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Model in Matrix Notation

5 F Distribution Let Qm   2 m and Qr   2 r be independent. The distribution of F = (Qm=m) = (Qr=r) is called the F distribution with degree of freedom parameters m and r, and we write F  Fm;r CHAPTER 5. NORMAL REGRESSION AND MAXIMUM LIKELIHOOD 150 Theorem 5.9 The density of Fm;r is f(x) = m r m=2 x m=21 m+r 2  m 2  r 2  1 + m r x (m+r)=2 ; x > 0: The proof is presented in Section 5.20. Plots of the Fm;r density for m = 2, 3, 6, 8, and r = 10 are displayed in Figure 5.4. 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 f(x) 0 1 2 3 4 5 F2,10 F3,10 F6,10 F8,10 Figure 5.4: F Density If m = 1 then we can write Q1 = Z 2 where Z  N(0; 1), and F = Z 2= (Qr=r) =  Z=p Qr=r2 = T 2 , the square of a student t with r degree of freedom. Thus the F distribution with m = 1 is equal to the squared student t distribution. In this sense the F distribution is a generalization of the student t. As a limiting case, as r ! 1 the F distribution simpliÖes to F ! Qm=m, a normalized  2 m. Thus the F distribution is also a generalization of the  2 m distribution. Theorem 5.10 Let fm;r(x) be the density of mF. As r ! 1, fm;r(x) ! fm(x), the density of  2 m: The proof is presented in Section 5.20. The F distribution was tabulated by Snedecor (1934). He introduced the notation F as the distribution is related to Sir Ronald Fisherís work on the analysis of CHAPTER 5. NORMAL REGRESSION AND MAXIMUM LIKELIHOOD 151 5.6 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