Chapter 7: Problem 11
Show that \(Y=|X|\) is a complete sufficient statistic for \(\theta>0\), where \(X\)
has the pdf \(f_{X}(x ; \theta)=1 /(2 \theta)\), for \(-\theta
Chapter 7: Problem 11
Show that \(Y=|X|\) is a complete sufficient statistic for \(\theta>0\), where \(X\)
has the pdf \(f_{X}(x ; \theta)=1 /(2 \theta)\), for \(-\theta
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Get started for freeLet \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with
pdf \(f(x ; \theta)=\theta^{2} x e^{-\theta x}, 0
Let a random sample of size \(n\) be taken from a distribution of the discrete type with pmf \(f(x ; \theta)=1 / \theta, x=1,2, \ldots, \theta\), zero elsewhere, where \(\theta\) is an unknown positive integer. (a) Show that the largest observation, say \(Y\), of the sample is a complete sufficient statistic for \(\theta\). (b) Prove that $$\left[Y^{n+1}-(Y-1)^{n+1}\right] /\left[Y^{n}-(Y-1)^{n}\right]$$ is the unique MVUE of \(\theta\).
Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a normal distribution with mean zero and variance \(\theta, 0<\theta<\infty\). Show that \(\sum_{1}^{n} X_{i}^{2} / n\) is an unbiased estimator of \(\theta\) and has variance \(2 \theta^{2} / n\)
The pdf depicted in Figure \(7.9 .1\) is given by
$$f_{m_{2}}(x)=e^{x}\left(1+m_{2}^{-1} e^{x}\right)^{-\left(m_{2}+1\right)},
\quad-\infty
. 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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