21MLE(1) - c 2011Dept.Statistics(IowaStateUniversity...

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Unformatted text preview: MAXIMUMLIKELIHOODandREMLESTIMATION INTHEGENERALLINEARMODEL c 2011Dept.Statistics(IowaStateUniversity) Stat511section21 1/23 Suppose f ( w | θ ) istheprobabilitydensityfunction ( pdf ) or probabilitymassfunction ( pmf ) ofarandomvector w ,where θ isa k × 1 vectorofparameters. Givenavalueoftheparametervector θ , f ( w | θ ) isareal-valued functionof w . Thelikelihoodfunction L ( θ | w )= f ( w | θ ) isareal-valuedfunctionof θ foragivenvalueof w . L ( θ | w ) isnotapdf. θ L ( θ | w ) d θ = 1 . Foranypotentialobservedvalue w ,define ˆ θ ( w ) tobeaparameter valueatwhich L ( θ | w ) attainsitsmaximumvalue. ˆ θ ( w ) isa maximumlikelihoodestimator (MLE)of θ . InvariancepropertyofMLES:TheMLEofafunctionof θ ,say g ( θ ) ,isthefunctionevaluatedattheMLEof θ : g ( θ ) = g ( ˆ θ ) c 2011Dept.Statistics(IowaStateUniversity) Stat511section21 2/23 Oftenmuchmoreconvenienttoworkwith l ( θ | w )= log L ( w | θ ) . If l ( θ | w ) isdifferentiable,candidatesfortheMLEof θ canbefound byequatingthescorefunction ∂ l ( θ | w ) ∂ θ ≡ ⎡ ⎢⎢ ⎣ ∂ l ( θ | w ) ∂ θ .. . ∂ l ( θ | w ) ∂ θ k ⎤ ⎥⎥ ⎦ to andsolvingfor θ Thescoreequationsare ∂ l ( θ | w ) ∂ θ = ⇒ ∂ l ( θ | w ) ∂θ j = ∀ j = 1 ,..., k OnestrategyforobtaininganMLEistofindsolution(s)ofthe scoreequationsandverifythatatleastonesuchsolution maximizes l ( θ | w ) . Ifthesolution(s)tothescoreequationslieoutsidetheappropriate parameterspace,theyarenotMLE’s c 2011Dept.Statistics(IowaStateUniversity) Stat511section21 3/23 Example:NormalTheoryGauss-MarkovLinearModel y n × 1 = X n × p β p × 1 + n × 1 ∼ N ( ,σ 2 I ) θ ( p + 1 ) × 1 = β σ 2 f ( y | θ )= exp { − 1 2 ( y − x β ) ( σ 2 I ) − 1 ( y − x β ) } ( 2 π ) n / 2 | σ 2 I | 1 / 2 = 1 ( 2 πσ 2 ) n / 2 exp − 1 2 σ 2 ( y − x β ) ( y − x β ) l ( θ | y )= − n 2 log2 πσ 2 − 1 2 σ 2 ( y − x β ) ( y − x β ) Thescorefunctionis ∂ l ( θ | y ) ∂ θ = ∂ l ( θ | y ) ∂ β ∂ l ( θ | y ) ∂σ 2 = 1 σ 2 ( x y − x x β ) ( y − x β ) ( y − x β ) 2 σ 4 − n 2 σ 2 Thescoreequationsare ∂ l ( θ | y ) ∂ θ = ⇔ x x β = x y σ 2 = ( y − x β ) ( y − x β ) n c 2011Dept.Statistics(IowaStateUniversity) Stat511section21 4/23 Anysolutionliesintheappropriateparameterspace: β ∈ IR p , σ 2 ∈ IR + . Asolutiontothescoreequationsis ˆ β ( y − x ˆ β ) ( y − x ˆ β ) n , where ˆ β isasolutiontothenormalequations Needtoshowthisisamaximumofthelikelihoodfunction Wealreadyknowthatanysolutiontothenormalequations, β = ˆ β ,minimizes ( y − X β ) ( y − X β ) for β ∈ IR p . Thus, ∀ σ 2 > , l ˆ β σ 2 | y ≥ l β σ 2 | y ∀...
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21MLE(1) - c 2011Dept.Statistics(IowaStateUniversity...

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