Math Stat Final-2008 - Mathematical Statistics-2008 Final...

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Unformatted text preview: Mathematical Statistics-2008 Final Exam 6/17/2008 1. State the definitions of the following terms and give an interpretation of each of the terms. (1) Consistent estimator. (2) Sufficient statistics. (3) The maximum likelihood estimator. (4) The minimum variance unbiased estimator. (5) The most powerful test. 2. Let X , X , … , X be a random sample of size n from the following density: ; 1 ,0 1, 0. (1) Find the MLE of θ. Find the distribution of the MLE. (2) Find the MLE of P X . (3) Find a sufficient statistic for θ. Is the MLE of θ a function of your sufficient statistic? (4) Find a method of moment estimator of θ. 3. Let X , X , … , X be a random sample of size n from the Poisson distribution with mean λ. (1) Show that the statistic T ∑ X (and hence X) is sufficient for the parameter λ by (i) The definition of a sufficient statistic. (ii) The factorization theorem. (2) Show that X is an unbiased minimum variance estimator of λ by comparing the variance of X to the Crámer-Rao lower bound of the variance of an unbiased estimator. 4. Let X , X , … , X be a random sample of size n. (1) State and prove the Neyman-Pearson Lemma. (2) Suppose that the random sample is from , 1 . Find the uniformly most powerful test for testing H : µ 0 vs H : µ 0. 5. Let X be a single observation from against H : θ 5, . Consider the problem of testing H : θ . The following table gives the probabilities under two hypotheses: f x; θ f x; f x; 0 1 2 3 4 5 Find the best critical regions by comparing the powers of all possible critical regions for both α and α . ...
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