# Examples of estimation problems

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## Summary

Two examples of estimation problems are given. In the first example,*X* _{1},*X* _{2} and*X* _{3} are independent random variables with*X* _{1} having a Poisson distribution with mean θ_{1},*X* _{2} being N(θ_{1},1) and X_{3}/θ_{3} having a chi-square distribution with*n* degrees of freedom. Based on these three observations, an estimator of (θ_{1},θ_{2},θ_{3}), 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 classification

Primary 62C15, 62F10 Secondary 62H99, 39A10### Key words and phrases

Admissibility loss function differential inequality difference inequality Poisson distribution normal distribution and chi-square distribution## Preview

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### References

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