Loading...

PPT – The multivariate normal distribution PowerPoint presentation | free to download - id: 53f0f5-NTZhM

The Adobe Flash plugin is needed to view this content

The multivariate normal distribution

- Characterizing properties of the univariate

normal distribution - Different definitions of normal random vectors
- Conditional distributions
- Independence
- Cochrans theorem

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.

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

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

The univariate normal distribution- defining

properties

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

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

Illustrations of independent and dependent normal

distributions

Illustrations of independent and dependent normal

distributions

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?

Fundamental properties of covariance matrices

- Let ? be a covariance matrix of a random vector

X - Then ? is symmetric
- Moreover, ? is nonnegative-definite, i.e.

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 ?

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 ?

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.

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.

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.

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

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

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

The multivariate normal vector- uncorrelated and

independent components

- The components of a normal random vector are

independent if and only if they are uncorrelated

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

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

(No Transcript)

Decomposition theorem for nonnegative-definite

quadratic forms

- Let
- where
- Then there exists an orthogonal matrix C such

that with x Cy - (y Cx)

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

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

Exercises Chapter V

- 5.2, 5.3, 5.7, 5.16, 5.25, 5.30