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| ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETER OF ONE-DIMENSIONAL DISCRETE EXPONENTIAL FAMILIES |
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Citation： |
Chen Xiru.ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION FOR PARAMETER OF ONE-DIMENSIONAL DISCRETE EXPONENTIAL FAMILIES[J].Chinese Annals of Mathematics B,1983,4(1):41~50 |
| Page view： 755
Net amount： 797 |
Authors： |
Chen Xiru; |
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| Abstract： |
Consider the discrete exponential family written in the form $\[{P_\theta }(X = x) = h(x)\beta (\theta ){\theta ^x},x = 0,1,2, \cdots ,\]$, where $\[h(x) > 0,x = 0,1,2, \cdots .\]$. The prior distribuiion of $\[\theta \]$ belongs to the family $\[ = \{ G:\int_0^\infty {{\theta ^2}dG} < \infty \} \]$. Denote by $\[{\delta _n}(x) = {\delta _n}({x_1}, \cdots ,{x_n};x) = \frac{{h(x)}}{{h(x + 1)}}\frac{{{u_n}(x + 1)}}{{{u_n}(x)}}\]$ the
Robbins' EB Estimate of $\[\theta \]$ under the square loss $\[{(\theta - d)^2}\]$ where $\[{u_n}(i)\]$ is the number of elements equaling to i in $\[{x_1}, \cdots ,{x_n}\]$. Under a quite general assumption imposed upon h, it is shown that $\[{\delta _n}\]$ is an asymptotically optimal EB estimate of $\[\theta \]$ relative to the whole family F. Further, the condition imposed on h mentioned above can be dispensed with by slightly modifying the definition of $\[{\delta _n}\]$. Also the case that h assumes the value zero is discussed. |
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