slides_4_ranvecs

A 1 0 x 1 a 2 a 1 x 2 a 3 a 1 x 3 a m a 1 x m c this

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a 1 0, X 1 a 2 / a 1 X 2 a 3 / a 1 X 3 ... a m / a 1 X m c . This is a degenerate situation that we often rule out, unless it holds by construction. 78

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Special Structures An important special case for Var X is a diagonal structure, Var X 1 2 0 0 0 2 2 0 0 0 m 2 This holds if and only if the X j : j 1,..., m are pairwise uncorrelated , so Cov X i , X j 0 all i j . For reasons having to do with the multivariate normal distribution (later), this is also called a spherical variance matrix . 79
We sometimes write Var X diag 1 2 , 2 2 ,..., m 2 as a shorthand. When Var X is spherical, variances of linear combinations are easy to compute because there are no covariance terms. Var j 1 m a j X j j 1 m a j 2 j 2 An important special case is Var j 1 m X j j 1 m Var X j , so the variance of the sum is the sum of the variance for pairwise uncorrelated RVs. 80

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A further specialization is when j 2 2 for all j . Then Var X 2 0 0 0 2 0 0 0 2 2 I m , where I m is the m m identity matrix. When Var X 2 I m we say that the variance matrix is a scalar variance matrix . 81
If Var X has a scalar structure then choosing a 1,1,...,1 gives Var j 1 m X j j 1 m 2 m 2 EXAMPLE : We can easily find Var Y , where Y ~ Binomial n , p , by writing Y X 1 ... X n , where the X j are independent Bernoulli p random variables. Because independence implies zero correlation (when second moments are finite) and Var X i p 1 p , Var Y Var i 1 n X i np 1 p 82

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Correlation Matrix The notion of a correlation matrix is also important. Let ij Corr X i , X j for i j . Then Corr X 1 12 1 m 12 1 2 m 1 m 2 m 1 (Note that Corr X j , X j 1 for all j .) The case Corr X I m corresponds to the RVs being pairwise uncorrelated. 83
It is often useful to write the variance-covariance matrix in terms of the correlation matrix and a diagonal matrix that includes only the variances. Define D 1 2 0 0 0 2 2 0 0 0 m 2 ; generally D is not equal to Var X . 84

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The matrix square root, D 1/2 , is sensibly defined as the diagonal matrix D 1/2 1 0 0 0 2 0 0 0 m (Because D 1/2 D 1/2 D .) Straightforward multiplication shows that Var X D 1/2 Corr X  D 1/2 85
An important structure in various branches of applied statistics is the equicorrelation structure , where all pairwise correlations are the same: Corr X 1 1 1 where 1 1. Notice this allows the variances to be the same or different.

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