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

Let $Y_{1}<Y_{2}<Y_{3}$ be the order statistics of a random sample of size 3 from the uniform distribution having pdf $f(x ; \theta)=1 / \theta, 0<x<\theta, 0<\theta<\infty$, zero elsewhere. Show that $4 Y_{1}, 2 Y_{2}$, and $\frac{4}{3} Y_{3}$ are all unbiased estimators of $\theta .$ Find the variance of each of these unbiased estimators.

Key Concepts

Order Statistics

Order statistics are the sorted values obtained from a random sample. They are useful in estimating parameters because their distributions can often be derived in closed?form. Understanding the behavior of order statistics, such as their expectations and variances, allows for the construction of improved estimators by applying functions to the sample extremes or other ranks.

Uniform Distribution

The uniform distribution is a continuous probability distribution where all outcomes in the interval are equally likely. In problems involving estimation of the parameter defining the upper limit of the distribution, properties such as the constant density and the relationship between the order statistics and the upper bound help develop estimators and determine their biases and variances.

Unbiased Estimation

Unbiased estimation refers to the property of an estimator whereby its expected value equals the parameter being estimated. This is a desirable attribute in estimation theory because, on average, the estimator neither overestimates nor underestimates the true parameter value. Establishing the unbiasedness of an estimator typically involves demonstrating that the expectation, computed via the relevant sampling distribution, equals the parameter.

Variance of Estimators

The variance of an estimator measures how much the estimates vary from sample to sample. Even when an estimator is unbiased, high variance may lead to imprecise estimates. Analyzing the variance is crucial for assessing the efficiency and reliability of an estimator, and comparisons are often made among several unbiased estimators to choose the one with the smallest variance.

Transformation of Random Variables

Transformation of random variables is a technique used to derive the distribution of a function of a random variable. When order statistics are scaled to obtain unbiased estimators, it is necessary to apply transformation rules to compute the new expectation and variance. This method is central to adjusting biased measures or standardizing estimators to achieve desirable statistical properties.

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