Let X1,X2, . . . , Xn be a random sample from the uniform [θ1,θ2], where 0 ≤θ1 ≤ θ2. (a) Show that Y1 = min(Xi) and Yn = max(Xi), the joint sufficient statistics for θ1 and θ2, are complete. (b) Find the MSEs of θ1 and θ2
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We have a random sample \( X_1, X_2, \ldots, X_n \) from a uniform distribution on the interval \([ \theta_1, \theta_2 ]\). We need to show that \( Y_1 = \min(X_i) \) and \( Y_n = \max(X_i) \) are joint sufficient statistics for \( \theta_1 \) and \( \theta_2 \), Show more…
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Key Concepts
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Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample from the normal distribution $N\left(\theta_{1}, \theta_{2}\right),-\infty<\theta_{1}<\infty, 0<\theta_{2}<\infty$. Show that the joint complete sufficient statistics $\bar{X}=\bar{Y}$ and $S^{2}$ for $\theta_{1}$ and $\theta_{2}$ are independent of each of $\left(Y_{n}-\bar{Y}\right) / S$ and $\left(Y_{n}-Y_{1}\right) / S$
Sufficiency
Sufficiency, Completeness, and Independence
Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from the uniform distribution with pdf $f\left(x ; \theta_{1}, \theta_{2}\right)=1 /\left(2 \theta_{2}\right), \theta_{1}-\theta_{2}<x<\theta_{1}+\theta_{2}$, where $-\infty<\theta_{1}<\infty$ and $\theta_{2}>0$ and the pdf is equal to zero elsewhere. (a) Show that $Y_{1}=\min \left(X_{i}\right)$ and $Y_{n}=\max \left(X_{i}\right)$, the joint sufficient statistics for $\theta_{1}$ and $\theta_{2}$, are complete. (b) Find the MVUEs of $\theta_{1}$ and $\theta_{2}$.
The Case of Several Parameters
Let $Y_{1}<Y_{2}<\cdots<Y_{n}$ be the order statistics of a random sample of size $n$ from the uniform distribution over the closed interval $[-\theta, \theta]$ having pdf $f(x ; \theta)=(1 / 2 \theta) I_{[-\theta, \theta]}(x) .$ (a) Show that $Y_{1}$ and $Y_{n}$ are joint sufficient statistics for $\theta$. (b) Argue that the mle of $\theta$ is $\hat{\theta}=\max \left(-Y_{1}, Y_{n}\right)$ (c) Demonstrate that the mle $\hat{\theta}$ is a sufficient statistic for $\theta$ and thus is a minimal sufficient statistic for $\theta$.
Minimal Sufficiency and Ancillary Statistics
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