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

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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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