**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 ....

View
Full Document

- Fall '08
- Staff
- Statistics, Normal Distribution, Maximum likelihood, Estimation theory, maximum likelihood estimator, mle, multivariate statistical inference