Examples of estimation problems
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Two examples of estimation problems are given. In the first example,X1,X2 andX3 are independent random variables withX1 having a Poisson distribution with mean θ1,X2 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 (X1,X2,X3/(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
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