S231 Lecture 4 Cyntha

Statistics and actuarial science model fitting

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Unformatted text preview: ulation and that Y Exponential( ), having p.d.f. f (y ; ) = 1 e for y &gt; 0, where &gt; 0. y / A random sample of light bulbs is tested and the lifetimes y1 , . . . , yn are observed Likelihood function for is L( ) = n i =1 1 yi / e = 1 n exp 1 i =1 n yi / n for &gt; 0. Log-likelihood `( ) = d `/d = 0 for = 1 n n n log yi = y . i =1 yi for &gt; 0. i =1 Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 24 / 45 Likelihood for an Exponential Population Suppose Y = lifetime of a randomly selected light bulb in a large population and that Y Exponential( ), having p.d.f. f (y ; ) = 1 e for y &gt; 0, where &gt; 0. y / A random sample of light bulbs is tested and the lifetimes y1 , . . . , yn are observed Likelihood function for is L( ) = n i =1 1 yi / e = 1 n exp 1 i =1 n yi / n for &gt; 0. Log-likelihood `( ) = d `/d = 0 for = 1 n n n log yi = y . i =1 yi for &gt; 0. i =1 A ...rst derivative test veri...es `( ) has a maximum value at = y and ^ so = y is the m.l. estimate of . Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 24 / 45 Likelihood for a Gaussian Population As an example involving more than one parameter, suppose Y has a h i 1 1 Gaussian distribution with p.d.f. f (y ; , ) = p2 exp )2 2 (y 2 for &lt; y &lt; . Let = (, ). Random sample y1 , ..., yn from f (y ; , ) gives L() = L(, ) = f (yi ; , ) i =1 n n = (2 ) for n/2 exp 1 n (yi 22 i =1 )2 &lt; &lt; , &gt; 0. Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 25 / 45 Relative Likelihood Function for Osteoporosis data Relative Likelihood L ( , ; y1 ,... y n ) L ( , ; y1 ,... y n ) ^ ^ 50 Shades of Grey Finding the Maximum Likelihood Estimates (Gaussian case) The log-likelihood is `() = `(, ) = n log 1 22 i =1 (yi n )2 (n/2) log(2 ). To maximize `(, ) with respect to both parameters and , solve the two equations ` ` and simultaneously. = = 1 n (yi ) = 0 2 i =1 n 1 n + 3 (yi )2 = 0 i =1 Statistics and Actuarial Science () Model Fitting, Estimation and Checking Jan 2013 28 / 45 Finding the Maximum Likelihood E...
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