note3(1) - STAT5044: Regression and Anova Inyoung Kim...

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Unformatted text preview: STAT5044: Regression and Anova Inyoung Kim Outline 1 Matrix Expression 2 Linear and quadratic forms 3 Properties of quadratic form 4 Properties of estimates 5 Distributional properties 6 Distributional properties Matrix Expression If we have p variables x i 1 ,..., x ip for the i the observation, variable Observation 1 2 p 1 x 11 x 12 x 1 p 2 x 21 x 22 x 2 p . . . . . . . . . . . . n x n 1 x n 2 x np x 1 x 2 x p Notation: let x 1 , x 2 ,..., x p be the column vectors . Matrix Expression X represents the matrix: element x ij corresponds to the ij th element X = { x ij } which is n p matrix, i = 1 ,..., n and j = 1 ,..., p x j = column vector of the j th variable capital=matrix, bold=vector, not bold=scalar We typically think of vectors as elements of real numbers, i.e., x R p Matrix Special matrices Square matrix: n n or p p Symmetric matrix: X = X t Diagonal matrix: square matrix with zeros except possibly on the diagonal D = d 1 d 2 d 3 d 4 Special matrices Identity matrix: square matrix with ones on the diagonal, 0s off diagonal Special matrices Identity matrix: square matrix with ones on the diagonal, 0s off diagonal I = 1 1 1 1 Special matrices 1 n = 1 1 . . . 1 = 1 J n n = 1 1 1 1 1 1 1 1 . . . . . . . . . . . . 1 1 1 1 J = J nn = Special matrices 1 n = 1 1 . . . 1 = 1 J n p = 1 1 1 1 1 1 1 1 . . . . . . . . . . . . 1 1 1 1 J = J nn = 1 n 1 t n = 11 t Special matrices X = 1 x 1 1 x 2 ....
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note3(1) - STAT5044: Regression and Anova Inyoung Kim...

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