Mathematical Statistics-2010 Midterm #4 6/24/2010 1.State the definitions of the following terms. (1)Type I error and type II error. (2) Power function and the relationship with (1) (3)Sufficient statistics. (4) The most powerful size αtest. (5) The likelihood ratio (LR) test. 2.Let Xଵ, Xଶ, … , X୬be a random sample of size n from the following density: ݂ሺݔ; ߠሻ ൌ ሺ1 ߠሻݔఏ, 0 ݔ 1, ߠ 0. (1) Find the MLE of θ. Find the distribution of the MLE. (2) Find the MLE of PሺX ଵଶሻ. 3.Let Xଵ, Xଶ, … , X୬be a random sample of size n from the Bernoulli distribution with parameter p. (1) Show that the statistic T ൌ ∑X୧୬୧ୀଵ(and hence Xഥ) is sufficient for the parameter pby (i) The definition of a sufficient statistic (ii) The factorization theorem. (2) Find the maximum likelihood estimator and the method of moment estimator of p4.Let Xଵ, Xଶ, … , X୬be a random sample of size n from ܰሺߤ, 1ሻ.
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