21MLE - MAXIMUM LIKELIHOOD and REML ESTIMATION IN THE...

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Unformatted text preview: MAXIMUM LIKELIHOOD and REML ESTIMATION IN THE GENERAL LINEAR MODEL c 2011 Dept. Statistics (Iowa State University) Stat 511 section 21 1 / 23 I Suppose f ( w | θ ) is the probability density function ( pdf ) or probability mass function ( pmf ) of a random vector w , where θ is a k × 1 vector of parameters. I Given a value of the parameter vector θ , f ( w | θ ) is a real-valued function of w . I The likelihood function L ( θ | w ) = f ( w | θ ) is a real-valued function of θ for a given value of w . I L ( θ | w ) is not a pdf. R θ L ( θ | w ) d θ 6 = 1 . I For any potential observed value w , define ˆ θ ( w ) to be a parameter value at which L ( θ | w ) attains its maximum value. ˆ θ ( w ) is a maximum likelihood estimator (MLE) of θ . I Invariance property of MLES: The MLE of a function of θ , say g ( θ ) , is the function evaluated at the MLE of θ : d g ( θ ) = g ( ˆ θ ) c 2011 Dept. Statistics (Iowa State University) Stat 511 section 21 2 / 23 I Often much more convenient to work with l ( θ | w ) = log L ( w | θ ) . I If l ( θ | w ) is differentiable, candidates for the MLE of θ can be found by equating the score function ∂ l ( θ | w ) ∂ θ ≡ ∂ l ( θ | w ) ∂ θ . . . ∂ l ( θ | w ) ∂ θ k to and solving for θ I The score equations are ∂ l ( θ | w ) ∂ θ = ⇒ ∂ l ( θ | w ) ∂θ j = ∀ j = 1 ,..., k I One strategy for obtaining an MLE is to find solution(s) of the score equations and verify that at least one such solution maximizes l ( θ | w ) . I If the solution(s) to the score equations lie outside the appropriate parameter space, they are not MLE’s c 2011 Dept. Statistics (Iowa State University) Stat 511 section 21 3 / 23 I Example: Normal Theory Gauss-Markov Linear Model y |{z} n × 1 = X |{z} n × p β |{z} p × 1 + |{z} n × 1 ∼ N ( ,σ 2 I ) θ |{z} ( 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 log 2 πσ 2- 1 2 σ 2 ( y- x β ) ( y- x β ) I The score function is ∂ 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 # I The score equations are ∂ l ( θ | y ) ∂ θ = ⇔ x x β = x y σ 2 = ( y- x β ) ( y- x β ) n c 2011 Dept. Statistics (Iowa State University) Stat 511 section 21 4 / 23 I Any solution lies in the appropriate parameter space: β ∈ IR p , σ 2 ∈ IR + . I A solution to the score equations is " ˆ β ( y- x ˆ β ) ( y- x ˆ β ) n # , where ˆ β is a solution to the normal equations I Need to show this is a maximum of the likelihood function I We already know that any solution to the normal equations, β = ˆ β , minimizes ( y- X β ) ( y- X β ) for β ∈ IR p ....
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This note was uploaded on 02/11/2012 for the course STAT 511 taught by Professor Staff during the Spring '08 term at Iowa State.

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21MLE - MAXIMUM LIKELIHOOD and REML ESTIMATION IN THE...

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