MHB How to Find the Conditional Density for an Improved Estimator?

AI Thread Summary
The discussion focuses on finding the conditional density needed to derive an improved estimator using the Rao-Blackwell theorem. The initial density function presented, f(x,θ) = exp(-√x)/θ, was questioned for its validity as a family of densities. The user initially claimed that √X1/2 is an unbiased estimator and that T(X) = √X1 + ... + √Xn is a sufficient statistic for θ. After some confusion about the density function, the user ultimately found a way to address the question. The conversation highlights the importance of correctly identifying density functions in statistical estimation.
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Consider a family of densitites $f(x,\theta)=\frac{exp(-{\sqrt{x}})}{{\theta}}$. Let $X_{1}$ be a single observation from this family. I have shown that ${\sqrt{X_{1}}}/2$ is an unbiased estimator. Now consider $n$ observations $X_{1},..X_{n}$. I have shown that $T(X)={\sqrt{X_{1}}}+..+{\sqrt{X_{n}}}$ is a sufficient statistic for $\theta$. Now use the Rao Blackwell theorem to find an improved estimator.

The improved estimator is $E({\sqrt{x}}/2|T)$. For this I need the conditional density. How do I find this?

Thanks
 
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Are you sure $$f(x,\theta)$$ is a family of densities?
 
stainburg said:
Are you sure $$f(x,\theta)$$ is a family of densities?

I got the density completely wrong. Anyway I have just found a way to do the question.
 

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