Question

Suppose $X_1,X_2,\dots ,X_n$ is a random sample from a distribution with pdf \(f(x ; \theta)=(1 / 2) \theta^{3} x^{2} e^{-\theta x}, 0

Short answer

The short answer cannot be given until exact mathematical operations are performed on the given functions.

Step-by-step solution

Calculation of MLE ($\hat{\theta }$)

The likelihood function is given by $L(\theta )={\mathrm{\Pi }}_{i=1}^{n}f(x_i;\theta )$, where $f(x;\theta )=(1/2){\theta }^3{x}^2{e}^{-\theta x}$ is the pdf. We derive and solve the log-likelihood equation $\frac{d}{d\theta }log(L(\theta ))=0$ to find the mle $\hat{\theta }$.

Checking for unbiasedness

To check if $\hat{\theta }$ is unbiased, we calculate the expected value $E(\hat{\theta })$ and verify if it is equal to the true population parameter $\theta$.

Checking for sufficiency and completeness

By definition, $Y=\sum _{i=1}^n{X}_i$ is a sufficient statistic if the conditional probability distribution of the data, given the statistic, does not depend on the unknown parameters. For completeness, $Y$ is complete if any function $g(Y)$ with zero expectation implies the function is zero almost everywhere.

Finding the MVUE

The MVUE is an estimator that is unbiased and has minimum variance. We may need the Lehmann–Scheffe theorem which states that if a statistic is unbiased, complete, and sufficient, then it is the MVUE.

Proving independence of $X1/Y$ and $Y$

Two random variables $X$ and $Y$ are independent if their joint probability distribution is the product of their marginal probability distributions. In other words, $P(X=x,Y=y)=P(X=x)P(Y=y)$. We need to prove this for $X1/Y$ and $Y$.

Finding the distribution of $X1/Y$

To find the distribution of $X1/Y$, we need to find the pdf of $X1/Y$ by using the formula for the transformation of random variables.

Key concepts

Unbiased Estimator

An unbiased estimator is a statistical term that refers to an estimate that, on average, hits the parameter it is estimating. This means that if we took multiple samples and calculated their estimates, those estimates would average to the true parameter value.
In mathematical terms, an estimator $\hat{\theta }$ is unbiased if $E(\hat{\theta })=\theta$. To determine whether $\hat{\theta }$ is unbiased for $\theta$, we calculate its expected value and check if it equals the parameter.
In our original exercise, we need to verify if the maximum likelihood estimator (MLE), $\hat{\theta }$, of $\theta$ is unbiased. By computing the expected value of $\hat{\theta }$ and comparing it to the parameter $\theta$, we can ascertain its unbiasedness.

Complete Sufficient Statistic

A complete sufficient statistic is a concept used to summarize data with the least amount of information loss. If a statistic is sufficient, it retains all the information needed to estimate a parameter. This sufficiency implies that the conditional distribution of the sample data, given the statistic, is independent of the parameter.
Completeness adds another layer, ensuring that no non-trivial functions of the statistic have an expected value of zero for all parameter values, unless the function itself is almost surely zero.

• Sufficiency means no loss of information.
• Completeness ensures maximal inferential power.

In the exercise, $Y=\sum _{i=1}^n{X}_i$ is shown to be a complete sufficient statistic for $\theta$. This means $Y$ contains all the necessary information to estimate $\theta$ without redundancy.

Minimum Variance Unbiased Estimator

The Minimum Variance Unbiased Estimator (MVUE) is an estimator that not only provides unbiased estimates of a parameter but also does so with the smallest possible variance among the class of unbiased estimators. This means that while other unbiased estimators might exist, none will be more precise than the MVUE.
To find the MVUE, we often rely on the Lehmann–Scheffe theorem. This theorem points out that if you have a complete sufficient statistic, any function of this statistic that is unbiased will be the MVUE.
In solving the exercise, once we determine $Y$ is a complete sufficient statistic, finding a function of $Y$ that estimates $\theta$ with unbiasedness helps us identify the MVUE.

Independence of Random Variables

Independence between random variables indicates that the occurrence or value of one variable does not affect the other. Mathematically, two random variables $X$ and $Y$ are independent if their joint probability distribution factors into the product of their marginal distributions: $P(X=x,Y=y)=P(X=x)P(Y=y)$.
In the context of the problem, proving independence between $X_1/Y$ and $Y$ involves verifying this factorization for the joint distribution of these particular transformations.
Independence is crucial because it allows separate consideration of random variables without accounting for possible interactions. In our exercise, demonstrating the independence of $X_1/Y$ and $Y$ shows that the distribution of their ratio does not depend on the summation statistic, simplifying further analysis.