🤖 AI Summary
Traditional goodness-of-fit tests exhibit imbalanced power against diverse alternative hypotheses involving location shifts, scale changes, heavy tails, or asymmetry. This work proposes a sequential conditional calibration framework that integrates a primary test statistic—such as the Kolmogorov–Smirnov statistic—with multiple secondary statistics (e.g., variance, skewness) in a stepwise manner. The rejection regions at each stage are mutually exclusive, enabling a multiplicative decomposition of the overall Type I error rate, while the significance level of the primary test can be explicitly adjusted in subsequent stages. The method maintains high power against location alternatives and substantially enhances detection capability for scale deviations, heavy-tailed distributions, and asymmetric departures, offering an ordered decomposition of first-rejection power across stages.
📝 Abstract
Goodness-of-fit statistics may have markedly different power against different types of alternatives. We propose a sequential procedure for augmenting a primary goodness-of-fit statistic with an ordered collection of secondary statistics. At each stage, the acceptance region of the current statistic is calibrated under the null distribution conditional on acceptance at all preceding stages. This conditional calibration gives a simple multiplicative decomposition of the overall Type~I error and allows the primary-stage level to be adjusted explicitly after the secondary-stage levels have been selected. The disjoint stagewise rejection regions also provide an ordered first-rejection decomposition of power.
We illustrate the method by augmenting the Kolmogorov--Smirnov statistic with sample variance and sample skewness. In simulations under a standard normal null, the resulting chain procedures retain nearly all of the primary test's power against location alternatives while substantially improving power against scale, heavy-tailed, and asymmetric alternatives. Reversing the order of the secondary statistics produces nearly identical total power in the experiment, although the stagewise attribution of power can change considerably.