mvnormal.pdf - Basic Multivariate Normal...

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Basic Multivariate Normal Theory [ Prerequisite probability background: Univariate theory of random variables, expectation, vari- ance, covariance, moment generating function, independence and normal distribution. Other requirements: Basic vector-matrix theory, multivariate calculus, multivariate change of vari- able.] 1 Random Vector A random vector U R k is a vector ( U 1 , U 2 , · · · , U k ) T of scalar random variables U i defined over a common probability space. The expectation and variance of U are defined as: E U := E U 1 E U 2 . . . E U k , Var U := E( { U - E U }{ U - E U } T ) = Var U 1 Cov( U 1 , U 2 ) . . . Cov( U 1 , U k ) Cov( U 2 , U 1 ) Var U 2 . . . Cov( U 2 , U k ) . . . . . . . . . . . . Cov( U k , U 1 ) Cov( U k , U 2 ) . . . Var U k . It is easy to check that if X = a + BU where a R p and B is a p × k matrix, then X is a random vector in R p with E X = a + B E U , Var X = B Var( U ) B T . You should also note that if Σ = Var U for some random vector U , then Σ is a k × k non-negative definite matrix, because for any a R k , a T Σ a = Var( a T U ) 0. 2 Multivariate Normal Definition 1 . A random vector U R k is called a normal random vector if for every a R k , a T U is a (one dimensional) normal random variable. Theorem 1. A random vector U R k is a normal random vector if and only if one can write U = m + AZ for some m R k and k × k matrix A where Z = ( Z 1 , · · · , Z k ) T with Z i IID Normal (0 , 1) . Proof. “If part” . Suppose U = m + AZ with m , A and Z as in the statement of the theorem. Then for any a R k , a T U = a T m + a T AU = b 0 + n i =1 b i Z i for some scalars b 0 , b 1 , · · · , b k . But a linear combination of independent (one dimensional) normal variables is another normal, so a T U is a normal variable. “Only if part” Suppose U is a normal random vector. It suffices to show that a V = m + AZ with Z as in the statement of the theorem, and suitably chosen m and A , has the same distribution as U . For any a R k , the moment generating function M U ( a ) of U at a is 1
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E e a T U = E e X where X = a T U is normally distributed by definition. But, by one dimensional normal distribution theory, E e X = e E X + 1 2 Var X = e a T E U + 1 2 a T (Var U ) a = e a T μ + a T Σ a where we denote E U by μ and Var U by Σ. Note that Σ is non-negative definite and thus can be written as Σ = AA T for some k × k matrix A . Write V = μ + AZ where Z = ( Z 1 , · · · , Z k ) T with Z i IID Normal (0 , 1). Then, by the “if part”, V is a normal random vector, and because Z i ’s are IID with mean 0 and variance 1, E V = μ and Var V = Σ. So by above discussion M V ( a ) = e a T μ + 1 2 a T Σ a = M U ( a ). So U and V have the same moment generating function. Because this moment generating function is defined for all a R k , it uniquely determines the associated probability distribution. That is, V and U have the same distribution.
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