Statistical Inference for Models with Multivariate t-Distributed Errors

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, (θ, σ²), with pdf

(2.1.1)

where θ is the mean and σ² is the variance of this distribution.

It is well known that

(i) If Y is N(θ, σ²), then .

(ii) If Y is (θ, σ²) and φ(Y) is a differentiable function satisfying |φ′(Y)| < ∞, then

(2.1.2)

For more information see Stein (1981).

(iii) If Z is N(θ, 1), then

(2.1.3)

where Φ(.) is the cumulative distribution function (cdf) of the standard normal (zero mean and unit variance) distribution. ...