3547 part 18 maximum likelihood estimation properties

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Part 18: Maximum Likelihood Estimation Properties of the Maximum Likelihood Estimator We will sketch formal proofs of these results: The log-likelihood function, again The likelihood equation and the information matrix. A linear Taylor series approximation to the first order conditions: g ( ML ) = 0 g ( ) + H( ) ( ML - ) (under regularity, higher order terms will vanish in large samples.) Our usual approach. Large sample behavior of the left and right hand sides is the same. A Proof of consistency . (Property 1) The limiting variance of n( ML - ). We are using the central limit theorem here. Leads to asymptotic normality (Property 2). We will derive the asymptotic variance of the MLE. Estimating the variance of the maximum likelihood estimator. Efficiency (we have not developed the tools to prove this.) The Cramer-Rao lower bound for efficient estimation (an asymptotic version of Gauss-Markov). Invariance . (A VERY handy result.) Coupled with the Slutsky theorem and the delta method, the invariance property makes estimation of nonlinear functions of parameters very easy. ™  36/47
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Part 18: Maximum Likelihood Estimation Regularity Conditions p Deriving the theory for the MLE relies on certain “regularity” conditions for the density. p What they are n 1. logf(.) has three continuous derivatives wrt parameters n 2. Conditions needed to obtain expectations of derivatives are met. (E.g., range of the variable is not a function of the parameters.) n 3. Third derivative has finite expectation. p What they mean n Moment conditions and convergence. We need to obtain expectations of derivatives. n We need to be able to truncate Taylor series. n We will use central limit theorems ™  37/47
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Part 18: Maximum Likelihood Estimation The MLE ™  38/47 ( 29 ( 29 ( 29 ( 29 ( 29 The results center on the first order conditions for the MLE log ˆ = = ˆ Begin with a Taylor series approximation to the first derivatives: ˆ ˆ = + [+ terms o(1/n) that v MLE MLE MLE MLE L - g 0 g 0 g H θ θ θ θ θ θ θ anish] ˆ The derivative at the MLE, , is exactly zero. It is close to zero at the ˆ true , to the extent that is a good estimator of . Rearrange this equation and make use of the Slutsky theo MLE MLE θ θ θ θ ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 1 2 rem ˆ In terms of the original log likelihood ˆ log ( ) log ( ) where and MLE n n MLE i i i i i i i i f f - - = = - - - - = = ∂ ∂ H g H g g H θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ
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Part 18: Maximum Likelihood Estimation Consistency ™  39/47 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 1 ˆ Divide both sums by the sample size. 1 1 1 ˆ = o The approximation is now exact because of the higher order term. As n n n MLE i i i i n n MLE i i i i n n n - = = - = = - - - - + ÷ → ∞ H g H g θ θ θ θ θ θ θ θ ( 29 ( 29 { } ( 29 ( 29 1 1 1 1 , the third term vanishes. The matrices in brackets are sample means that converge to their expectations.
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