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Chapter 7: Problem 7

Let \(Y_{1}

Short Answer

Expert verified
The variables \(\left(Y_{n}-\bar{Y}\right) / S\) and \(\left(Y_{n}-Y_{1}\right) / S\) are proven to be independent from \(\bar{X}=\bar{Y}\) and \(S^{2}\) by showing that their higher-order moments do not depend on or involve \(\bar{Y}\) and \(S\).

Step by step solution

01

Define the Variables

Firstly, make sure you understand the notations. Here, \(Y_{1}<Y_{2}<\cdots<Y_{n}\) represent the order statistics from the normal distribution \(N(\theta_{1},\theta_{2})\), where \(\theta_{1}\) and \(\theta_{2}\) are parameters of the normal distribution. The variables \(\bar{X}=\bar{Y}\) and \(S^{2}\) represent the joint-complete sufficient statistics. \(\left(Y_{n}-\bar{Y}\right) / S\) and \(\left(Y_{n}-Y_{1}\right) / S\) are the variables that we need to prove as independent from the \(\bar{X}, S^{2}\).
02

Use Properties of Normal Distribution

We know that mean and variance are both sufficient statistics for the normal distribution. Also, \(Y_{1}\) and \(Y_{n}\) are independent of mean \(\bar{Y}\) and \(S\). Hence, the variables \(\left(Y_{n}-\bar{Y}\right) / S\) and \(\left(Y_{n}-Y_{1}\right) / S\) should be independent of any function containing \(\bar{Y}\) and \(S\).
03

Proving Independence

To prove independence, show that the third moment about the origin of the variables does not involve any \(\bar{Y}\) or \(S\). Specifically, compute the third central moment of \(\left(Y_{n}-\bar{Y}\right) / S\) and \(\left(Y_{n}-Y_{1}\right) / S\) and ensure that their values are independent from the same central moment calculated for \(\bar{X}=\bar{Y}\) and \(S^{2}\). As the moments are independent, so are the variables.

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Most popular questions from this chapter

Let \(Y_{1}

Let a random sample of size \(n\) be taken from a distribution that has the pdf \(f(x ; \theta)=(1 / \theta) \exp (-x / \theta) I_{(0, \infty)}(x) .\) Find the mle and MVUE of \(P(X \leq 2)\)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\theta^{2} x e^{-\theta x}, 00\) (a) Argue that \(Y=\sum_{1}^{n} X_{i}\) is a complete sufficient statistic for \(\theta\). (b) Compute \(E(1 / Y)\) and find the function of \(Y\) which is the unique MVUE of \(\theta\).

Show that the mean \(\bar{X}\) of a random sample of size \(n\) from a distribution having pdf \(f(x ; \theta)=(1 / \theta) e^{-(x / \theta)}, 0

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a Poisson distribution with parameter \(\theta>0\) (a) Find the MVUE of \(P(X \leq 1)=(1+\theta) e^{-\theta}\). Hint: \(\quad\) Let \(u\left(x_{1}\right)=1, x_{1} \leq 1\), zero elsewhere, and find \(E\left[u\left(X_{1}\right) \mid Y=y\right]\), where \(Y=\sum_{1}^{n} X_{i}\) (b) Express the MVUE as a function of the mle of \(\theta\). (c) Determine the asymptotic distribution of the mle of \(\theta\). (d) Obtain the mle of \(P(X \leq 1)\). Then use Theorem \(5.2 .9\) to determine its asymptotic distribution.
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