Unbiased Estimators and Their Applications, 1993
Volume 1: Univariate Case
Mathematics and Its Applications Series, Vol. 263

Statistical inferential methods are widely used in the study of various physical, biological, social, and other phenomena. Parametric estimation is one such method. Although there are many books which consider problems of statistical point estimation, this volume is the first to be devoted solely to the problem of unbiased estimation. It contains three chapters dealing, respectively, with the theory of point statistical estimation, techniques for constructing unbiased estimators, and applications of unbiased estimation theory. These chapters are followed by a comprehensive appendix which classifies and lists, in the form of tables, all known results relating to unbiased estimators of parameters for univariate distributions. About one thousand minimum variance unbiased estimators are listed. The volume also contains numerous examples and exercises.
This volume will serve as a handbook on point unbiased estimation for researchers whose work involves statistics. It can also be recommended as a supplementary text for graduate students.

I. Elements of the Theory of Point Statistical Estimation.- 1. The Problem of Point Statistical Estimation.- 2. Risk of a Statistical Estimator.- 3. Consistency.- 4. Unbiasedness.- 5. Examples.- 6. Order Statistics.- 7. Empirical Distribution Function.- 8. Sufficient Statistics.- 9. The Information Inequality.- 10. The Rao—Blackwell—Kolmogorov Theorem.- 11. Maximum Likelihood Estimation.- 12. The Method of Moments.- 13. Equivariant Estimators.- 14. Techniques for Improving Estimators.- 15. Unbiasedness in the Mean, and Other Concepts of Unbiasedness.- 16. Some Problems of Unbiased Estimation for the Normal Distribution.- I. Elements of the Theory of Point Statistical Estimation.- 1. Introducing a Factor Deleting the Bias.- 2. The Rao—Blackwell—Kolmogorov Method.- 3. Estimators Based on the Unbiased Probability Density Function Estimator.- 4. Techniques Based on the Equation of Unbiasedness.- 5. Unbiased Estimators of Parameters of Discrete Distributions.- 6. The Abbey and David Techniques.- III. Applications of Unbiased Estimation Theory.- 1. Unbiased Estimators of Parameters of Truncated Exponential and Normal Distributions, with Applications.- 2. Some Statistical Models and Estimators Useful in Integrated Circuit Quality Control.- 3. Unbiased Estimators in the Classical Occupancy Problem with Applications.- 4. Some Unbiased Estimators of the Reliability of Systems.- 5. Unbiasedness, Sufficiency, and a Chi-Square Goodness-of-Fit Test for Discrete Exponential Distributions of Rank One.- Tables of Unbiased Estimators.- A1. Unbiased Estimators of Functions of Parameters of the Normal Distribution.- A2. Unbiased Estimators of Functions of the Parameter ? of the Uniform Distribution.- A3. Unbiased Estimators of the Functions of the Parameter ? of the DiscreteUniform Distribution (Sampling with Replacement).- A4. Unbiased Estimators of Functions of the Parameter ? of the Discrete Uniform Distribution (Sampling without Replacement).- A5. Unbiased Estimators of Functions of the Parameter ? of the Gamma-Distribution.- A6. Unbiased Estimators of Functions of Parameters ? and ? of the Inverse Gaussian Distribution.- A7. Unbiased Estimators of Functions of Parameters of the Weibull Distribution.- A8. Unbiased Estimators of Functions of Parameters of the Rayleigh Distribution.- A9. Unbiased Estimators of Functions of Parameters of some Independent Normal Distributions.- A10. Unbiased Estimators of Functions of a Paramater of the Seminormal Distribution.- A11. Unbiased Estimators of Functions of Parameters of the Left-Truncated Exponential Distribution (? and ? are Unknown).- A12. Unbiased Estimators of Functions of Parameters of the Left-Truncated Exponential Distribution (? is Known).- A13. Unbiased Estimators of Functions of Parameters of the Paretian Law.- A14. Unbiased Estimators of Functions of Parameters of the Left-Truncated Exponential Distribution of a General Type.- A15. Unbiased Estimators of Functions of a Parameter of the Right-Truncated Exponential Distributions.- A16. Unbiased Estimators of Functions of Parameters of the Uniform Distribution with Two Unknown Parameters.- A17. Unbiased Estimators for Functions of a Parameter ? of the Extreme Value Distribution.- A18. Unbiased Estimators of Functions of Parameters of the Burr Distribution.- A19. Unbiased Estimators of Functions of Parameters of the Logarithmic-Power Distribution.- A20. Unbiased Estimators for Functions of Parameters ? and r of the Left-Truncated Power Series Distribution.- A21. Unbiased Estimators of Functions of Parameters ? and r of theLeft-Truncated Generalized Logarithmic Series Distribution.- A22. Unbiased Estimators of Functions of Parameters ? and r of the Left-Truncated Logarithmic Series Distribution.- A23. Unbiased Estimators of Functions of Parameters of the Truncated Geometrical Distribution.- A24. Unbiased Estimators of Functions of a Parameter ? of the Binomial Distribution.- A25. Unbiased Estimators of Functions of a Parameter ? of the Negative Binomial Distribution.- A26. Unbiased Estimators of Functions of Parameters of the Left-Truncated Generalized Negative Binomial Distribution.- A27. Unbiased Estimators of Functions of Parameters of the Left-Truncated Generalized Poisson and the Borel-Tanner Distribution.- A28. Unbiased Estimators of Functions of the Parameter ? of the Poisson Distribution.- A29. Unbiased Estimators of Functions of Parameters of the Left-Truncated Poisson Distribution.- A30. Unbiased Estimators of Functions of the Parameter ? of the Hypergeometric Distribution.- A31. Unbiased Estimators of Functions of the Parameter ? of the Negative Hypergeometric Distribution.- A32. Unbiased Estimators of Functions of Parameters of the Lognormal Distribution.- A33. UMVUE’s of Parameters for a Subclass of Stable Distributions.- A34. UMVUE’s of Parameters for the Stirling Distribution of the Second Kind.- References.