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S231 Lecture 4 Cyntha

# For continuous distributions p y y is unsuitable as

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Unformatted text preview: elihood function similarly but with the p.f. P (Y = y; ) replaced by the p.d.f. evaluated at the observed values. For independent observations Yi , i = 1, 2, ..., n from the same p.d.f. f (y ; ), the joint p.d.f. of (Y1 , Y2 , ..., Yn ), i =1 f (yi ; ) is used for the likelihood function. n For n independent observations y1 , y2 , ..., yn from a continuous p.d.f. f (y ; ), we de...ne the likelihood function as L ( ) = L (; y) = f (yi ; ) i =1 Statistics and Actuarial Science () Model Fitting, Estimation and Checking n for 2 . Jan 2013 23 / 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 > 0, where > 0. y / A random sample of light bulbs is tested and the lifetimes y1 , . . . , yn are observed 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 > 0, where > 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 i =1 yi / n for > 0. 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 > 0, where > 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 > 0. Log-likelihood `( ) = n log i =1 yi for > 0. 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 pop...
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