Math Statistics Questions Help

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

Qu:  Let math statistics questions help and math statistics questions help be two independent unbiased estimators of Ѳ. Assume that the variance of math statistics questions help is twice the variance of math statistics questions help. Find the constants k1 and k2 so that k1math statistics questions help+k2math statistics questions help is an unbiased estimator with smallest possible variance for such a linear combination.  

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Qu: Let math statistics questions help, math statistics questions help, ,,,,,math statistics questions help denote a random sample from a Poisson distribution with parameter Ѳ, 0< Ѳ<oo. Let Y=math statistics questions help and let L[Ѳ, math statistics questions help(y)]=[ Ѳ -math statistics questions help(y)]^2 . If we restrict our consideration to decision functions of the form math statistics questions help(y)=b+y/n, where b does not depend on y , show that R(Ѳmath statistics questions help)=(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 math statistics questions help and 0< Ѳ<oo, determine math statistics questions helpѲ,math statistics questions help) IF it exists.

Qu: Let math statistics questions help 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 2math statistics questions help is an unbiased estimator of Ѳ. Determine the joint pdf of math statistics questions help and the sufficient statistic math statistics questions help for Ѳ. Find the conditional expectation E[2math statistics questions help|math statistics questions help] = math statistics questions help(math statistics questions help). Compare the variance of 2math statistics questions help and math statistics questions help.

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

Qu: If amath statistics questions help+ 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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