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Chapter 7: Problem 2

Let \(Y_{1}

Short Answer

Expert verified
Through calculation, it was shown that distribution of \(Z=Y_{n}-\bar{X}\) does not depend upon \(\theta\) as \(\mathbb{E}(Z) = 0\) and \(\mathbb{Var}(Z) = \mathbb{Var}(Y_n) + \mathbb{Var}(\bar{X})\), both of which do not involve \(\theta\). Hence, \(\bar{Y}\), a complete sufficient statistic for \(\theta\), is indeed found to be independent of \(Z\), as per the condition that the marginal and joint distributions of these statistics are equal.

Step by step solution

01

Representation of Variable \(Z\)

Represent \(Z\) as difference between \(Y_n\) (the maximum order statistic) and the \(\bar{X}\) (the sample mean). \(Z = Y_{n} - \bar{X}\) where \(\bar{X} = \frac{\sum_{i=1}^{n} Y_{i}}{n}\)
02

Calculation of Expected Value \(\mathbb{E}(Z)\)

The expected value can be expressed as the difference in the expected values of \(Y_n\) and \(\bar{X}\). From the properties of the normal distribution, the expectation of any normally distributed random variable is \(theta\). Therefore, \(\mathbb{E}(Z) = \mathbb{E}(Y_n) - \mathbb{E}(\bar{X}) = \theta - \theta = 0\)
03

Calculation of Variance \(\mathbb{Var}(Z)\)

Calculate the variance of \(Z\), \(\mathbb{Var}(Z) = \mathbb{Var}(Y_n - \bar{X})\). The variance of the sum or difference of two independent random variables is the sum of their variances, i.e., \(\mathbb{Var}(Y_n - \bar{X}) = \mathbb{Var}(Y_n) + \mathbb{Var}(\bar{X})\). The variances of \(Y_n\) and \(\bar{X}\) do not depend on \(\theta\) as they are properties of the normal distribution.
04

Proof of \(Y_n, \bar{X}\) and \(Z\) Independence

If the distribution of \(Z\) does not depend on \(\theta\), it should be independent of the sufficient statistic \(\bar{Y}\). To demonstrate this, it can be shown that the join distribution of \(\bar{Y}\) and \(Z\) equals the product of their marginal distributions, which means \(\bar{Y}\) and \(Z\) are independent. This can be written as \(f(\bar{Y}, Z) = f(\bar{Y})f(Z)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Uniform Distribution
The uniform distribution is a type of probability distribution in which all outcomes are equally likely. When we say that a random variable has a uniform distribution over an interval \( [-\theta, \theta] \) with a probability density function (pdf) \( f(x ; \theta)=(1 / 2 \theta) I_{[-\theta, \theta]}(x) \), it means that the variable can take any value within this interval with constant likelihood. The pdf represents the constant value that is characteristic of the uniform distribution over the specified range.

Understanding the uniform distribution is crucial when dealing with order statistics, which are the sampled values in increasing order. If you were to sample several numbers from the uniform distribution and arrange them from least to greatest, those arranged numbers would be your order statistics. In the context of the exercise provided, \( Y_{1} \) is the smallest statistic (the minimum), and \( Y_{n} \) is the largest statistic (the maximum) of the sampled data from this uniform distribution.
Maximum Likelihood Estimation
Maximum Likelihood Estimation (MLE) is a method used for estimating the parameters of a statistical model. MLE is based on finding the parameter values that maximize the likelihood function, which measures how well the model explains the observed data.

In the exercise, to find the MLE of \(\theta\), students are instructed to consider \(\hat{\theta}=\max \left(-Y_{1}, Y_{n}\right)\). This estimation makes intuitive sense as the absolute values of the smallest and largest statistics from the sample will be closest to the edges of the uniform distribution range. The MLE aims to determine the value of \(\theta\) that makes the observed data most probable within the context of the uniform distribution, which in this case, will be the maximum of the observed absolute values.
Order Statistics
Order statistics refer to the statistics that are derived from the sample data by arranging the data in ascending order. The order statistics provide important insights into the sample data and are particularly useful in non-parametric statistics, which do not assume a particular distribution.

In our exercise, \(Y_{1}\) and \(Y_{n}\) are the first and the last order statistics, representing the minimum and maximum values in the sample, respectively. These statistics are particularly informative in the case of uniform distribution and are used to identify sufficient statistics for \(\theta\). A sufficient statistic is a function of the data that captures all the information needed to estimate a parameter effectively. In the exercise, it is shown that \(Y_{1}\) and \(Y_{n}\) are joint sufficient statistics for \(\theta\), meaning that these two values alone are enough to estimate the parameter \(\theta\) with no loss of information.

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Most popular questions from this chapter

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.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample of size \(n\) from a geometric distribution that has pmf \(f(x ; \theta)=(1-\theta)^{x} \theta, x=0,1,2, \ldots, 0<\theta<1\), zero elsewhere. Show that \(\sum_{1}^{n} X_{i}\) is a sufficient statistic for \(\theta\).

Let \(X_{1}, \ldots, X_{n}\) be iid with pdf \(f(x ; \theta)=1 /(3 \theta),-\theta0\). (a) Find the mle \(\hat{\theta}\) of \(\theta\). (b) Is \(\widehat{\theta}\) a sufficient statistic for \(\theta\) ? Why? (c) Is \((n+1) \widehat{\theta} / n\) the unique MVUE of \(\theta\) ? Why?

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 \(\theta\).
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