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# Example Class8 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 8 1. Let X 1 , ..., X n denote a random sample from the distribution that has the p.d.f. f ( x | θ ) = 1 2 π exp - ( x - θ ) 2 2 , -∞ < x < . Find the critical region for the most powerful size α test of H 0 : θ = 0 vs H 1 : θ = 1. 2. A random variable X follows Gamma distribution with probability density function f ( x | θ ) = x θ - 1 e - x / Γ( θ ) , x > 0 . The θ is an unknown parameter. Suppose X 1 , . . . , X n are i.i.d. with the distribution of X . (a) Give an expression for the p-value of the likelihood ratio test of H 0 : θ θ 0 against H 1 : θ < θ 0 , where θ 0 is a fixed constant. (b) Describe the critical region of the size α likelihood ratio test of H 0 against H 1 as specified in (a). 3. Let X 1 , ..., X n be i.i.d. sample from an exponential distribution with p.d.f. f ( x | λ ) = λe - λx , λ > 0 , x > 0 . Derive a likelihood ratio test of H 0 : λ = λ 0 against H 1 : λ 6 = λ 0 , where λ 0 is a fixed constant, and show that the critical region is of the form { ¯

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• Spring '11
• Dr.Yun
• Probability distribution, Probability theory, probability density function, University of Hong Kong, likelihood ratio

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