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

Index 125always nonnegative definite 10of the prediction error 32, 78partial covariance 26crazy, still vCronbach’s alpha 118Ddata matrixX y D .X 0 W y/ 6best rank-k approximation 79keeping data as a theoretical distribution 8decomposition see eigenvalue decomposition,see full rank decomposition, seesingular value decompositionHartwig–Spindelböck 112derivatives, matrix derivatives 120determinant† 22 25Laplace expansion 86recursive decomposition of det.S/ 34recursive decomposition of det.†/ 34Schur complement 87DFBETA i 45diagonalizability 103discrete uniform distribution 23discriminant function 79disjointnessC .A/ \ C .B/ D f0g, several conditions84C .VH/ \ C .VM/ 64C .X 0 .1/ / \ C .X0 .2/ / 47C .X/ \ C .VM/ 53C .X 1 / \ C .X 2 / 12estimability in outlier testing 46distance see Cook’s distance, see Mahalanobisdistance, see statisticaldistancedistribution see Hotelling’s T 2 , see normaldistributionF -distribution 28t-distribution 28Bernoulli 23binomial 23chi-squared 27, 28discrete uniform 23Hotelling’s T 2 29of MHLN 2 .x; ; †/ 27of y 0 .H J/y 28of y 0 .I H/y 28of z 0 Az 27, 28of z 0 † 1 z 27Wishart-distribution 28, 77dominant eigenvalue 101, 112Draper, Norman R. vDurbin–Watson test statistic 36Dürer’s Melencolia I 113EEckart–Young theoremnon-square matrix 111symmetric matrix 105efficiency of OLSE see Watson efficiency,Bloomfield–Watson efficiency, Rao’sefficiencyeigenvalue, dominant 101, 112eigenvaluesdefinition 100.; w/ as an eigenpair for .A; B/ 106ch.† 22 / 25ch.A; B/ D ch.B 1 A/ 106ch.A 22 / 100ch 1 .A; B/ D ch 1 .B C A/ 108jA Ij D 0 100jA Bj D 0 105nzch.A; B/ D nzch.AB / 108nzch.AB/ D nzch.BA/ 103algebraic multiplicity 103characteristic polynomial 100eigenspace 103geometric multiplicity 103intraclass correlation 101, 102of ˛2I C a1 0 C 1a 0 105of cov.X Qˇ/ 108of P A P B 82, 96, 97of P K P L P F 82spectral radius 101elementary column/row operation 103equalityof BLUPs of y f under two models 71of BLUPs under two mixed models 74of OLSE and BLUE, several conditions 58of OLSE and BLUE, subject toC .U/ C .X/ 60of the BLUEs under two models 59of the OLSE and BLUE of ˇ2 61Oˇ.M .i/ / D Qˇ.M .i/ / 66Oˇ2.M 12 / D Qˇ2.M 12 / 61Qˇ2.M 12 / D Qˇ2. M 12 / 67BLUP.y N f / D BLUE.X f ˇ/ 70BLUP.y f / D OLSE.X f ˇ/ 70SSE.V / D SSE.I / 57X Oˇ D X Qˇ 58Ay D By 54P XIV1 D P XIV2 59estimability