How to Find the Conditional Density for an Improved Estimator?

In summary, the conversation discusses a family of densities with the function f(x,\theta)=\frac{exp(-{\sqrt{x}})}{{\theta}} and its application to finding an unbiased estimator. It is shown that ${\sqrt{X_{1}}}/2$ is an unbiased estimator and that $T(X)={\sqrt{X_{1}}}+..+{\sqrt{X_{n}}}$ is a sufficient statistic for $\theta$. The Rao Blackwell theorem is then mentioned as a means to find an improved estimator, which is found to be $E({\sqrt{x}}/2|T)$. The question of finding the conditional density is raised.
  • #1
Fermat1
187
0
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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  • #2
Are you sure \(\displaystyle f(x,\theta)\) is a family of densities?
 
  • #3
stainburg said:
Are you sure \(\displaystyle f(x,\theta)\) is a family of densities?

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

FAQ: How to Find the Conditional Density for an Improved Estimator?

What is conditional expectation?

Conditional expectation is a statistical concept that represents the expected value of a random variable given that another related random variable has taken on a specific value or falls within a certain range of values. It is a measure of the average outcome of a random variable under a specific condition.

How is conditional expectation calculated?

The conditional expectation is calculated by taking the expected value of the random variable of interest, given the specific condition, and then multiplying it by the conditional probability of that condition. In mathematical notation, it is written as E(X|Y) = E(X|Y=y) * P(Y=y).

What is the difference between conditional expectation and unconditional expectation?

Conditional expectation takes into account a specific condition or event, while unconditional expectation considers all possible outcomes of a random variable. In other words, conditional expectation factors in the relationship between two random variables, while unconditional expectation does not.

What is the importance of conditional expectation in statistics?

Conditional expectation is important in statistics because it allows us to make more accurate predictions and decisions by taking into account relevant information or conditions. It also helps in understanding the relationship between two random variables and how they affect each other.

In what situations is conditional expectation commonly used?

Conditional expectation is commonly used in many fields of study, including economics, finance, and machine learning. It is often used in predicting future outcomes, making decisions based on uncertain information, and analyzing complex data sets.

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