On the Large-sample Size Critical Values of the Maximum Absolute Internally Studentized Residuals

The maximum absolute internally studentized residual is a regular diagnostic measure for identification of a single outlying observation in the response variable in linear regression models. However, due to the daunting and formidable nature of the probability density function of this statistic, exact critical values are tough to compute. The Bonferroni inequality and intensive simulations are the only tools for determining its critical values as a means for detecting a single outlying observation in a linear regression model. In this paper, we present a straightforward alternative technique for obtaining asymptotic critical values of this statistic. The technique can be applied to any linear regression model and is convenient for routine use. The asymptotic distribution of this statistic is derived and used in obtaining the upper bounds for its critical values. It is shown that the proposed technique does not depend on the number of independent variables or the number of regression parameters in the model. Thus, the computational cumbersomeness and tedium imposed by the complexity associated with the distribution of this statistic and the use of the Bonferroni inequality are circumvented. The main advantages of the proposed procedure are its computational simplicity and efficiency to handle large datasets in high dimension. The asymptotic critical values of this statistic obtained by the proposed method are almost identical to those obtained by other authors, even though the techniques and principles employed in this work are entirely different from that employed by them.