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Class Notes in Statistics and Econometrics Part 5

CHAPTER 9 Random Matrices. The step from random vectors to random matrices (and higher order random arrays) is not as big as the step from individual random variables to random vectors. We will first give a few quite trivial verifications that the expected value operator is indeed a linear operator | CHAPTER 9 Random Matrices The step from random vectors to random matrices and higher order random arrays is not as big as the step from individual random variables to random vectors. We will first give a few quite trivial verifications that the expected value operator is indeed a linear operator and them make some not quite as trivial observations about the expected values and higher moments of quadratic forms. 9.1. Linearity of Expected Values Definition 9.1.1. Let Z be a random matrix with elements zj. Then E Z is the matrix with elements E zij . 245 246 9. RANDOM MATRICES X E X Y E Y THEOREM 9.1.2. If A B and C are constant matrices then E AZB C A E Z B C. Proof by multiplying out. Theorem 9.1.3. E ZT E Z T E tr Z trE Z . Theorem 9.1.4. For partitioned matrices E Special cases If C is a constant then E C C E AX BY A E X B E Y and E a X b Y a X b E Y . If X and Y are random matrices then the covariance of these two matrices is a four-way array containing the covariances of all elements of X with all elements of Y . Certain conventions are necessary to arrange this four-way array in a twodimensional scheme that can be written on a sheet of paper. Before we develop those we will first define the covariance matrix for two random vectors. Definition 9.1.5. The covariance matrix of two random vectors is defined as 9.1.1 C x y E x - E x y - E y T . Theorem 9.1.6. C x y E xyT - E x E y T. Theorem 9.1.7. C Ax b Cy d AC x y CT. 9.1. LINEARITY OF EXPECTED VALUES 247 PROBLEM 152. Prove theorem 9.1.7 Theorem 9.1.8. C x y u 1 C x u C x v V J C y u C y v _ Special case C Ax By Cu Dv AC x u CT AC x v DT BC y u CT BC y v DT. To show this express each of the arguments as a partitioned matrix then use theorem 9.1.7. Definition 9.1.9. V x C x x is called the dispersion matrix. It follows from theorem 9.1.8 that 9.1.2 V x var xi cov x2 X1 . . COv xi X2 var x2 . . . cov xi xn COv x2 X . . . cov xn X1 . cov xn x2 . var xn Theorem 9.1.10. V Ax AV x AT. From this follows that V x is .