# CHAPTER 3

# LOCATION MODEL

## Outline

**3.1** Model Specification

**3.2** Unbiased Estimates of *θ* and σ^{2} and Test of Hypothesis

**3.3** Estimators

**3.4** Bias and MSE Expressions of the Location Estimators

**3.5** Various Estimates of Variance

**3.6** Problems

In this chapter, we discuss some basic results on the location model under the assumption that the error-vector is distributed according to the multivariate t-distribution.

# 3.1 Model Specification

Consider the location model with the response vector *Y* = (*Y*_{1}, ···, *Y*_{n})′ such that it satisfies the relation

(3.1.1)

where **1**_{n} = (1, ···, 1)′ is an *n*-tuple of 1’s, and the error vector ε follows the multivariate *M*^{(n)}_{t}(**0**, σ^{2}**V**_{n} γ_{o}) with pdf

where σ > 0, **V**_{n} is a positive definite matrix of rank *n* and γ_{o} > 2.

The mean of ε is the zero-vector and the covariance-matrix of ε is

(3.1.2)

The plan of this chapter is as follows. In sections 2 and 3, unbiased estimators of *θ* and σ^{2}_{ε} are proposed along with the test statistic for testing the hypothesis *H*_{0} : *θ* = *θ*_{0}. In addition, we propose some improved estimates of location parameter. Section 4 contains some important theorems for the bias and MSE expressions of the proposed estimators of *θ* and their mathematical characteristics. ...