Cumulative chi-squared statistic as a tool for testing goodness of fit

The cumulative chi-squared statistic in the one-way analysis of variance model is characterized as the weighted sum of independent one degree of freedom chi-squared random variables, which are linear, quadratic, and so on, components in the sense of Čebyšev's orthogonal polynomials for finite points. This suggests that the cumulative chi-squared should be an appropriate goodness-of-fit test statistic for a hypothesized model against a systematic departure that is mainly but not exclusively linear. This, together with the follow-up analysis of residuals, will give a good statistical technique for analysing data. As a natural extension of the cumulative chi-squared, a new statistic is proposed for testing for linearity against convexity.