410Hw07ans.pdf - STAT 410 Spring 2019 A Stepanov Homework#7(due Friday April 26 by 4:00 p.m Please include your name with your last name underlined your

# 410Hw07ans.pdf - STAT 410 Spring 2019 A Stepanov...

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STAT 410 Homework #7 (due Friday, April 26, by 4:00 p.m.) Spring 2019 A. Stepanov Please include your name ( with your last name underlined ) , your NetID, and your section number at the top of the first page. No credit will be given without supporting work. 1. The Weibull distribution has many applications in reliability engineering, survival analysis, and general insurance. Let > 0, > 0. Consider the probability density function δ 1 β δ , δ δ ; β β x e x x f , x > 0, zero otherwise. Recall: W = X has an Exponential ( = 1 β ) = Gamma ( = 1, = 1 β ) distribution. Let X 1 , X 2 , … , X n be a random sample from the above probability distribution. Y = n i i 1 δ X = n i i 1 W has a Gamma ( = n , = 1 β ) distribution. ! ! ! Suppose is known. m) Find a sufficient statistic Y = u ( X 1 , X 2 , … , X n ) for . f ( x 1 , x 2 , x n ; ) = f ( x 1 ; ) f ( x 2 ; ) f ( x n ; ) = n i x i i e x 1 δ 1 β δ δ β = 1 1 δ δ δ β β n i i n x n x i e . By Factorization Theorem, Y = n i i 1 δ X is a sufficient statistic for .
OR f ( x ; ) = exp { x + ln + ln + ( – 1 ) ln x } . K ( x ) = x . Y = n i i 1 δ X is a sufficient statistic for . n) Find the Fisher information I ( ) . ln f ( x ; ) = x + ln + ln + ( – 1 ) ln x . β ln f ( x ; ) = x + 1 β .

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