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Anskey3

# Anskey3 - x i Second derivatives H = 2 4 2 L&& 2 2...

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Answer Key for Homework 3 1. Report descriptive statistics for this data set. What is the fraction of duration that are censored? duration d age educ white rate min 8 0 17 9 0 1 max 1174 1 33 17 1 6 mean 229.3 0.34 22.52 12 .5 3.58 median 140 0 22 12 .5 3 std 256.2 .48 3.52 1.43 051 1.25 34% is censored 2. Compute the MLE of ° 0 if ° 1 = 0 . Do this with and without accounting for censored observations. ° Without accounting for censored observations Log likelihood function is L ( ° ) = n X i =1 ln f ( y j x ) = n X i =1 [( ° 0 + ° 1 x i ) ± y i exp( ° 0 + ° 1 x i )] = n X i =1 [ ° 0 ± y i exp( ° 0 )] From @L ( ° ) 0 = 0 ; ^ ° 0 = ln 0 @ n X y i 1 A = ± 5 : 435 ° With accounting for censored observations L ( ° ) = n X i =1 [(1 ± d i ) ln f ( y i j x ) + S ( y j x )] ; where d = 1 if censored = n X i =1 [(1 ± d i )( ° 0 + ° 1 x i ) ± y i exp( ° 0 + ° 1 x i )] = n X i =1 [(1 ± d i ) ° 0 ± y i exp( ° 0 )] (*) From @L ( ° ) 0 = 0 ; ^ ° 0 = ln 0 @ X (1 ± d i ) X y i 1 A = ± 6 : 514 1

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3. Show that the log likelihood function at ° 0 = ° 1 = 0 is equal to -11465. Remember to take account of the censoring At ° 0 = ° 1 = 0 ; From (*), L ( ° ) = ± X y i 4. Write a program for the °rst and second derivatives of the log likelihood. L ( ° ) = n X i =1 [(1 ± d i )( ° 0 + ° 1 x i ) ± y i exp( ° 0 + ° 1 x i )] @L ( ° ) 0 = n X i =1 [(1 ± d i ) ± y i exp( ° 0 + ° 1 x i )] @L ( ° ) 1 = n X i =1 [(1 ± d i ) x i ± x i y i exp(
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Unformatted text preview: x i )] Second derivatives H = 2 4 @ 2 L ( & ) @& 2 @ 2 L ( & ) @& & 1 @ 2 L ( & ) @& 1 & @ 2 L ( & ) @& 2 1 3 5 = 2 6 6 6 6 4 & n X i =1 [ y i exp( & + & 1 x i )] & n X i =1 [ x i y i exp( & + & 1 x i )] & n X i =1 [ x i y i exp( & + & 1 x i )] & n X i =1 & x 2 i y i exp( & + & 1 x i ) ± 3 7 7 7 7 5 5. Maximize the log likelihood function using the Newton-Raphson algorithm with anlytic derivatives. Use zeros as starting values. Also use the MLE for & obtained above and & 1 = 0 : ^ & mle = ( & 5 : 6139 ; & : 0198) 6. Compute the standard errors of the MLE. AV ar ( & ) = ( & E [ H ]) & 1 =n standard errors = (1 : 349 ; : 112) 2...
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Anskey3 - x i Second derivatives H = 2 4 2 L&& 2 2...

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