Examples of estimation problems
- 16 Downloads
Two examples of estimation problems are given. In the first example,X 1,X 2 andX 3 are independent random variables withX 1 having a Poisson distribution with mean θ1,X 2 being N(θ1,1) and X3/θ3 having a chi-square distribution withn degrees of freedom. Based on these three observations, an estimator of (θ1,θ2,θ3), strictly better than the standard one (X 1,X 2,X 3/(n+2)), is constructed by solving an inequality. In the second example, we establish a counter-example to the assertion that the lack of a nontrivial solution to a difference inequality (corresponding to the problem of improving upon an estimator δ through an identity of Hudson's (1974,Technical Report No. 58, Stanford University), and Stein's type (1973,Proc. Prague Symp. Asymptotic Statist., 345–381)) implies the admissibility of δ. Implications of these two examples are discussed.
AMS 1970 subject classificationPrimary 62C15, 62F10 Secondary 62H99, 39A10
Key words and phrasesAdmissibility loss function differential inequality difference inequality Poisson distribution normal distribution and chi-square distribution
Unable to display preview. Download preview PDF.
- Brown, L. D. (1979). Counterexample—An inadmissible estimator which is generalized Bayes for a prior with “light” tails,J. Multivariate Anal.,6, 256–264.Google Scholar
- Hodges, J. L., Jr. and Lehman, E. L. (1951). Some applications of the Cramer-Rao inequality,Proc. Second Berkeley Symp. Math. Statist. Prob., University of California Press, 13–22.Google Scholar
- Hudson, M. (1974). Empirical Bayes estimation,Technical Report No. 58, Department of Statistics, Stanford University.Google Scholar
- James, W. and Stein, C. (1960). Estimation with quadratic loss,Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, Univ. of California Press, 361–379.Google Scholar
- Peng, J. C. M. (1975). Simultaneous estimation of the parameters of independent Poisson distributions,Technical Report No. 78, Department of Statistics, Stanford University.Google Scholar
- Stein, C. (1973). Estimation of the mean of a multivariate distribution,Proc. Prague Symp. Asymptotic Statist., 345–381.Google Scholar