Statistical Inference for Models with Multivariate t-Distributed Errors

CHAPTER 9

MULTIVARIATE MODELS

Outline

9.1 Location Model

9.2 Testing of Hypothesis and Several Estimators of Local Parameter

9.3 Bias, Quadratic Bias, MSE, and Risk Expressions

9.4 Risk Analysis of the Estimators

9.5 Simple Multivariate Linear Model

9.6 Problems

Multivariate statistical analysis of multidimensional data dominates the literature based on multivariate normal distribution like the normal distribution in the case of univariate problems. In this chapter, we consider the statistical theory based on multivariate t-distribution to increase the scope of applications. We consider only two models, namely, (i) the location and (ii) the simple linear regression models and discuss the test of hypothesis and propose several estimators for two models with details of the dominance properties.

9.1 Location Model

Let Y₁, Y₂, …, Y_N be N observation vectors of p-dim satisfying the model

(9.1.1)

where Y_α = (Y_α1, …, Y_αp)′, θ = (θ₁,…, θ_p)′ is the location vector parameter, and ε_α = (ε_α1,,…, ε_αp)′ and {ε_α|α = 1,…, N} are distributed as M_t^(p) (0, ∑, γ_o) for each α = 1,…, N.

The unrestricted estimator (UE) of θ is , and the exact distribution of _N is .

Consider a statistic of the form

It is ...