual densities fi(x), i = 1,2. This led Cox (1966),/27) to model the "posterior odds" by a simple function. Specifically,Cox's logistic discrimination (LGD) rule is then based on the statistic a + 'ÃŸx. This has the advantage that individual densities f.(x) need not be known and we only need to estimate the parameters a and B.Another advantage, which is claimed , is that the family of densitics satisfying is "quite wide". It is this richness of the family that we intend to explore, since, beyond thec multivariate normal distribution and multivariate discrete distributions following the log-lincar model, no para- metric description of this class is available. Some of the ideas developed here have been introduced by us, in (191),[92).We introduce the following definitions.Definition 1.1.1 Let a distribution possess a density fi(x) with re- spect to mcasure v. fi(x) â‚¬ C, some specified class, will be said to(i) obey the LGD relationship if 3 a density f2(x) with respect to some measure u, satisfying (1.1).(ii) admit the LGD if f:(x) given by (1.1) is also in the class C.Clearly a rule capable of discriminating (in probability) between a pair of populations only when they belong to entirely two different classes is of limited use if the class of such pairs is large, with at least any one component of the pair corresponding to an usually encountered distribu- tion. It is thus cssential and important to characterize such pairs (fi, fa) or cquivalently fi. In this pursuit, a general method of characterizing familics admitting and/ or obeying Cox's [27] LGD for a given density is obtained through functional equations in characteristic type functions. The result is applied to different types and classes of univariate as well as multivariate distributions. This characterization enables us to generalize Cox's LGD to familics even not representable by densitics with respect to any measure v , c.g. to the stable and proper infinitely divisible familics. We observe that almost all distributions f1 usually encbuntered in practice obey the LGD relationship. Of notable exceptions are the Cauchy etc. However the class of distributions for f1 admitting LGD is somewhat restrictive. We recommend that we should not use LGD when any of these distributions is suspected for f1. Rather a preliminary data analysis, should be carried out to identify fi more clearly and then accordingly decide on the use of LGD.Our characterization exposes a functional relationship among the parameters which renders the validity of the usually proposed likelihood based estimation procedures (Anderson, (1982),[5]) theoretically suspect. We present the proper likelihood equations and propose a scheme to evaluate the parameters. The performance of our rule is established to be quite satisfactory through extensive simulations. LGD is illustrated through two well-known real life data sets.We also suggest and explore some new discrimination rules in the con- text of stable distributions, directional data, and neural networks. For the case of stable distributions the performance of the rule is studied by using a new rcal life data set while for that of directional data we use well- known real-life data set available in the literature. We provide a listing of the source code for various programs developed by us which include both programs written in C as well as in SPLUS.

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:28842934

ISILib-TH157