Study on efficacy of Anderson-Darling test statistic in multiple nonparametric regression using adaptive splines

Abstract

The Anderson–Darling (AD) statistic is a well-known goodness-of-fit metric that is valued for its sensitivity to tail behaviour. The standard AD test may be distorted by the bias and variability of nonparametric estimation; however, residual-based tests are essential in multiple nonparametric regression to confirm assumptions such as homoscedasticity and normality. This paper suggests using adaptive spline smoothing to estimate conditional mean and variance functions prior to applying the AD statistic to standardised residuals. Adaptive splines adjust locally to capture nonlinear structure while minimising bias, and the test is calibrated through a bootstrap procedure. Together, these processes yield more reliable inference. This approach improves the power and accuracy of the AD test in complex regression scenarios, as shown by both theory and simulations.