1. Let.K = (Xl' X 2' ••• , X n) be a random sample from a discrete uniform distribution with support
S(O) = {1,2, ... ,0} with unknown 0 (0 ~ 1).
1) Find the maximum likelihood estimator (MLE) OMLE of O.
2) Find the sufficient statistic for 0, call TUD.
3) Find the distribution (pmf) of T('K) .
4) Show that T(K) is a complete sufficient statistic for O.
5) Find the UMVUE for O.
2. The number of breakdowns X per day for a certain machine is a Poisson RV with unknown mean 0
(0
1) Find a minimal sufficient statistic for parameter 0 .
2) Find the maximum likelihood estimator (MLE) OMLE of 0, and show that it is efficient estimator for O.
3) The daily cost of repairing the breakdowns is given by Y = 3X2 .
Find MLE and UMVUE for expected value of Y.
4) Find the asymptotic distribution of e-OMLE as n ~ oo?
n " 5) Find the limiting distribution (as n ~ 00 ) of the sequence of random variables w" = -,,-(OMLE - 0)2.
°MLE
6) Use the pivotal method to construct a confidence interval for 0 with coverage probability 1-a .
7) Find the variance stabilizing transformation for the underlying Poisson family.
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MA 781/782 Qualifier 2011 1. (a) Carefully state the Lehmann–Scheff´e theorem. (b) Let X1, ...,Xn be i.i.d. random variables from Binomial(r, ?) distribution where 0 <><> 0. For this model X and cS are unbiased estimators of ?, where X is the sample mean,S is the sample standard deviation, and c is a constant given by c = pn-1??((n-1)/2) p2??(n/2) . (a) Consider an estimator of the form T1 = a1X + a2(cS), where a1 + a2 = 1. Show that the estimator T1 is an unbiased estimator of ? for any choices of {(a1, a2) : a1 + a2 = 1} and find the best estimator in this class that minimizes the variance. Let’s call this estimator T1 . (b) Now consider another estimator of ? of the form T2 = a1X+a2(cS), where we do NOT assume that a1 + a2 = 1. Find the best...