OPTIMAL HETEROSKEDASTICITY TESTING IN NONPARAMETRIC REGRESSION

• 成果类型: Article
• 署名作者: Kotekal, Subhodh; Kundu, Soumyabrata
• 署名单位: University of Chicago
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/24-AOS2467
• 发表日期: 2025
• 页码: 295-321
• 关键词: optimal adaptive estimation; variance function; detecting heteroscedasticity; signal-detection; parametric form; consistent test; Minimax Rates; estimator

摘要

Heteroskedasticity testing in nonparametric regression is a classic statistical problem with important practical applications, yet fundamental limits are unknown. Adopting a minimax perspective, this article considers the testing problem in the context of an alpha-H & ouml;lder mean and a beta-H & ouml;lder variance function. For alpha > 0 and beta is an element of (0, 1/2), the sharp minimax separation rate n(-4 alpha) + n(-4 beta /(4 beta+1)) + n(-2 beta )is established. To achieve the minimax separation rate, a kernel-based statistic using first-order squared differences is developed. Notably, the statistic estimates a proxy rather than a natural quadratic functional (the squared distance between the variance function and its best L-2 approximation by a constant) suggested in previous work. The setting where no smoothness is assumed on the variance function is also studied; the variance profile across the design points can be arbitrary. Despite the lack of structure, consistent testing turns out to still be possible by using the Gaussian character of the noise, and the minimax rate is shown to be n(-4 alpha )+ n(-1/2). Exploiting noise information happens to be a fundamental necessity, as consistent testing is impossible if nothing more than zero mean and unit variance is known about the noise distribution. Furthermore, in the setting where the variance function is beta-H & ouml;lder but heteroskedasticity is measured only with respect to the design points, the minimax separation rate is shown to be n(-4 alpha )+n(-((1/2)boolean OR(4 beta /(4 beta+1)))) when the noise is Gaussian and n(-4 alpha) + n(-4 beta /(4 beta+1)) + n(-2 beta) when the noise distribution is unknown.