Lecture 5 - Estimation Elementary Multivariate Statistical...

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Elementary Multivariate Statistical Inference Estimation II.B. Estimation When Data Are Normal 1. The summary statistics X and S are sufficient . i.e., there is no loss of information when you reduce the data to X and S . Warning : There may be serious loss of information if the (normality) assumption failed! 2. Distribution of X and S . (a) X and S are independent. (b) X N p ( μ , Σ /n ) . (c) The multivariate SS = ( n - 1) S = n i =1 ( X i - X )( X i - X ) 0 has the Wishart distribution with ( n - 1) degrees of freedom. This is a generalization of the χ 2 distribution to p dimensions. 3. Maximum Likelihood Estimator (MLE) (a) Definition : The maximum likelihood estimator of an unknown parameter θ is the value of θ that makes the data most likely to occur. (b) MLE of μ is X and MLE of Σ is S n = n - 1 n S . Idea for derivation: likelihood function = joint p.d.f. of X 1 , . . . , X n which is (2 π ) - np 2 | Σ | - n 2 exp ( - 1 2 n X i =1 ( x i - μ ) 0 Σ - 1 ( x i - μ ) ) . Now, find the values of μ and Σ that maximize the above function. Hence, get MLE’s,
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Unformatted text preview: b μ = X , and c Σ = S n . (c) Invariance property of MLE’s i. Idea : The MLE of a relationship h ( μ , Σ ) is h ( X , S n ) . ii. Example : Let SS ij be the sum of squares between the i th and the j th coordinates. We know c Σ = S n = 1 n SS 11 SS 12 ··· SS 1 p SS 21 SS 22 ··· SS 2 p . . . . . . . . . . . . SS p 1 SS p 2 ··· SS pp = ˆ σ 11 ˆ σ 12 ··· ˆ σ 1 p ˆ σ 21 ˆ σ 22 ··· ˆ σ 2 p . . . . . . . . . . . . ˆ σ p 1 ˆ σ p 2 ··· ˆ σ pp . So ˆ ρ ij = ˆ σ ij √ ˆ σ ii q ˆ σ jj = SS ij √ SS ii q SS jj = r ij . Fact : In practice S n and S are usually quite close and so S is used to estimate Σ ....
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  • Fall '08
  • Staff
  • Statistics, Normal Distribution, Maximum likelihood, Estimation theory, maximum likelihood estimator, mle, multivariate statistical inference

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