MAST90082_HW2.pdf - MAST 90082 Mathematical Statistics Assignment 2 Due Tuesday 11th June Please present your solution in details the mark is

MAST90082_HW2.pdf - MAST 90082 Mathematical Statistics...

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MAST 90082 Mathematical Statistics Assignment 2 Due: Tuesday, 11th June Please present your solution in details, the mark is distributed to essential steps. 1. Let X 1 , . . . , X n be a random sample from the Pareto distribution with pdf f ( x | θ ) = θx - ( θ +1) , x 1 , 0 , x < 1 , where θ > 0 is unknown. (a) Find a uniformly most powerful (UMP) test of size α for testing H 0 : θ θ 0 versus H 1 : θ > θ 0 , where θ 0 > 0 is a fixed real number. (Use quantiles of chi-square distributions to express the test) (b) Find a confidence interval for θ with confidence coefficient 1 - α by pivoting a ran- dom variable based on T = n i =1 log X i . (Use quantiles of chi-square distributions to express the confidence interval and use equal-tail confidence interval) (c) Find a confidence interval for θ with confidence coefficient 1 - α by pivoting the cdf of X (1) = min { X 1 , . . . , X n } . (Use equal-tail confidence interval) 2. Let X 1 , . . . , X n be a random sample from the distribution with pdf f ( x | μ, θ ) = θ - 1 e - ( x - μ ) , x μ, 0 , x < μ, where μ ∈ R and θ > 0. 1
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(a) If θ is known but μ is unknown, find a likelihood ratio test (LRT) of size α for testing H 0 : μ μ 0 versus H 1 : μ > μ 0 , where μ 0 is a known constant.
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