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

Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample from a distribution with the pdf $$ f\left(x ; \theta_{1}, \theta_{2}\right)=\frac{1}{\theta_{2}} \exp \left(-\frac{x-\theta_{1}}{\theta_{2}}\right) $$ $\theta_{1}<x<\infty$, zero elsewhere, where $-\infty \leq \theta_{1} \leq \infty, 0<\theta_{2}<\infty$. Show that the joint complete sufficient statistics $Y_{1}$ and $\bar{X}=\bar{Y}$ for the parameters $\theta_{1}$ and $\theta_{2}$ are independent of $\left(Y_{2}-Y_{1}\right) / \sum_{1}^{n}\left(Y_{i}-Y_{1}\right)$

Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample from a distribution with the pdf
$$
f\left(x ; \theta_{1}, \theta_{2}\right)=\frac{1}{\theta_{2}} \exp \left(-\frac{x-\theta_{1}}{\theta_{2}}\right)
$$
$\theta_{1}<x<\infty$, zero elsewhere, where $-\infty \leq \theta_{1} \leq \infty, 0<\theta_{2}<\infty$. Show that the
joint complete sufficient statistics $Y_{1}$ and $\bar{X}=\bar{Y}$ for the parameters $\theta_{1}$ and $\theta_{2}$ are independent of $\left(Y_{2}-Y_{1}\right) / \sum_{1}^{n}\left(Y_{i}-Y_{1}\right)$

Key Concepts

Exponential Family

The exponential family is a broad class of probability distributions characterized by a specific functional form that allows for systematic identification of sufficient statistics. Distributions in the exponential family often have natural sufficient statistics that are also complete, making them amenable to the application of results like Basu’s Theorem. The given distribution in the problem is an exponential distribution, a member of this family, which underpins the derivation of the complete sufficient statistic and subsequent independence results.

Basu’s Theorem

Basu’s Theorem is a fundamental result in statistical theory stating that any complete sufficient statistic is independent of any ancillary statistic. This theorem is instrumental in the problem because it directly provides the theoretical justification for the independence between the complete sufficient statistic (Y1, ?) and the ancillary statistic (Y2 - Y1) / [?(Yi - Y1)].

Order Statistics

Order statistics are the sorted values from a sample. They are pivotal in statistical inference because the extreme values (such as the minimum or maximum) often convey essential information about the underlying distribution. In the context of this problem, the minimum observation (Y1) is used as part of the complete sufficient statistic, and the differences between order statistics (such as Y2 - Y1) help form ancillary statistics that are used in establishing independence.

Complete Sufficient Statistics

A complete sufficient statistic is a function of the data that captures all the information about the parameters of interest, and no non-trivial function of it has an expected value independent of the parameter. In this problem, the pair (Y1, ?) is identified as a joint complete sufficient statistic for the parameters. This fact is central to various theoretical results in inference, such as those that guarantee the uniqueness of minimal unbiased estimators.

Ancillary Statistics

Ancillary statistics are statistics whose distribution does not depend on the parameters of the model. They provide useful information in the assessment of the reliability of inference procedures, as they encapsulate randomness that is free from parameter influence. In the given problem, the ratio (Y2 - Y1) / [?(Yi - Y1)] is an ancillary statistic, meaning its distribution remains invariant with respect to ?1 and ?2.

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