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## The multivariate normal distribution

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### cf. Cauchy distributions Probability theory 2011 The univariate normal distribution - defining properties Probability theory 2011 The multivariate normal ... – PowerPoint PPT presentation

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Title: The multivariate normal distribution

1
The multivariate normal distribution
• Characterizing properties of the univariate
normal distribution
• Different definitions of normal random vectors
• Conditional distributions
• Independence
• Cochrans theorem

2
The univariate normal distribution- defining
properties
• A distribution is normal if and only if it has
the probability density
• where ? ? R and ? gt 0.
• A distribution is normal if and only if the
sample mean
• and the sample variance
• are independent for all n.

3
The univariate normal distribution- defining
properties
• Suppose that X1 and X2 are independent of each
other, and that the same is true for the pair
• where no coefficient vanishes. Then all four
variables are normal.
• Corollary A two-dimensional random vector that
preserves independence under rotation must be
normal

x2
x1
4
The univariate normal distribution- defining
properties
• Let F be a class of distributions such that
• X ? F ? a bX ? F
• Can F be comprised of distributions other than
the normal distributions?
• cf. Cauchy distributions

5
The univariate normal distribution- defining
properties

6
The multivariate normal distribution- a first
definition
• A random vector is normal if and only if every
linear combination of its components is normal
• Immediate consequences
• Every component is normal
• The sum of all components is normal
• Every marginal distribution is normal
• Vectors in which the components are independent
normal random variables are normal
• Linear transformations of normal random vectors
give rise to new normal vectors

7
The multivariate normal distribution- a first
definition
• Every component is normal
• The sum of all components is normal
• Every marginal distribution is normal
• Vectors in which the components are independent
normal random variables are normal
• Linear transformations of normal random vectors
give rise to new normal vectors

8
Illustrations of independent and dependent normal
distributions
9
Illustrations of independent and dependent normal
distributions
10
Parameterization of the multivariatenormal
distribution
• Is a multivariate normal distribution uniquely
determined by the vector of expected values and
the covariance matrix?
• Is there a multivariate normal distribution for
any covariance matrix?

11
Fundamental properties of covariance matrices
• Let ? be a covariance matrix of a random vector
X
• Then ? is symmetric
• Moreover, ? is nonnegative-definite, i.e.

12
Factorization of covariance matrices
• Let ? be a covariance matrix.
• Because ? is symmetric there exists an
orthogonal matrix C
• (CC C C I) such that
• C ? C D and ? CD C
• where D is a diagonal matrix.
• Beacuse ? is also nonnegative-definite, the
diagonal elements of D must be non-negative.
Consequently, there exists a symmetric matrix B
such that
• B B ?
• B is often called the square root of ?

13
Construction of a random vector with a given
covariance matrix
• Let ? be a covariance matrix.
• Derive a matrix B such that
• B B ?
• If X has independent components with variance 1,
• then
• Y BX
• has covariance matrix B B ?

14
The multivariate normal distribution- a second
definition
• A random vector is normal if and only if it has a
characteristic function of the form
• where ? is a nonnegative-definite, symmetric
matrix and ? is a vector of constants
• Proof of the equivalence of definition I and II
• Let X?N(? , ?) according to definition I, and
set Z tX. Then E(Z) tu and Var(Z) t? t,
and ?Z(1) gives the desired expression.
• Let X?N(? , ?) according to definition II. Then
we can derive the characteristic function of any
linear combination of its components and show
that it is normally distributed.

15
The multivariate normal distribution- a third
definition
• Let Y be normal with independent standard normal
components and set
• Then
• provided that the determinant is non-zero.

16
The multivariate normal distribution- a fourth
definition
• Let Y be normal with independent standard normal
components and set
• Then X is said to be a normal random vector.

17
The multivariate normal distribution-
conditional distributions
• All conditional distributions in a multivariate
normal vector
• are normal
• The conditional distribution of each component is
equal to that of a linear combination of the
other components plus a random error

18
The multivariate normal distribution-
conditional distributions and optimal predictors
• For any random vector X it is known that E(Xn
X1, , Xn-1) is an optimal predictor of Xn based
on X1, , Xn-1 and that
• Xn E(Xn X1, , Xn-1) ?
• where ? is uncorrelated to the conditional
expectation.
• For normal random vectors X, the optimal
predictor E(Xn X1, , Xn-1) is a linear
expression in X1, , Xn-1

19
The multivariate normal distribution-
calculation of conditional distributions
• Let X?N (0 , ?) where
• Determine the conditional distribution of X3
given X1 and X2
• Set Z a X1 bX2 c
• Minimize the variance of the prediction error Z
- X3

20
The multivariate normal vector- uncorrelated and
independent components
• The components of a normal random vector are
independent if and only if they are uncorrelated

21
The multivariate normal distribution- orthogonal
transformations
• Let X be a normal random vector with independent
standard normal components, and let C be an
orthogonal matrix.
• Then
• Y CX
• has independent, standard normal components

22
Quadratic forms of the components of a
multivariate normal distribution one-way
analysis of variance
• Let Xijij, i 1, , k, j 1, , ni , be k
samples of observations. Then, the total
variation in the X-values can be decomposed as
follows

23
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24
Decomposition theorem for nonnegative-definite
• Let
• where
• Then there exists an orthogonal matrix C such
that with x Cy
• (y Cx)

25
Decomposition theorem for nonnegative-definite
• Let X1, , Xn be independent and N(0 ?2) and
suppose that
• where
• Then there exists an orthogonal matrix C such
that with X CY (Y CX)
• Furthermore, Q1, , Qp are independent and
??2?2-distrubuted with r1, rp degrees of freedom

26
Quadratic forms of the components of a
multivariate normal distribution one-way
analysis of variance
• Let Xijij, i 1, , k, j 1, , ni , be
independent and N(? ,?2). Then, the total sum of
squares can be decomposed into three