Question

. Suppose the Y1, Y2, · · · , Yn denote a random sample from a population with Rayleigh distribution (Weibull distribution with parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ > 0, y > 0

Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn}, and ˆθ2 = 1 n Xn i=1 Y 2 i .

ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased estimators, and in the case that they are not find a multiple of them that make them unbiased.

iii) (5 points) Find the efficiency between the unbiased estimators

iv) (5 points) Are the unbiased estimators consistent?

Problem 5. Suppose the Yı, Y2,...,Ydenote a random sample from a population with Rayleigh distribution (Weibull distribution with parameters 2,0) with density function f(yle) = { evle, 6 >0, y>0 0, elsewhere i) (10 points) Prove that if Y have Rayleigh distribution find its distribution function and prove that Y2 has exponential distribution with mean e. Consider the estimators Ôn = Y(u) = min{Yı, Yz, ... ,Yn}, and ôz = ŻY? ii) (10 points) Determine if Ôn and 2 are unbiased estimators, and in the case that they are not find a multiple of them that make them unbiased. iii) (5 points) Find the efficiency between the unbiased estimators. iv) (5 points) Are the unbiased estimators consistent?

Answer 1

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