Transcription of Topic 15 Maximum Likelihood Estimation
1 Fisher InformationExampleTopic 15 Maximum Likelihood EstimationMultidimensional Estimation1 / 10 Fisher InformationExampleOutlineFisher InformationExampleDistribution of Fitness EffectsGamma Distribution2 / 10 Fisher InformationExampleFisher InformationFor a multidimensional parameter space = ( 1, 2,.., n), the Fisher informationI( ) is a with one-dimensional case, theij-th entry has two alternativeexpressions, namely,I( )ij=E [ ilnL( |X) jlnL( |X)]= E [ 2 i jlnL( |X)].Rather than taking reciprocals to obtain an estimate of the variance, we find thematrix inverseI( ) 1.
2 The diagonal entries ofI( ) 1gives estimates of variances. The off-diagonal entries ofI( ) 1give estimates of / 10 Fisher InformationExampleFisher InformationTo be precise, fornobservations, let i,n(X) be the Maximum Likelihood estimator ofthei-th ( i,n(X)) 1nI( ) 1iiCov ( i,n(X), j,n(X)) 1nI( ) thei-th parameter is i, the asymptotic normality and efficiency can beexpressed by noting that thez-scoreZi,n= i(X) i I( ) 1 approximately a standard we saw in one dimension, we can replace theinformation matrix with the observed information matrix,J( )ij= 2 i jlnL( (X)|X).
3 4 / 10 Fisher InformationExampleDistribution of Fitness EffectsWe return to the model of the gamma distribution for the distribution of fitness effectsof deleterious obtain the Maximum Likelihood estimate for the gammafamily of random variables, write the likelihoodL( , |x) =( ( )x 11e x1) ( ( )x 1ne xn)=( ( ))n(x1x2 xn) 1e (x1+x2+ +xn).and its logarithmlnL( , |x) =n( ln ln ( )) + ( 1)n i=1lnxi n i= score function is a vector( lnL( , |x), lnL( , |x)).5 / 10 Fisher InformationExampleGamma DistributionlnL( , |x) =n( ln ln ( )) + ( 1)n i=1lnxi n i= zeros of the components of the score function determine the Maximum , to determine these parameters, we solve the equations lnL( , |x) =n(ln dd ln ( )) +n i=1lnxi= 0and lnL( , |x) =n n i=1xi= 0,or x=.
4 Substituting = / xinto the first equationresults the following relationship for .n(ln ln x dd ln ( ) +lnx) = 06 / 10 Fisher InformationExampleGamma DistributionThis can be solved deriva-tive of the logarithm of the gamma function ( ) =dd ln ( )is know as the digamma function and iscalled in R the example for the distribution of fit-ness effects in humans,a simulated dataset (rgamma(500, , ))yields = and = for Maximum likeli-hood : ln ln x dd ln ( ) +lnxicrossesthe horizontal axis at = / 10 Fisher InformationExampleGamma DistributionExercise.
5 To determine the variance of these estimators, compute the appropriatesecond ( , )11= 2 2lnL( , |x) =nd2d 2ln ( ),I( , )22= 2 2lnL( , |x) =n 2,I( , )12= 2 lnL( , |x) = n1 .This give a Fisher information matrixI( , ) =n(d2d 2ln ( ) 1 1 2)I( , ) = 500( ).NB. 1( ) =d2ln ( )/d 2is known as the trigamma function and is called in / 10 Fisher InformationExampleGamma DistributionThe inverse matrixI( , ) 1=1500( ).Thus,Var( ) 10 5 ( ) this with the method of moments estimators Estimate the correlation ( , ).
6 9 / 10 Fisher InformationExampleGamma DistributionalphabetaloglikeliFigure: The log- Likelihood surface. The domainis and 5 : Graphs of vertical slices through thelog- Likelihood function surface through theMLE. (top) = (bottom) = / 10
