By Bacco M., Mocellin V.
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In P˝ /X D 0), are called the residuals. y θ/ is called the residual sum of squares or RSS. , Seber and Lee 2003, chapter 10) which we shall not consider here apart from a few properties. We note that yO D P˝ y, where the projection matrix P˝ is usually referred to as the hat matrix. 6(v) in Sect. 7). D P˝ ? / has rank n dimŒ˝ < n, and is therefore a singular matrix. pij /, the diagonal elements pii are called the hat matrix diagonals. 7 Assumptions and Residuals 39 and constant variance, for looking for outliers, and for checking on the linearity of the model.
Sect. In P˝ /θ D 0. θ; v/ with respect to v and θ gives us the equations (cf. y θ/ O D v; θ/ O giving us the same estimates, as expected. References Atiqullah, M. (1962). The estimation of residual variance in quadratically balanced least squares problems and the robustness of the F test. Biometrika, 49, 83–91. Cook, R. , & Weisberg, S. (1982). Residuals and influence in regression. New York: Chapman & Hall. Rao, C. R. (1952). Some theorems on minimum variance estimation. Sankhy¯a, 12, 27–42. Seber, G.
In P! In P! Xr βr C Xp r βp r / P! 4 with ˝ ? replaced by ! In P! /Xp r with rank p r as CŒXp r \ ! In P! /Xp r is non-singular. 9) is established. In P! In P! /Xp and it follows from Eq. In P˝ P! In P! In r P! /, that P! In P! 10) which can be used for a Wald test. 1, and we wish to test Aβ D b, where A is q p of rank q. Let β0 be any solution of Aβ D b, put z D y Xβ0 and let γ D β β0 . Then our original model and hypothesis are equivalent to z D Xγ C ε, where ε is Nn Œ0; 2 In , and ! W Aγ D 0.