Let Y1<Y2<Y3<Y4<Y5 be the order statistics of a random...

Let $Y_{1}<Y_{2}<Y_{3}<Y_{4}<Y_{5}$ be the order statistics of a random sample of size 5 from the uniform distribution having pdf $f(x ; \theta)=1 / \theta, 0<x<\theta, 0<\theta<\infty$ zero elsewhere. Show that $2 Y_{3}$ is an unbiased estimator of $\theta .$ Determine the joint pdf of $Y_{3}$ and the sufficient statistic $Y_{5}$ for $\theta$. Find the conditional expectation $E\left(2 Y_{3} \mid y_{5}\right)=\varphi\left(y_{5}\right)$. Compare the variances of $2 Y_{3}$ and $\varphi\left(Y_{5}\right)$

Key Concepts

Variance Comparison

The comparison of variances between different estimators is a way to assess their efficiency. Although both 2Y? and the conditional estimator ?(Y?) are unbiased for ?, they may have different variances. In estimation theory, under certain regularity conditions, an estimator that is a function of a sufficient statistic will have a lower variance. Comparing these variances provides insight into the quality of the estimators and illustrates how conditioning on sufficient statistics can yield estimators that are not only unbiased but also more precise.

Conditional Expectation

Conditional expectation involves finding the expected value of one statistic given another statistic. In this problem, the conditional expectation E(2Y? | Y? = y?) is computed to form an improved estimator that uses the information in the sufficient statistic Y?. Conditioning can reduce variance by taking into account known information about the parameter and is a common technique in the theory of estimator improvement.

Joint Probability Density Function

The joint probability density function (pdf) of two order statistics, such as Y? and Y?, is derived from the joint distribution of order statistics in a sample. This requires combining the individual distribution functions with the combinatorial counts of ways to index the order statistics. Deriving the joint pdf is helpful for understanding the dependence between the statistics and is a preliminary step for further analysis such as conditional expectation.

Sufficient Statistic

A sufficient statistic is a function of the data that contains all the information needed about the parameter of interest. For samples from a uniform distribution with unknown upper bound ?, the maximum observation (Y? in this case) is a sufficient statistic. This means that, given the maximum, no additional information about ? can be gained from the rest of the sample. Sufficiency is a key concept because it can simplify both estimation and inference by reducing the data without losing information about the parameter.

Order Statistics

Order statistics are the sorted values from a random sample. In statistical inference, the k-th order statistic (for example, the median or a particular percentile) has its own probability distribution. For a sample from a continuous distribution—like the Uniform(0, ?) distribution—the order statistics are commonly used to construct estimators for the population parameters. When the original sample has a known distribution, like the uniform distribution, the pdf of a given order statistic can often be derived in closed?form using combinatorial arguments.

Unbiased Estimator

An unbiased estimator is a statistic used to estimate a parameter such that its expected value equals the true value of the parameter. In this context, showing that 2Y? is unbiased for ? means verifying that the expectation of 2 times the third order statistic (Y?) equals ?. This involves calculating the expected value of Y? under its distribution (which for a uniform sample has a beta distributed form) and checking that multiplying by 2 gives the parameter being estimated.

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