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Mathematical Statistics Midterm Exam#4-2010

# Mathematical Statistics Midterm Exam#4-2010 - Mathematical...

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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 α test. (5) The likelihood ratio (LR) test. 2. Let X , X , … , X be a random sample of size n from the following density: ݂ሺݔ; ߠሻ ൌ ሺ1 ൅ ߠሻݔ , 0 ൑ ݔ ൑ 1, ߠ ൐ 0 . (1) Find the MLE of θ . Find the distribution of the MLE. (2) Find the MLE of PሺX ൐ . 3. Let X , X , … , X be a random sample of size n from the Bernoulli distribution with parameter p . (1) Show that the statistic T ൌ ∑ X ୧ୀଵ (and hence X ) is sufficient for the parameter p by (i) The definition of a sufficient statistic (ii) The factorization theorem. (2) Find the maximum likelihood estimator and the method of moment estimator of p 4. Let X , X , … , X be a random sample of size n from ܰሺߤ, 1ሻ .
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