BIOS 2083 Linear Models Abdus S. Wahed. Chapter 2 84

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1 Chapter 2 84

2 Chapter 3 Random Vectors and Multivariate Normal Distributions 3.1 Random vectors Definition Random vector. Random vectors are vectors of random variables. For instance, X = X 1 X 2. X n, where each element represent a random variable, is a random vector. Definition Mean and covariance matrix of a random vector. The mean (expectation) and covariance matrix of a random vector X is de- 85

3 fined as follows: E [X 1 ] E [X E [X] = 2 ],. E [X n ] and cov(x) = E [{X ] E (X)}{X E (X)} T σ1 2 σ σ 1n σ = 21 σ σ 2n,.... σ n1 σ n2... σ 2 n (3.1.1) where σ 2 j = var(x j)andσ jk = cov(x j, X k )forj, k =1, 2,...,n. Properties of Mean and Covariance. 1. If X and Y are random vectors and A, B, C and D are constant matrices, then E [AXB + CY + D] =AE [X] B + CE[Y]+D. (3.1.2) Proof. Left as an exercise. Chapter 3 86

4 2. For any random vector X, the covariance matrix cov(x) is symmetric. Proof. Left as an exercise. 3. If X j,j =1, 2,...,n are independent random variables, then cov(x) = diag(σj 2,j =1, 2,...,n). Proof. Left as an exercise. 4. cov(x + a) = cov(x) for a constant vector a. Proof. Left as an exercise. Properties of Mean and Covariance (cont.) 5. cov(ax) =Acov(X)A T for a constant matrix A. Proof. Left as an exercise. 6. cov(x) is positive semi-definite. Proof. Left as an exercise. 7. cov(x) =E[XX T ] E[X] {E[X]} T. Proof. Left as an exercise. Chapter 3 87

5 Definition Correlation Matrix. A correlation matrix of a vector of random variable X is defined as the matrix of pairwise correlations between the elements of X. Explicitly, corr(x) = 1 ρ ρ 1n ρ ρ 2n.... ρ n1 ρ n2... 1, (3.1.3) where ρ jk = corr(x j, X k )=σ jk /(σ j σ k ),j,k =1, 2,...,n. Example If only successive random variables in the random vector X are correlated and have the same correlation ρ, then the correlation matrix corr(x) isgivenby corr(x) = 1 ρ ρ 1 ρ ρ , (3.1.4) Example If every pair of random variables in the random vector X Chapter 3 88

6 have the same correlation ρ, then the correlation matrix corr(x) isgivenby corr(x) = 1 ρ ρ... ρ ρ 1 ρ... ρ ρ ρ 1... ρ,..... (3.1.5) ρ ρ ρ... 1 and the random variables are said to be exchangeable. 3.2 Multivariate Normal Distribution Definition Multivariate Normal Distribution. A random vector X =(X 1, X 2,...,X n ) T is said to follow a multivariate normal distribution with mean μ and covariance matrix Σ if X canbeexpressedas X = AZ + μ, where Σ = AA T and Z =(Z 1, Z 2,...,Z n )withz i,i=1, 2,...,n iid N(0, 1) variables. Definition Multivariate Normal Distribution. A random vector X =(X 1, X 2,...,X n ) T is said to follow a multivariate normal distribution with mean μ and a positive definite covariance matrix Σ if X has the density [ 1 f X (x) = (2π) n/2 Σ 1/2exp 1 ] 2 (x μ)t Σ 1 (x μ) (3.2.1) Chapter 3 89

7 BIOS 2083 Linear Models Abdus S. Wahed 0.25 Bivariate normal distribution with mean (0, 0) T and covariance matrix Probability Density x x Properties 1. Moment generating function of a N(μ, Σ) random variable X is given by { M X (t) =exp μ T t + 1 } 2 tt Σt. (3.2.2) Chapter 3 90

8 2. E(X) =μ and cov(x) =Σ. 3. If X 1, X 2,...,X n are i.i.d N(0, 1) random variables, then their joint distribution can be characterized by X =(X 1, X 2,...,X n ) T N(0, I n ). 4. X N n (μ, Σ) if and only if all non-zero linear combinations of the components of X are normally distributed. Linear transformation 5. If X N n (μ, Σ) anda m n is a constant matrix of rank m, theny = Ax N p (Aμ, AΣA T ). Proof. Use definition or property 1 above. Orthogonal linear transformation 6. If X N n (μ, I n )anda n n is an orthogonal matrix and Σ = I n,then Y = Ax N n (Aμ, I n ). Marginal and Conditional distributions Suppose X is N n (μ, Σ) andx is partitioned as follows, X = X 1, Chapter 3 91 X 2

9 where X 1 is of dimension p 1 andx 2 is of dimension n p 1. Suppose the corresponding partitions for μ and Σ are given by μ = μ 1, and Σ = Σ 11 Σ 12 μ 2 Σ 21 Σ 22 respectively. Then, 7. Marginal distribution. X 1 is multivariate normal - N p (μ 1, Σ 11 ). Proof. Use the result from property 5 above. 8. Conditional distribution. The distribution of X 1 X 2 is p-variate normal - N p (μ 1 2, Σ 1 2 ), where, and provided Σ is positive definite. μ 1 2 = μ 1 + Σ 12 Σ 1 22 (X 2 μ 2 ), Σ 1 2 = Σ 11 Σ 12 Σ 1 22 Σ 21, Proof. See Result , page 156 (Ravishanker and Dey). Uncorrelated implies independence for multivariate normal random variables 9. If X, μ, andσ are partitioned as above, then X 1 and X 2 are independent if and only if Σ 12 =0=Σ T 21. Chapter 3 92

10 Proof. We will use m.g.f to prove this result. Two random vectors X 1 and X 2 are independent iff M (X1,X 2 )(t 1,t 2 )=M X1 (t 1 )M X2 (t 2 ). 3.3 Non-central distributions We will start with the standard chi-square distribution. Definition Chi-square distribution. If X 1,X 2,...,X n be n independent N(0, 1) variables, then the distribution of n i=1 X2 i is χ 2 n (ch-square with degrees of freedom n). χ 2 n-distribution is a special case of gamma distribution when the scale parameter is set to 1/2 and the shape parameter is set to be n/2. That is, the density of χ 2 n is given by f χ 2 n (x) = (1/2)n/2 Γ(n/2) e x/2 x n/2 1, x 0; n =1, 2,...,. (3.3.1) Example The distribution of (n 1)S 2 /σ 2,whereS 2 = n i=1 (X i X) 2 /(n 1) is the sample variance of a random sample of size n from a normal distribution with mean μ and variance σ 2, follows a χ 2 n 1. Chapter 3 93

11 The moment generating function of a chi-square distribution with n d.f. is given by M χ 2 n (t) =(1 2t) n/2,t<1/2. (3.3.2) The m.g.f (3.3.2) shows that the sum of two independent ch-square random variables is also a ch-square. Therefore, differences of sequantial sums of squares of independent normal random variables will be distributed independently as chi-squares. Theorem If X N n (μ, Σ) and Σ is positive definite, then (X μ) T Σ 1 (X μ) χ 2 n. (3.3.3) Proof. Since Σ is positive definite, there exists a non-singular A n n such that Σ = AA T (Cholesky decomposition). Then, by definition of multivariate normal distribution, X = AZ + μ, where Z is a random sample from a N(0, 1) distribution. Now, Definition Non-central chi-square distribution. Suppose X s are as in Definition (3.3.1) except that each X i has mean μ i, i =1, 2,...,n. Equivalently, suppose, X =(X 1,...,X n ) T be a random vector distributed as N n (μ, I n ), where μ =(μ 1,...,μ n ) T. Then the distribution of n i=1 X2 i = X T X is referred to as non-central chi-square with d.f. n and non-centrality Chapter 3 94

12 λ= λ= λ= λ=6 λ=8 λ= Figure 3.1: Non-central chi-square densities with df 5 and non-centrality parameter λ. parameter λ = n i=1 μ2 i /2= 1 2 μt μ. The density of such a non-central chisquare variable χ 2 n (λ) can be written as a infinite poisson mixture of central chi-square densities as follows: e λ λ j (1/2) (n+2j)/2 f χ 2 n (λ)(x) = j! Γ((n +2j)/2) e x/2 x (n+2j)/2 1. (3.3.4) j=1 Properties 1. The moment generating function of a non-central chi-square variable χ 2 n (λ) isgivenby M χ 2 n (n,λ)(t) =(1 2t) n/2 exp { } 2λt,t<1/2. (3.3.5) 1 2t Chapter 3 95

13 2. E [ χ 2 n (λ)] = n +2λ. 3. Var [ χ 2 n(λ) ] =2(n +4λ). 4. χ 2 n(0) χ 2 n. 5. For a given constant c, (a) P (χ 2 n (λ) >c) is an increasing function of λ. (b) P (χ 2 n(λ) >c) P (χ 2 n >c). Theorem If X N n (μ, Σ) and Σ is positive definite, then X T Σ 1 X χ 2 n (λ = μt Σ 1 μ/2). (3.3.6) Proof. Since Σ is positive definite, there exists a non-singular matrix A n n such that Σ = AA T (Cholesky decomposition). Define, Y = {A T } 1 X. Then, Definition Non-central F -distribution. If U 1 χ 2 n 1 (λ) andu 2 χ 2 n 2 and U 1 and U 2 are independent, then, the distribution of F = U 1/n 1 U 2 /n 2 (3.3.7) is referred to as non-central F -distribution with df n 1 and n 2, and noncentrality parameter λ. Chapter 3 96

14 λ=0 0.5 λ= λ=4 λ=6 λ=8 λ= Figure 3.2: Non-central F-densities with df 5 and 15 and non-centrality parameter λ. Chapter 3 97

15 λ=0 λ=2 λ=4 λ=6 λ=8 0.1 λ= Figure 3.3: Non-central t-densities with df 5 and non-centrality parameter λ. Definition Non-central t-distribution. If U 1 N(λ, 1) and U 2 χ 2 n and U 1 and U 2 are independent, then, the distribution of T = U 1 U2 /n (3.3.8) is referred to as non-central t-distribution with df n and non-centrality parameter λ. Chapter 3 98

16 3.4 Distribution of quadratic forms Caution: We assume that our matrix of quadratic form is symmetric. Lemma If A n n is symmetric and idempotent with rank r, thenr of its eigenvalues are exactly equal to 1 and n r are equal to zero. Proof. Use spectral decomposition theorem. (See Result on page 51 of Ravishanker and Dey). Theorem Let X N n (0, I n ). The quadratic form X T AX χ 2 r iff A is idempotent with rank(a) =r. Proof. Let A be (symmetric) idempotent matrix of rank r. Then, by spectral decomposition theorem, there exists an orthogonal matrix P such that P T AP = Λ = I r (3.4.1) Define Y = P T X = PT 1 X P T 2 X = Y 1 Y 2, sothatp T 1 P 1 = I r. Thus, X = Chapter 3 99

17 PY and Y 1 N r (0, I r ). Now, X T Ax = (PY) T APY = Y T I r 0 Y 0 0 = Y1 T Y 1 χ 2 r. (3.4.2) Now suppose X T AX χ 2 r. This means that the moment generating function of X T AX is given by M XT AX(t) =(1 2t) r/2. (3.4.3) But, one can calculate the m.g.f. of X T AX directly using the multivariate normal density as M XT AX(t) = E [ exp { (X T AX)t }] = exp { (X T AX)t } f X (x)dx = exp { (X T AX)t } [ 1 (2π) n/2exp 1 ] 2 xt x dx [ 1 = (2π) n/2exp 1 ] 2 xt (I n 2tA)x dx = I n 2tA 1/2 n = (1 2tλ i ) 1/2. (3.4.4) i=1 Chapter 3 100

18 Equate (3.4.3) and (3.4.4) to obtain the desired result. Theorem Let X N n (μ, Σ) where Σ is positive definite. The quadratic form X T AX χ 2 r(λ) where λ = μ T Aμ/2, iffaσ is idempotent with rank(aσ) = r. Proof. Omitted. Theorem Independence of two quadratic forms. Let X N n (μ, Σ) where Σ is positive definite. The two quadratic forms X T AX and X T BX are independent if and only if AΣB =0=BΣA. (3.4.5) Proof. Omitted. Remark Note that in the above theorem, the two quadratic forms need not have a chi-square distribution. When they are, the theorem is referred to as Craig s theorem. Theorem Independence of linear and quadratic forms. Let X N n (μ, Σ) where Σ is positive definite. The quadratic form X T AX and the linear form BX are independently distributed if and only if BΣA =0. (3.4.6) Proof. Omitted. Chapter 3 101

19 Remark Note that in the above theorem, the quadratic form need not have a chi-square distribution. Example Independence of sample mean and sample variance. Suppose X N n (0, I n ). Then X = n i=1 X i/n = 1 T X/n and SX 2 = n i=1 (X i X) 2 /(n 1) are independently distributed. Proof. Theorem Let X N n (μ, Σ). Then E [ X T AX ] = μ T Aμ + trace(aσ). (3.4.7) Remark Note that in the above theorem, the quadratic form need not have a chi-square distribution. Proof. Theorem Fisher-Cochran theorem. Suppose X N n (μ, I n ).Let Q j = X T A j X,j =1, 2,...,k be k quadratic forms with rank(a j )=r j such that X T X = k j=1 Q j. Then, Q j s are independently distributed as χ 2 r j (λ j ) where λ j = μ T A j μ/2 if and only if k j=1 r j = n. Proof. Omitted. Theorem Generalization of Fisher-Cochran theorem. Suppose X N n (μ, I n ). Let A j,j =1, 2,...,k be kn n symmetric matrices with rank(a j )=r j such that A = k j=1 A j with rank(a) =r. Then, Chapter 3 102

20 1. X T A j X s are independently distributed as χ 2 r j (λ j ) where λ j = μ T A j μ/2, and 2. X T AX χ 2 r (λ) where λ = k j=1 λ j if and only if any one of the following conditions is satisfied. C1. A j Σ is idempotent for all j and A j ΣA k =0for all j<k. C2. A j Σ is idempotent for all j and AΣ is idempotent. C3. A j ΣA k =0for all j<kand AΣ is idempotent. C4. r = k j=1 r j and AΣ is idempotent. C5. the matrices AΣ, A j Σ,j =1, 2,...,k 1 are idempotent and A k Σ is non-negative definite. 3.5 Problems 1. Consider the matrix A = Chapter 3 103

21 (a) Find the rank of this matrix. (b) Find a basis for the null space of A. (c) Find a basis for the column space of A. 2. Let X i,i =1, 2, 3 are independent standard normal random variables. Show that the variance-covariance matrix of the 3-dimensional vector Y, defined as 5X 1 Y = 1.6X 1 1.2X 2, 2X 1 X 2 is not positive definite. 3. Let X = X 1 X 2 X 3 N 3 μ 1 μ 2 μ 3, 1 ρ 0 ρ 1 ρ 0 ρ 1. (a) Find the marginal distribution of X 2. (b) What is the conditional distribution of X 2 given X 1 = x 1 and X 3 = x 3? Under what condition does this distribution coincide with the marginal distribution of X 2? 4. If X N n (μ, Σ), then show that (X μ) T Σ 1 (X μ) χ 2 n. 5. Suppose Y =(Y 1,Y 2,Y 3 ) T be distributed as N 3 (0,σ 2 I 3 ). (a) Consider the quadratic form: Q = (Y 1 Y 2 ) 2 +(Y 2 Y 3 ) 2 +(Y 3 Y 1 ) 2. (3.5.1) 3 Chapter 3 104

22 Write Q as Y T AY where A is symmetric. Is A idempotent? What is the distribution of Q/σ 2? Find E(Q). (b) What is the distribution of L = Y 1 + Y 2 + Y 3? Find E(L) andvar(l). (c) Are Q and L independent? Find E(Q/L 2 ) 6. Write each of the following quadratic forms in X T AX form: (a) 1 6 X X X X 1X X 1X X 2X 3 (b) n X 2,where X =(X 1 + X X n )/n. (c) n i=1 X2 i (d) n i=1 ( Xi X ) 2 (e) 2 i=1 2 j=1 (f) 2 2 i=1 ( Xij X ) 2 i.,where X i. = X i1+x i2 2 ( Xi. X ) 2..,where X.. = X 11+X 12 +X 21 +X (g) 2 ( X1. X.. ) 2 +3 ( X2. X.. ) 2,where X 1. = X 11+X 12 2, X 2. = X 21+X 22 +X 23 3, X.. = 2 X X In each case, determine if A is idempotent. If A is idempotent, find rank(a). 7. Let X N 2 (μ, Σ), where μ = μ 1, andσ= Show that Q 1 = μ (X 1 X 2 ) 2 and Q 2 =(X 1 + X 2 ) 2 are independently distributed. Find the distribution of Q 1, Q 2,and Q 2 3Q Assume that Y N 3 (0,I 3 ). Define Q 1 = Y T AY and Q 2 = Y T BY,where A = 1 1 0, and,b = (3.5.2) Chapter 3 105

23 Are Q 1 and Q 2 independent? Do Q 1 and Q 2 follow χ 2 distribution? 9. Let Y N 3 (0,I 3 ). Let U 1 = Y T A 1 Y, U 2 = Y T A 2 Y,andV = BY where 1/2 1/2 0 1/2 1/2 0 ( A 1 = 1/2 1/2 0,A 2 = 1/2 1/2 0, and,b = ). (a) Are U 1 and U 2 independent? (b) Are U 1 and V independent? (c) Are U 2 and V independent? (d) Find the distribution of V. (e) Find the distribution of U 2 U 1. (Include specific values for any parameters of the distribution.) 10. Suppose X =(X 1,X 2,X 3 ) T N 3 (μ, σ 2 V ), where, μ =(1, 1, 0) T,and V = (a) What is the joint distribution of (X 1,X 3 )? (b) What is the conditional distribution of X 1,X 3 X 2? (c) For what value or values of a does L =(ax X aX 1 X 3 )/σ 2 follow a chi-square distribution? (d) Find the value of b for which L in (c) and M = X 1 + bx 3 are independently distributed. Chapter 3 106

24 11. Suppose X N 3 (μ, Σ), where μ = μ 1 μ 2, and Σ = σ σ μ σ 2 3 Find the distribution of Q = 3 Xi 2 i=1. Express the parameters of its distribution in σi 2 terms of μ i and σi 2,i=1, 2, 3. What is the variance of Q? 12. Suppose X N(0, 1) and Y = UX,whereU follows a uniform distribution on the discrete space { 1, 1} independently of X. (a) Find E(Y )andcov(x, Y ). (b) Show that Y and X are not independent. 13. Suppose X N 4 (μ, I 4 ), where X = X 11 X 12 X 21,μ= α + a 1 α + a 1 α + a 2. X 22 α + a 2 (a) Find the distribution of E = 2 2 ( i=1 j=1 Xij X ) 2,where i. Xi. = X i1+x i2. 2 (b) Find the distribution of Q =2 2 ( i=1 Xi. X ) 2,where.. X.. = X 11+X 12 +X 21 +X (c) Use Fisher-Cochran theorem to prove thate and Q are independently distributed. (d) What is the distribution of Q/E? Chapter 3 107