Formulas Useful for Linear Regression Analysis and Related Matrix ...

130 Indexof det.R/ 89of det.S/ 34of det.†/ 34reduced modelR 2 .M 121 / 21M 121 61M 1H 65AVP 22SSE.M 121 / 21reducible matrix 112reflection 112regression coefficients 14old ones do not change 15standardized 16regression function 27regression towards mean vrelative efficiency of OLSE see Watsonefficiencyresidualy BLP.yI x/ 26, 32e i1:::i 1 34after elimination of X 1 20externally Studentized 43, 46internally Studentized 43of BLUE 55, 56of OLSE 13, 43predicted residual 45scaled 43residual mean square, MSE 19residual sum of squares, SSE 19rotation 111of observations 79SSamuelson’s inequality 118Schur complement 87determinant 87MP-inverse 93rank additivity 87Schur’s triangularization theorem 111Seber, George A. F. v, vishorted matrix 115similarity 103Simon, Paul vsimultaneous diagonalizationby a nonsingular matrix 107by an orthogonal matrix 107of commuting matrices 107singular value decomposition 109skew-symmetric: A 0 D A 1Smith, Harry vsolutionto A.X W VX ? / D .X f W V 21 X ? / 70to A.X W VX ? / D .X f W V 21 X ? / 70to AX D B 91to AXB D C 91to Ax D y 91to A 91to G.X W VM/ D .X W 0/ 53to X 0 Xˇ D X 0 y 13to X.A W B/ D .0 W B/ 98spectral radius 101, 112spectrum see eigenvalues, 100square root of nnd A 101standard error of Oˇi 18statistical distance 9Stigler, Stephen M. vstochastic matrix 113stochastic restrictions 74sufficiencylinear 67linear prediction 72linearly minimal 68sum of products SP xy D t xy 8sum of squareschange in SSE 20, 37, 39change in SSE(V) 41predicted residual 45SS y D t yy 8SSE 19SSE under M 121 21SSE, various representations 20SSR 19SST 19weighted SSE 40, 56sum of squares and cubes of integers 23Survo viSylvester’s inequality 84TtheoremAlbert 114Cochran 28Courant–Fischer 104Eckart–Young 105, 111Frisch–Waugh–Lovell 15, 22, 61Gauss–Markov 53interlacing 103Lehmann–Scheffé 68Perron–Frobenius 112Poincaré separation 104Schur’s triangularization 111Wedderburn–Guttman 87Thomas, Niels Peter viTiritiri Island 122total sum of squares, SST 19