Qu: Let , , ,,,,, denote a random sample from a normal distribution with mean zero and variance Ѳ, 0< Ѳ<oo. is an unbiased estimator of Ѳ. Show that has variance

Qu: Let and be two independent unbiased estimators of Ѳ. Assume that the variance of is twice the variance of . Find the constants k1 and k2 so that k1+k2 is an unbiased estimator with smallest possible variance for such a linear combination.

Qu: Let , , ,,,,, denote a random sample from a Poisson distribution with parameter Ѳ, 0< Ѳ<oo. Let Y= and let L[Ѳ, (y)]=[ Ѳ -(y)]^2 . If we restrict our consideration to decision functions of the form (y)=b+y/n, where b does not depend on y , show that R(Ѳ)=(b^2)+ Ѳ/n. What decision function of this form yields a uniformly smaller risk than every other decision function of this form ? With this solution, say and 0< Ѳ<oo, determine Ѳ,) IF it exists.

Qu: Let be the order statistic of a random sample of size 5 from the uniform distribution having pdf f(x; Ѳ) = 1/ Ѳ , 0<x< Ѳ, 0< Ѳ<oo, zero elsewhere. Show that 2 is an unbiased estimator of Ѳ. Determine the joint pdf of and the sufficient statistic for Ѳ. Find the conditional expectation E[2|] = (). Compare the variance of 2 and .

Qu: If , is a random sample of size 2 from a distribution having pdf f(x; Ѳ) =(1/ Ѳ), 0<x<oo, 0< Ѳ<oo, zero elsewhere, find the joint pdf of the sufficient statistic + for Ѳ and =. Show that is an unbiased estimator of Ѳ with variance Ѳ^2. Find E[|] = () and the variance of ().

Qu: If a+ bz + c = 0 for more than two values of z, then a=b=c=0 . Use this result to show that the family {b(2, Ѳ) : 0< Ѳ<1 } is complete.

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