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Unbiased estimators and their applications. Vol. I: Univariate case. Transl. from the Russian and updated. (English) Zbl 0832.62019

Mathematics and its Applications (Dordrecht). 263. Dordrecht: Kluwer Academic Publishers. xii, 521 p. $225.00; Dfl. 365.00; £145.50 (1993).
[For the review of the original Russian edition see Zbl 0665.62025).]
The book under review consists of three chapters and an appendix.
Chapter 1 (Elements of the theory of point statistical estimation) describes the basic concepts of point statistical estimation theory with a focus on the problems of unbiased estimation. The topics discussed here are: the problem of point statistical estimation; risk of a statistical estimator; consistency; unbiasedness; order statistics; empirical distribution function; sufficient statistics; the information inequality; the Rao-Blackwell-Kolmogorov theorem; maximum likelihood estimation; the method of moments; equivariant estimators; techniques for improving estimators; unbiasedness in the mean and other concepts of unbiasedness; some problems of unbiased estimation for the normal distribution.
Chapter 2 (Techniques for constructing unbiased estimators) presents widely used methods for deriving unbiased estimators. The topics of this chapter are: introducing a factor deleting the bias; the Rao-Blackwell- Kolmogorov method; estimators based on the unbiased probability density function estimator; techniques based on the equation of unbiasedness; unbiased estimators of parameters of discrete distributions; the Abbey and David techniques [see J. L. Abbey and H. T. David, Ann. Math. Stat. 41, 1217-1226 (1970; Zbl 0225.62034)].
Chapter 3 (Applications of unbiased estimation theory) provides illustrations of employment of unbiased estimators in practice. The topics considered are: unbiased estimators of parameters of truncated exponential and normal distributions with applications; some statistical models and estimators useful in integrated circuit quality control; unbiased estimators in the classical occupancy problem with applications; some unbiased estimators of the reliability of systems; unbiasedness, sufficiency and a chi-square goodness-of-fit test for discrete exponential distributions of rank one.
The Appendix (Tables of unbiased estimators) is a collection of 34 tables of unbiased estimators of parameters and their functions for many probability distributions. The list of references consists of 366 items.

MSC:

62F10 Point estimation
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics