# CHAPTER 4

# SIMPLE REGRESSION MODEL

## Outline

**4.1** Introduction

**4.2** Estimation and Testing of **η**

**4.3** Properties of Intercept Parameter

**4.4** Comparison

**4.5** Numerical Illustration

**4.6** Problems

In this chapter, we discuss some of the basic results from a simple regression model under the assumption that the error-vector is distributed according to the multivariate t-distribution. For normal errors see Saleh (2006) and for t-errors see Khan and Saleh (1997, 2008).

# 4.1 Introduction

Consider a simple linear model

(4.1.1)

where *Y* = (*Y*_{1}, *Y*_{2}, …, *Y*_{n})′ is the response vector and *x* = (*x*_{1}, *x*_{2}, …, *x*_{n})′ is a fixed vector of known constants, while ε = (ε_{1}, ε_{2}, …, ε_{n})′ is the error vector distributed according to the low belonging to the class of multivariate t-distributions, say, *M*^{(n)}_{t} (**0**, σ^{2}**V**_{n}, γ_{o}).

As in Chapter 3, the covariance matrix is formulated as

(4.1.2)

# 4.2 Estimation and Testing of η

In this section, we consider LSE of **η** and test of hypothesis, *H*_{0} : **η** = **η**_{o} vs *H*_{A} : **η** ≠ **η**_{o}.

## 4.2.1 Estimation of η

For the LSE of **η**, we minimize

to obtain the LSE of **η** as

(4.2.1)

where

(4.2.2)

with *K*_{1} = **1′V**^{−1}_{n}**1**, *K*