Εκτιμητές τύπου Strawderman για παραμέτρους κλίμακας

Abstract

This PhD thesis deals with the study of the problem of point estimation of a scale parameter from the decision theoretic point of view. Chapter 1 contains an overview of the problem of point estimation of a scale parameter from the decision theoretic point of view as well as a concise presentation of the results of this thesis per chapter. In Chapter 2, some basic notions and definitions are given and, for the sake of completeness, relative known results about improved estimators of scale parameters and ratio of scale parameters are presented. All the results of the remaining chapters are new. In Chapter 3, Strawderman’s (1974) result for estimating the variance of a normal distribution is extended to estimating a general scale parameter in the presence of a nuisance parameter under both quadratic and entropy losses. Application of this general result to the exponential distribution gives new sufficient conditions, i.e., different from those available in the literature, for improving upon the best affine equivariant estimator. Also, new classes of estimators satisfying the above conditions are constructed. In Chapter 4, Strawderman’s (1974) result for estimating the variance of a normal distribution is extended to estimating the reciprocal of a general scale parameter in the presence of a nuisance parameter under both quadratic and entropy losses. The results are analogous to those of Chapter 3. In addition to their own value, the results of this chapter – as well as those of Chapter 3 – are also useful (essentially, necessary) for the construction of Strawderman (1974)type estimators for the ratio of scale parameters of two independent populations, which is studied in the next two chapters. In Chapter 5, new classes of improved, Strawderman (1974)type, estimators for the ratio of the variances, ?2 2=?2 1, of two normal populations are constructed under both quadratic and entropy losses. The method of proof is not the typical one for this kind of problem which requires a twosample extension of respective onesample arguments. In contrast, the methodology of Iliopoulos and Kourouklis (1999) is applied which reduces the twosample problem of estimating ?2 2=?2 1 to two onesample problems, namely, one of estimating ?2 2 and another of estimating 1=?2 1. In Chapter 6, new classes of improved, Strawderman (1974)type, estimators for the ratio of the scale parameters of two exponential distributions are constructed. The results are analogous to those of Chapter 5. Finally, the thesis is completed with the Appendix which provides auxiliary results