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.
Consider the regression model
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 ×