CHAPTER 2

PRELIMINARIES

Outline

2.1 Normal Distribution

2.2 Chi-Square Distribution

2.3 Student's t-Distribution

2.4 F-Distribution

2.5 Multivariate Normal Distribution

2.6 Multivariate t-Distribution

2.7 Problems

In this chapter, we discuss some basic results on various distributions, particularly the normal, chi-square, Student's t-, multivariate t-distributions.

2.1 Normal Distribution

The most basic distribution in statistical theory is the normal distribution, (?, s2), with pdf

(2.1.1)

where ? is the mean and s2 is the variance of this distribution.

It is well known that

(2.1.2)

(2.1.3)

(2.1.4)

2.2 Chi-Square Distribution

If Z is (0, 1), then Z2 follows the central chi-square distribution with one degree of freedom (d.f.). However, if Z is (?, s2), then follows the noncentral chi-square distribution with one d.f. and noncentrality parameter , with . The pdf/cdf of this noncentral chi-square variable with one d.f. is given by

(2.2.1)

where

(2.2.2)

and the cdf of the chi-square distribution is given by

(2.2.3)

where H1+2r(c; 0) is the cdf of a central chi-square distribution with 1 + 2r d.f. An important identity w.r.t. the cdf is given by

(2.2.4)

A central and noncentral chi-square variable will be denoted by ?2?o and ?2?o(?2), respectively. In statistical theory, moment results involving chi-square variables are important and they are given below.

If Z ~ N(?, s2) and f(.) is a measurable function of Z2, then

(2.2.5)

If Z = (Z1,., Zp)´ be a Np(?, Ip) (see (2.5.1)) and f(Z´Z) is a measurable function of Z´Z, then

(2.2.6)

Further, if , then

(2.2.7)

For details, see Judge and Bock (1978) and Saleh (2006).

2.3 Student's t-Distribution

It is well known that if and U is independent of Z such that is a chi-square variable with ?o degrees of freedom, then follows the Student's t-distribution with ?o d.f. The pdf of this t-statistic is given by

(2.3.1)

The distribution f(u) of u may be obtained as the expected value of the pdf of (0, t-1), which is the conditional distribution of u given t with respect to the inverse gamma distribution, IG (t-1, ?o), with the pdf given by

(2.3.2)

Then,

(2.3.3)

(2.3.4)

since .

This distribution will be denoted by M(1)t (0, 1, ?o).

Clearly, E(u) = 0 and . Further, the odd central moments are zero while the even central moments are given by

(2.3.5)

Now consider the distribution of Z to be (?, t-1). The distribution of tZ2 is then the noncentral chi-square distribution with one d.f. having the pdf

(2.3.6)

where h1+2r(x; 0) is the pdf of a central chi-square variable with (1 + 2r) d.f. The cdf of this random variable (r.v.) is given by

(2.3.7)

where H1+2r(x; 0) is the cdf of a central chi-square variable with (1 + 2r) d.f.

Let f(.) be a measurable function, then

(2.3.8)

based on (2.3.6), where EN denotes getting expectation w.r.t. the normal theory, i.e., (?, t-1).

In this regard, if t-1 follows the inverse gamma distribution (2.3.2), then the exact distribution of tZ2 is given by the expectation of h1 (?2(?2t)) or H1 (x; ?2t) using (2.3.3). Thus, the unconditional distribution of tZ2 is given by the pdf and cdf, respectively, as

(2.3.9)

and

(2.3.10)

where the mixing distribution of r is given by

(2.3.11)

for ?o = 3. Let us define

(2.3.12)

Then we may write

(2.3.13)

where

(2.3.14)

Specifically, using the fact that

Make the transformation with the Jacobian J(t u) = to get

Now, we have the following theorem similar to the equations (2.2.5)-(2.2.6).

Theorem 2.3.1. If Z ~ M(1)t(?, 1, ?o) and f(Z2) is a measurable function, then

Proof:

(2.3.13).

by (2.2.7)

by (2.3.13).

by (2.2.5)

by (2.2.5)

2.4 F-Distribution

Let Z ~ (0, 1) and mU be the central chi-square variable with m d.f. Then it is well known that follows the central F-distribution with (1, m) d.f. In other words, u2 in the other section follows the central F-distribution with (1, m) d.f.

Now, consider two independent chi-square variables ?2?1 and ?2?1 with ?1 and ?2 d.f.s respectively. Then, the ratio follows the central F-distribution with (?1, ?2) d.f. The pdf of F is given by

(2.4.1)

and the cdf of F is

(2.4.2)

where Iy(a, b) is the incomplete beta (Pearson's regularized incomplete beta) function.

Further, consider two independent chi-square variables, where one is a noncentral chi-square variable ?2?1 (?2) with ?1 d.f. and noncentral parameter ?2, and the other, ?2?2, is a central chi-square variable with ?2 d.f. Then, F?1,?2 (?2) = follows the noncentral F-distribution with (?1, ?2) d.f. and noncentrality parameter ?2. The cdf of F?1,?2 (?2) is given by

(2.4.3)

with pdf

(2.4.4)

Let f(.) be measurable function of F?1,?2 (?2), then

(2.4.5)

Now, if Z ~ (?, t-1), then tZ2 follows the noncentral chi-square distribution with one d.f. and noncentrality parameter ?2t = t?2. Further, let mU be a central chi-square with m d.f. Then, follows the noncentral F-distribution with (1, m) d.f. and noncentrality parameter ?2t. Thus, for a measurable function f(.), we have

(2.4.6)

by (2.4.5).

Further, if Z ~ M(1)t (?, 1, ?o), then

(2.4.7)

where

(2.4.8)

See (2.3.13)-(2.3.14) for details with .

Thus, we have

(2.4.9)

(2.4.10)

similar to (2.3.15). Here, mU is a central chi-square variable with m d.f.

If Gq+i,m+j (c; ?2t) denote the cdf of a noncentral F-distribution with (q + i, m + j) d.f. with noncentrality parameter ?2t = t?2, then we write

(2.4.11)

where

(2.4.12)

2.5 Multivariate Normal Distribution

In the multivariate setup, multivariate normal distribution, , and S S(p) (space of all positive definite matrices of order p), is the basic distribution on which all statistical inference depends. The pdf of a p(?, S) is given by

(2.5.1)

Then,

(2.5.2)

Further, the distribution of Y´...