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

Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample of size $n$ from a distribution that has pdf $f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0<x<\infty, 0<\theta<\infty$, zero elsewhere. Show that the ratio $R=n Y_{1} / \sum_{1}^{n} Y_{i}$ and its denominator (a complete sufficient statistic for $\theta$ ) are independent. Use the result of the preceding exercise to determine $E\left(R^{k}\right), k=1,2,3, \ldots$

Key Concepts

Transformation of Random Variables and Moment Calculation

The problem involves finding the moments of the ratio of the minimum observation to the sum of observations. This requires transforming the joint distribution of the involved statistics and recognizing the resulting distribution form, often a Beta distribution. Calculating the k-th moment E(R^k) then typically involves the use of beta integrals or properties of the Beta function, which are common techniques in the computation of moments of ratio distributions.

Basu’s Theorem

Basu’s theorem states that any complete sufficient statistic is independent of any ancillary statistic—one whose distribution does not depend on the parameter. In the problem, the ratio involving the order statistic and the sum is an ancillary statistic, implying by Basu’s theorem that this ratio is independent of the complete sufficient statistic given by the sum.

Complete Sufficient Statistic

A complete sufficient statistic is a function of the sample that encapsulates all the information about a parameter and has the property that no non?trivial function of it has zero expectation for all parameter values. In the problem, the sum of the sample observations serves as a complete sufficient statistic for the scale parameter, meaning all inference about the parameter can be based on this sum.

Order Statistics

Order statistics refer to the sorted values from a random sample. In this context, the minimum value (the first order statistic) from an independent sample of exponential random variables has a known distribution that is itself exponential with a different rate. Understanding the behavior of order statistics is essential for studying functions of the sample, such as the ratio involving the minimum and the sum.

Exponential Distribution

The exponential distribution is a continuous probability distribution characterized by its constant hazard rate or memoryless property. It is widely used to model time until an event occurs, such as the failure of a component. Its density function, which is parameterized by a scale parameter, plays a key role in deriving many properties of related statistics, including order statistics and sums of independent exponential random variables.

Gamma Distribution

The sum of independent exponential random variables follows a Gamma distribution. The Gamma distribution is characterized by a shape and a scale parameter, and it provides a flexible model for the sum of waiting times. Its properties, especially moments and the form of its density function, are key in understanding the distribution of complete sufficient statistics derived from exponential samples.

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