北京化工大学2012-2013(1)高等数理统计试卷

北京化工大学2012-2013(1)《高等数理统计》试卷

1. Let n X X X ,,,21 be iid random samples with pdf given by: 1)|(-=θθθx x f for 10<θ is the parameter.

(1) Find the method of moment estimator (MME) for θ and denote it by θ~.

(2) Find the maximum likelihood estimator (MLE) of θ and denote it by 1ˆθ. Find the MLE of θ

1 and denote it by 2ˆθ. (3) Calculate the Cramer-Rao Lower Bound for the variance of any unbiased estimator of θ

1. (4) Is 2ˆθ unbiased estimator of θ1? If 2ˆθ is unbiased estimator of θ

1, is it the uniformly minimum variance estimator (UMVUE) of θ

1? Justify your answer. 2. Let n X X X ,,,21 be iid with pdf θθθ<<=x x f 0,/1)|(.

(1) Prove that i n i n X X ≤≤=1)(max a complete sufficient statistics for the parameter θ.

(2) Based on )(n X , find the uniformly minimum variance estimator (UMVUE) of θ.

3. Let n X X X ,,,21 be a random sample form a ),(2σθn population, where θ and 2

σ are unknown. We are interested in testing 00:θθ=H versus 01:θθ≠H ,

here 0θ is a specified value of θ.

(1) Show that the test that rejects 0H when

n S t X n /22/,10αθ->- is a test of size α, where 2/,1α-n t satisfies /2)t (/2,1αα=≥n-T P with 1~-n t T .

(2) Show that the test in part (1) can be derived as a likelihood ratio test (LRT).

4. Let n X X X ,,,21 be a random sample form a ),(2σθn population,2

σ is known. Consider testing 1:0=θH versus 2:1=θH . Construct a uniformly most powerful (UMP) test with the size of α.