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The multivariate normal distribution

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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
quadratic forms
  • Let
  • where
  • Then there exists an orthogonal matrix C such
    that with x Cy
  • (y Cx)

25
Decomposition theorem for nonnegative-definite
quadratic forms (Cochrans theorem)
  • 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
    quadratic forms
  • which are independent and ??2?2-distrubuted with
    1, k-1, and n-k degrees of freedom

27
Exercises Chapter V
  • 5.2, 5.3, 5.7, 5.16, 5.25, 5.30
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