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 with pdf $f(x ; \theta)=1 / \theta, 0<x<\theta$, zero elsewhere. By Example 7.4.2, the statistic $Y_{n}$ is a complete sufficient statistic for $\theta$ and it has pdf $$ g\left(y_{n} ; \theta\right)=\frac{n y_{n}^{n-1}}{\theta^{n}}, \quad 0<y_{n}<\theta $$ and zero elsewhere. (a) Find the distribution function $H_{n}(z ; \theta)$ of $Z=n\left(\theta-Y_{n}\right)$. (b) Find the $\lim _{n \rightarrow \infty} H_{n}(z ; \theta)$ and thus the limiting distribution of $Z$.

Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample of size $n$ from a distribution with pdf $f(x ; \theta)=1 / \theta, 0<x<\theta$, zero elsewhere. By Example 7.4.2, the statistic $Y_{n}$ is a complete sufficient statistic for $\theta$ and it has pdf
$$
g\left(y_{n} ; \theta\right)=\frac{n y_{n}^{n-1}}{\theta^{n}}, \quad 0<y_{n}<\theta
$$
and zero elsewhere.
(a) Find the distribution function $H_{n}(z ; \theta)$ of $Z=n\left(\theta-Y_{n}\right)$.
(b) Find the $\lim _{n \rightarrow \infty} H_{n}(z ; \theta)$ and thus the limiting distribution of $Z$.

Key Concepts

Order Statistics

Order statistics refer to the statistics obtained by arranging a random sample in increasing (or decreasing) order. They are used to describe sample extremes such as the minimum or maximum, and they play a key role in inference procedures by revealing information about the underlying distribution. The maximum order statistic, in particular, is often used in cases where the sample provides a natural bound on the parameter of interest.

Complete and Sufficient Statistics

A statistic is called sufficient if it encapsulates all the information in the sample relevant to the parameter of interest. A complete statistic not only is sufficient but has the property that no non?zero function of it has an expected value of zero for all parameter values. This concept is important because it validates the use of certain estimators and simplifies inference by reducing the complexity of the sample without losing information about the parameter.

Transformation of Random Variables

Transformation of random variables is a method used to derive the probability distribution of a function of an observed statistic. This involves applying a change-of-variable technique to the known probability density function (or cumulative distribution function) of the original variable, and it is a crucial tool in deriving the distribution of estimators and test statistics in statistical inference.

Asymptotic Distribution and Convergence in Distribution

Asymptotic distribution involves studying the behavior of a sequence of random variables as the sample size tends to infinity. Convergence in distribution means that the distribution of the random variable approaches a limiting form. This concept is essential in establishing approximate distributions for complex statistics, facilitating inference when exact finite-sample distributions are intractable.

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