Statistical Inference for Models with Multivariate t-Distributed Errors

CHAPTER 11

LINEAR PREDICTION MODELS

Outline

11.1 Model and Preliminaries

11.2 Distribution of SRV and RSS

11.3 Regression Model for Future Responses

11.4 Predictive Distributions of FRV and FRSS

11.5 An Illustration

11.6 Problems

The predictive inference had been the oldest form of statistical inference used in real life. In general, predictive inference is directed towards inference involving the observable rather than the parameters. However, recently, Khan (2002b, 2004, 2006b) proposed the prediction distribution for the future regression vector and residual sum of squares. Predictive inference for a set of future responses of the model, conditional on the realized responses from the same model, has been derived by many authors, including Aitchison and Sculthorpe (1965), Fraser and Haq (1969), Guttman (1970), Haq and Rinco (1973), Aitchison and Dunsmore (1975), Geisser (1993), Khan and Haq (1994), Khan (2002b, 2006b), and Ng (2010). Khan and Haq (1994), Anderson and Fang (1990), and Khan (2002a) provide predictive analyses of linear models with multivariate t and spherical errors. The contribution of Prof. Shahjahan Khan to this field should be acknowledged.

11.1 Model and Preliminaries

Consider the regression model

(11.1.1)

where the n-dim row vector y is the vector of the response variable; X is the p × n dimensional matrix of the values of the p regressors; e is the 1 ×