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Projection Matrix .

Chi-Square Distribution Many important distributions can be derived as transformation of multivariate normal random vectors, including the chi-square, the student t, and the F. In this section we introduce the chisquare distribution. Let X  N (0; Ir) be multivariate standard normal and deÖne Q = X0X. The distribution of Q is called chi-square with r degrees of freedom, written as Q   2 r . The mean and variance of Q   2 r are r and 2r, respectively. (See Exercise 5.11.) The chi-square distribution function is frequently used for inference (critical values and pvalues). In practice these calculations are performed numerically by statistical software, but for completeness we provide the density function. Theorem 5.6 The density of  2 r is f(x) = 1 2 r=2 r 2 x r=21 e x=2 ; x > 0 (5.2) where (t) = R 1 0 u t1 e udu is the gamma function (Section 5.19). The proof is presented in Section 5.20. Plots of the chi-square density function for r = 2; 3, 4, and 6 are displayed in Fig CHAPTER 5. NORMAL REGRESSION AND MAXIMUM LIKELIHOOD 148 0 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 f(x) χ2 2 χ3 2 χ4 2 χ6 2 0 1 2 3 4 5 6 7 8 9 10 Figure 5.2: Chi-Square Density 5.4 Student t Distribution Let Z  N (0; 1) and Q   2 r be independent, and deÖne T = Z=p Q=r. The distribution of T is called the student t with r degrees of freedom, and is written T  tr. Like the chi-square, the distribution only depends on the degree of freedom parameter r. Theorem 5.7 The density of T is f (x) = r+1 2  p r r 2   1 + x 2 r ( r+1 2 ) ; 1 < x < 1: The proof is presented in Section 5.20. Plots of the student t density function are displayed in Figure 5.3 for r = 1, 2, 5 and 1. The density function of the student t is bell-shaped like the normal density function, but the t has thicker tails. The t distribution has the property that moments below r are Önite, but absolute moments greater than or equal to r are inÖnite. The student t can also be seen as a generalization of the standard normal, for the latter is obtained as the limiting case where r is taken to inÖnity. Theorem 5.8 Let fr(x) be the student t density. As r ! 1, fr(x) ! CHAPTER 5. NORMAL REGRESSION AND MAXIMUM LIKELIHOOD 149 −4 −2 0 2 4 0.0 0.1 0.2 0.3 0.4 f(x) −5 −4 −3 −2 −1 0 1 2 3 4 5 t1 t2 t5 t∞ Figure 5.3: Student t Density The proof is presented in Section 5.20. This means that the t1 distribution equals the standard normal distribution. Another special case of the student t distribution occurs when r = 1 and is known as the Cauchy distribution. T