03_07ans - STAT 410 Examples for 03/07/2008 Spring 2008 1....

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STAT 410 Examples for 03/07/2008 Spring 2008 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; θ ) = ± ² ³ - otherwise 0 1 0 1 1 x x 0 < θ < . a) Obtain the method of moments estimator of θ , ~ . ( ) ( ) ´ ´ µ µ µ · ¸ ¸ ¸ ¹ º = = - - 1 0 1 X 1 X E ; dx x x dx x f x . = 1 1 0 1 1 1 1 1 1 1 1 1 0 1 + = µ µ µ · ¸ ¸ ¸ ¹ º + = µ µ µ · ¸ ¸ ¸ ¹ º + ´ x dx x . 1 1 X + = . X X 1 ~ - = . b) Obtain the maximum likelihood estimator of θ , ˆ . Likelihood function: L( θ ) = ( ) 1 1 1 X X 1 X ; - = = µ µ µ · ¸ ¸ ¸ ¹ º = n i n n i i i f . ln L( θ ) = » » = = µ µ · ¸ ¸ ¹ º - + - = - + - n i i n i i n n 1 1 X ln 1 1 X ln 1 ln ln . ( ) ( ) » = - - = n i i d d n 1 2 X ln ˆ 1 ˆ ˆ L ln = 0. ¼ » = - = n i i n 1 X ln 1 ˆ .
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c) Suppose n = 3, and x 1 = 0.2, x 2 = 0.3, x 3 = 0.5. Compute the values of the method of moments estimate and the maximum likelihood estimate for θ . 3
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03_07ans - STAT 410 Examples for 03/07/2008 Spring 2008 1....

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