26GLM - -Markovlinearmodel y = X β ∼ N,σ 2 I error...

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Unformatted text preview: GENERALIZEDLINEARMODELS ConsiderthenormaltheoryGauss-Markovlinearmodel y = X β + , ∼ N ( ,σ 2 I ) . Doesnothavetobewrittenasfunction+error Couldspecifydistributionandmodel(s)foritsparameters i.e., y i ∼ N ( μ i ,σ 2 ) , where μ i = X i β forall i = 1 ,..., n and y 1 ,..., y n independent. Thisisoneexampleofageneralizedlinearmodel. HereisanotherexampleofaGLM: y i ∼ Bernoulli ( π i ) ,where π i = exp ( X i β ) 1 + exp ( X i β ) forall i = 1 ,..., n and y 1 ,..., y n independent. c 2011Dept.Statistics(IowaStateUniversity) Stat511section26 1/16 Ineachexample,allresponsesareindependentandeach responseisadrawfromonetypeofdistributionwhose parametersmaydependonexplanatoryvariablesthrougha knownfunctionofalinearpredictor X i β . ThenormalandBernouliimodels(andmanyothers)arespecial casesofageneralizedlinearmodel. Thesearemodelswhere: Theparametersarespecifiedfunctionsof X β Thedistributionisintheexponentialscalefamily i.e., y i hasdensity(orp.m.f.) exp η ( θ ) T ( y i ) − b ( θ ) a ( φ ) + c ( y i ,φ ) (1) where η () , T () , a () , b () ,and c () areknownfunctionsand θ isa vectorofunknownparametersdependingon X β and φ iseithera knownorunknownparameter. Exponentialfamily/exponentialclassis(1)withoutthe a ( φ ) a ( φ ) includes“overdispersed”distributionsinthefamily c 2011Dept.Statistics(IowaStateUniversity) Stat511section26 2/16 Forexample,thepdfforanormaldistributioncanbewrittenas: exp − 1 2 σ 2 y 2 i + μ 2 σ 2 y i − μ 2 2 σ 2 − 1 2 log ( 2 πσ 2 ) fromwhich: η ( θ )= ( μ σ 2 , − 1 2 σ 2 ) and T ( y i )= ( y i , y 2 i ) Thisfamilyincludesmanycommondistributions: Distribution η ( θ ) T ( y i ) a ( φ ) Normal ( μ σ 2 , − 1 2 σ 2 ) ( y i , y 2 i ) 1 Bernoulii log π 1 − π y i 1 Poisson log λ y i 1 Overdisp.Poisson log λ y i φ Gamma ( − 1 θ , ( k − 1 ) ) ( y i , log y i ) 1 c 2011Dept.Statistics(IowaStateUniversity) Stat511section26 3/16 Alotofstattheoryresultsfollowimmediatelyfromtheexponential form e.g. T ( y i ) isthevectorofsufficientstatistics Alotoftheoryismuchnicerwhendistributionparameterizedin termsof η ( θ ) insteadof...
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26GLM - -Markovlinearmodel y = X β ∼ N,σ 2 I error...

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