312-final2sol-may 03

Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat312: Final Exam Solutions Moo K. Chung mchung@stat.wisc.edu May 16, 2003 1. We wish to fit paired sample ( x 1 , y 1 ) , ··· , ( x n , y n ) with a linear regression model Y = cx + ± . We assume that ± follows a normal distribution with zero mean and variance σ 2 . (a) Estimate c by minimizing the sum of the squared residuals (5pts). (b) Write the log-likelihood function as a func- tion of c and σ (5pts). (c) Find the likelihood estimator of c by differen- tiating the log-likelihood function in (b). De- rive everything (5pts). (d) Either prove or disprove unbiasedness of the estimator you computed in (c) (5pts, no point given if (c) is incorrect). (e) Compute the variance of the estimator you computed in (a) (5pts, no point given if (a) is incorrect). (f) If the sample correlation coefficient of the above paired data is 0 . 5 , what is the sam- ple correlation correlation coefficient of data ( x 1 - ¯ x, y 1 - ¯ y ) , ··· , ( x n - ¯ x, y n - ¯ y ) ? ¯ x and ¯ y are the respective sample means of x i ’s and y i ’s. Prove your result (5pts). Solution. (a) The sum of squared residuals SSE = n i =1 ( y i - cx i ) 2 (2pts). Letting ∂SSE/∂c = 0 , we get n i =1 x i ( y i - cx i ) = 0 . Solving this, ˆ c = n i =1 x i y i / n i =1 x 2 i (3pts). (b) Note that Y i = cx i and Var Y i = σ 2 for some σ . So Y i N ( cx i , σ 2 ) . Then the likelihood function is given by L ( c, σ ) = const. σ n exp ± n i =1 ( y i - cx i ) 2 2 σ 2 ² . The log-likelihood function is then log L ( c, σ ) = const. - n log σ + 1 2 σ 2 n X i =1 ( y i - cx i ) 2 .
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312-final2sol-may 03 - Stat312: Final Exam Solutions Moo K....

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