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Q3.Let X=(X1,…Xn)be a collection of i.i.d.random variables with probability density

By December 6, 2021No Comments

Q3.Let X=(X1,…Xn)be a collection of i.i.d.random variables with probability densityfunction (pdf):f(x;0)=2,0<0≤x<∞,where 00 is an unknown parameter.Let x (x1,...,xn)be a random sample observed from this distribution (this is arealisation of X).(a)Find a one-dimensional sufficient statistic T(X)for 0.[2](b)Find the Maximum Likelihood Estimator of 0.[5](c)Is the Maximum Likelihood Estimator of 0 unbiased for 0?Justify your answer.[5]A Bayesian statistician assumes that the prior distribution for 0 is given by 0~0,a,where a∈R+,a≠0.Suppose that a sample of X1,..,Xn,denoted x=(x1,..,xn),is observed.(d)Determine the posterior distribution of 0.[5]Suppose that the following sample of X1,...Xn is obtained,where n 8:14.8,7.2,5.9,5.4,6.1,5.2,9.4,5.5(e)Assuming that the prior distribution for is such that~U[o,,determine theBayes'estimate of 0 where the loss function is given byL(0,d)=(d-0)2 where d∈R.

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