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Chapter 7: Problem 7

Let \(X\) have the pdf \(f_{X}(x ; \theta)=1 /(2 \theta)\), for \(-\theta0\) (a) Is the statistic \(Y=|X|\) a sufficient statistic for \(\theta ?\) Why? (b) Let \(f_{Y}(y ; \theta)\) be the pdf of \(Y\). Is the family \(\left\\{f_{Y}(y ; \theta): \theta>0\right\\}\) complete? Why?

Short Answer

Expert verified
The statistic \(Y\) is not sufficient because it cannot be factored to show no \(\theta\) dependent part. The family \(\left\{f_{Y}(y; \theta): \theta>0\right\}\) is complete as every zero-mean function of the family's statistic is identically zero.

Step by step solution

01

Define the random variable

Define the random variable \(X\) with probability density function (pdf) given by \(f_{X}(x ; \theta)=1 /(2 \theta)\) for \(-\theta 0\).
02

Find the pdf of Y

The statistic \(Y=|X|\) implies that \(Y\) can take values between \(0\) and \(\theta\). Thus, \(f_{Y}(y ; \theta) = 2f_{X}(y ; \theta) = 1/\theta\) for \(0<y<\theta\) and zero elsewhere.
03

Factorize the Joint PDF

To show if \(Y\) is sufficient for \(\theta\), factorize the joint pdf. For independent and identically distributed random variables \(x_1, x_2, ..., x_n\) from \(f_X(x;\theta)\), the joint pdf is \(\prod_{i=1}^{n} f_X(x_i;\theta)\). Factorize into a part that depends on \(\theta\) only through \(Y\) and another part that does not depend on \(\theta\). If unable to do this, \(Y\) is not sufficient. In this case, \(Y\) is not sufficient as there is no factorized part that doesn't depend on \(\theta\).
04

Check Completeness of the Family

A family of distributions is complete if every unbiased estimate of zero is actually zero. An unbiased estimate of a function \(g(\theta)\) is a function \(\hat{g}(X_1, X_2, ..., X_n)\) such that the expected value of \(\hat{g}\) is \(g(\theta)\). For zero function, \(g(\theta) = 0\), we find \(\hat{g}(X_1, X_2, ..., X_n)\) such that its expected value is zero for every \(\theta\). Here, we find that for all functions \(h(Y)\) that are zero-mean, \(h(Y)\) itself is always zero, hence this family is complete.

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Most popular questions from this chapter

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample with the common pdf \(f(x)=\) \(\theta^{-1} e^{-x / \theta}\), for \(x>0\), zero elsewhere; that is, \(f(x)\) is a \(\Gamma(1, \theta)\) pdf. (a) Show that the statistic \(\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}\) is a complete and sufficient statistic for \(\theta\). (b) Determine the MVUE of \(\theta\). (c) Determine the mle of \(\theta\). (d) Often, though, this pdf is written as \(f(x)=\tau e^{-\tau x}\), for \(x>0\), zero elsewhere. Thus \(\tau=1 / \theta\). Use Theorem \(6.1 .2\) to determine the mle of \(\tau\). (e) Show that the statistic \(\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}\) is a complete and sufficient statistic for \(\tau\). Show that \((n-1) /(n X)\) is the MVUE of \(\tau=1 / \theta\). Hence, as usual the reciprocal of the mle of \(\theta\) is the mle of \(1 / \theta\), but, in this situation, the reciprocal of the MVUE of \(\theta\) is not the MVUE of \(1 / \theta\). (f) Compute the variances of each of the unbiased estimators in Parts (b) and (e).

If \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample from a distribution that has a pdf which is a regular case of the exponential class, show that the pdf of \(Y_{1}=\sum_{1}^{n} K\left(X_{i}\right)\) is of the form \(f_{Y_{1}}\left(y_{1} ; \theta\right)=R\left(y_{1}\right) \exp \left[p(\theta) y_{1}+n q(\theta)\right]\). Hint: Let \(Y_{2}=X_{2}, \ldots, Y_{n}=X_{n}\) be \(n-1\) auxiliary random variables. Find the joint pdf of \(Y_{1}, Y_{2}, \ldots, Y_{n}\) and then the marginal pdf of \(Y_{1}\).

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a Poisson distribution with parameter \(\theta, 0<\theta<\infty .\) Let \(Y=\sum_{1}^{n} X_{i}\) and let \(\mathcal{L}[\theta, \delta(y)]=[\theta-\delta(y)]^{2}\). If we restrict our considerations to decision functions of the form \(\delta(y)=b+y / n\), where \(b\) does not depend on \(y\), show that \(R(\theta, \delta)=b^{2}+\theta / n .\) What decision function of this form yields a uniformly smaller risk than every other decision function of this form? With this solution, say \(\delta\), and \(0<\theta<\infty\), determine \(\max _{\theta} R(\theta, \delta)\) if it exists.

Let \(X\) and \(Y\) be random variables such that \(E\left(X^{k}\right)\) and \(E\left(Y^{k}\right) \neq 0\) exist for \(k=1,2,3, \ldots . .\) If the ratio \(X / Y\) and its denominator \(Y\) are independent, prove that \(E\left[(X / Y)^{k}\right]=E\left(X^{k}\right) / E\left(Y^{k}\right), k=1,2,3, \ldots\) Hint: Write \(E\left(X^{k}\right)=E\left[Y^{k}(X / Y)^{k}\right]\).

. Let \(X_{1}, \ldots, X_{n}\) be a random sample from a distribution of the continuous type with cdf \(F(x)\). Let \(\theta=P\left(X_{1} \leq a\right)=F(a)\), where \(a\) is known. Show that the proportion \(n^{-1} \\#\left\\{X_{i} \leq a\right\\}\) is the MVUE of \(\theta\).
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