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Least Squares Residuals

Non-Central Chi-Square and F Distributions For some theoretical applications, including the study of the power of statistical tests, it is useful to deÖne a non-central version of the chi-square distribution. When X  N (; Ir) is multivariate normal, we say that Q = X0X has a non-central chi-square distribution, with r degrees of freedom and non-centrality parameter  =  0, and is written as Q   2 r (). The non-central chi-square simpliÖes to the central (conventional) chi-square when  = 0, so that  2 r (0) =  2 r . Theorem 5.11 The density of  2 r () is f(x) = X1 i=0 e =2 i!   2 i fr+2i(x); x > 0 (5.3) where fr+2i(x) is the  2 r+2i density function (5.2). The proof is presented in Section 5.20. Plots of the  2 3 () density for  = 0; 2, 4, and 6 are displayed in Figure 5.5. Interestingly, as can be seen from the formula (5.3), the distribution of  2 r () only depends on the scalar non-centrality parameter , not the entire mean vector . 0.00 0.05 0.10 0.15 0.20 0.25 f(x) 0 2 4 6 8 10 12 14 16 χ3 2 (0) χ3 2 (2) χ3 2 (4) χ3 2 (6) Figure 5.5: Non-Central Chi-Square Density A useful fact about the central and non-central chi-square distributions is that they also can be derived from multivariate normal distributions with general covariance matrices CHAPTER 5. NORMAL REGRESSION AND MAXIMUM LIKELIHOOD 152 Theorem 5.12 If X  N(; A) with A > 0, r r, then X0A1X   2 r (); where  =  0A1. The proof is presented in Section 5.20. In particular, Theorem 5.12 applies to the central chi-squared distribution, so if X  N(0; A) then X0A1X   2 r : Similarly with the non-central chi-square we deÖne the non-central F distribution. If Qm   2 m() and Qr   2 r are independent, then F = (Qm=m) = (Qr=r) is called a non-central F with degree of freedom parameters m and r and non-centrality parameter