PARAMETER ESTIMATION OF SOME POPULATIONS AFTER SELECTION

Abstract

Let Ilk be k populations with associated probability density function of the ith population being given by f (xf0i) where Oi's (i = 1, 2, . , k) are unknown parameters. Quite often, one is interested in selecting the best population or a subset of populations containing the best. The population is termed the best according to some characteristic such as the largest mean, the smallest variance etc. For example, Out of the several options available, an individual would like to buy that vehicle for himself, which can give him the maximum fuel efficiency. A farmer would like to sow those seeds in his farm, which he expects can provide him the maximum yield. A TV manufacturer would like to use those picture tubes for his brand of TV, which have maximum average life. A company owner would like to install those machines in his company which have maximum reliability for production. There are many such practical problems where one is interested in selecting the best from the several options available. These type of problems have been studied extensively by various statisticians over the past sixty years under the term, "Ranking and selection procedures". Once the selection is done, quite naturally a further problem of interest is to estimate the unknown(s) (Parameter(s)) of the selected population(s). Moving on the same examples as stated above, it is quite obvious that the vehicle buyer would like to have an estimate of the expected value for the fuel efficiency of the vehicle chosen. The farmer would naturally be interested to have an estimate of the yield in advance that he would be receiving by sowing the selected seeds. Similarly, the TV manufacturer will not only select the best brand of picture tubes, but would also like to have an estimate of the expected life of the selected brand and the company owner would like to have an estimate of the production he would get after installing the selected machine. These type of problems are termed as "Estimation. after selection problem" . This thesis, comprising of seven chapters, is concerned with the study of the problem of estimating parameters of the selected population for certain distributions. A chapter wise summary of the thesis is given below: Chapter 1 contains a brief review of the literature available on the problem of estimation after selection and a gist of the work presented in the thesis. Some basic definitions, results regarding statistical inference and decision theory and some techniques that are useful in the sequel are presented in Chapter 2. Chapter 3 deals with estimating the variance of the selected Rayleigh population. Suppose we have k populations H1,112, which are Rayleigh distributed with the probability density function of the ith population (Hi) given by y2 X f(x,bi) = - x > 0, i 1, , k. —14 e where I), is the scale parameter of the ith population. Suppose Xii, . . . , Xin be a random sample of size n available from the ith population, where i = 1, 2, ... , k and let us represent by Y. Following the natural selection rule, the population corresponding to the smallest Yi is selected. We consider estimation of the variance Vj of the selected population for the case of two populations, that is, for k = 2. In Section 3.2, the natural estimator Y(2) (= min{Yi, Y2}) is shown to be negatively biased. In section 3.3, we consider two expressions viz. cl)(y) and 0(y) for obtaining the uniformly minimum variance unbiased estimator (UMVUE) of V. These expressions are respectively the linear and non-linear functions of Y(1)(= max{Yi, Y2}) and Y(2). It is discussed that the expression 0(y) provides the UMVUE for n --= 1 and for n > 1, the UMVUE is obtained using the expression 0(y). In section 3.4, we derive the risk of the estimator with respect to the scale invariant loss. Admissible estimators are obtained in the class 7', = cY(2; by the application of the Brewster-Zidek [12] technique ii in section 3.5. In section 3.6, we find the risk of the estimator b with respect to the scale invariant loss. Further in section 3.7, taking different possible prior distributions, we obtain Bayes estimators for V. In section 3.8, we have discussed about the minimaxity results. Finally in section 3.9, we compared all the estimators, that were obtained in the preceding sections, on the basis of their bias and mean squared error(MSE) risk performance using Monte-Carlo simulation. In Chapter 4, we study the problem of estimation after selection for estimating the scale parameter of the selected Pareto population. Suppose 111 , 112, . , Ilk be k Pareto populations with ni having the density function = Si 4i x > ai, A > o, k xi3i+ 1 9 where a, and are respectively the scale and shape parameters of the ith population. Consider random samples (Xil, , Xin); i = I, , k , of size n from each of the k populations. Let the minimum observation of the ith sample be denoted by Xi. The population corresponding to the largest X2 is selected. In Section 4.2, for k = 2, we consider the estimation of the scale parameter. aj of the selected population for the case when the shape parameters /3,'s are known and equal. The estimators discussed therein are compared numerically, using Monte-Carlo simulation for two populations, on the basis of their bias and mean squared error(MSE) risk values. Section 4.3, deals with the case when the shape parameters Oi's are considered unknown. Some estimators analogous to the component problem are proposed and numerically compared. The comparison is made on the basis of the risk values obtained as a result of simulation on 5000 samples. In Chapter 5, we take up the problem of estimating the quantile of the selected nor-mal population. This problem has been studied by Sharma and Vellaisamy [95] for the normal populations having different means and equal variances. Further, Kumar and Kar [57] have studied it for populations having different means and different variances. In this chapter, we consider the case when mean is same but variances are different. iii Suppose independent random samples (Xii, Xin) ; i = 1, . , k, are available from k normal populations with same(and unknown) mean p, but different(and un-known) variances 01. For the ith sample, let us denote the mean and the sum of the squared deviations from the mean by Xi and Si respectively. That is, 1 — E'? Xi • n 1 and S = E;!'i(Xii 2 We select the population corresponding to the smallest Si , i = 1, . , k . That is, the population Hi is selected if Si = min{ S1, , SO-. The quantile of the selected population is given by 0 = o-j, where J = i if Si is the smallest among Si, . • • , Sk; and 77(74 0) is any constant. In Section 5.2, by applying the Brewster-Zidek technique [12], we derive the admissi-ble class of estimators in the class Ta given by Ta = ft+ 77 a S3 , where ft = X1 s2-vsi s2-+ x2 s1 The estimators {Ta : a E (0, c/,,,]} are found admissible among all Ta's, where do ( 2 n+j-1)! 7 `-'j 2=0 3. ("1-F.i)! E 2 3 j! In Section 5.3, we consider a more general class of estimators OA, Sj) = S3 0(Zj) 1 where Zj = μ sj 2 . A general inadmissibilty result for this class of estimators is derived and an improvement of 5 is obtained. In Section 5.4, the various estimators studied in section 5.2 and 5.3 are compared numerically with the help of Monte-Carlo simulation for k = 2. The comparison is done with respect to the bias and mean squared error risk performance. In Chapter 6, we extend the problem of estimation after selection for estimating the regression coefficient of the selected bivariate population. In Section 6.2, we formulate this problem for k bivariate distributions assuming that the linear regression model is valid for each of them_ In Section 6.3, for two bivariate populations (i.e. for k = 2), we demonstrate that the problem of estimation of the regression coefficient of the selected population can iv be reduced to the problem of estimation of the mean of the selected population as was studied by Dahiya [22]. In Section 6.4, an improvement of A, as given in [22] is obtained by the application of the Brewster-Zidek [12] technique. Finally in Section 6.5, we apply the concept to the portfolio theory of corporate finance and it is our expectation that the development of theoretical results in this direction will enhance the applicability of the results on estimation after selection. Chapter 7 concludes the thesis with analysis of the salient points of the work pre-sented in earlier chapters.