Weighted residual empirical processes, martingale transformations, and model specification tests for regressions with diverging number of parameters

TL;DR

This work develops model-specification tests for mean and conditional variance functions in regressions where the number of parameters diverges with sample size. It introduces weighted residual empirical processes to mitigate the curse of dimensionality, comparing tests without martingale transformations (which achieve the parametric local-alternative rate $n^{-1/2}$ but are not distribution-free) with martingale-transformed tests (which are distribution-free but have a slower, dimension-independent rate of $n^{-1/4}$). The authors prove limiting null distributions, establish smooth-residual bootstrap validity for $CvM_n$, and derive exact power properties under local alternatives, highlighting a surprising power advantage of the untransformed approach in high dimensions. They also propose practical weight-function constructions via Fourier-based estimation of the regression function and sufficient dimension reduction to maintain power in diverging-dimension settings, supported by extensive simulations and a real-data example. Overall, the paper provides scalable, theoretically grounded tools for checking regression models in high-dimensional contexts and clarifies the trade-offs between distribution-free transformations and untransformed residual-process tests.

Abstract

This paper explores hypothesis testing for the parametric forms of the mean and variance functions in regression models under diverging-dimension settings. To mitigate the curse of dimensionality, we introduce weighted residual empirical process-based tests, both with and without martingale transformations. The asymptotic properties of these tests are derived from the behavior of weighted residual empirical processes and their martingale transformations under the null and alternative hypotheses. The proposed tests without martingale transformations achieve the fastest possible rate of detecting local alternatives, specifically of order $n^{-1/2}$, which is unaffected by dimensionality. However, these tests are not asymptotically distribution-free. To address this limitation, we propose a smooth residual bootstrap approximation and establish its validity in diverging-dimension settings. In contrast, tests incorporating martingale transformations are asymptotically distribution-free but exhibit an unexpected limitation: they can only detect local alternatives converging to the null at a much slower rate of order $n^{-1/4}$, which remains independent of dimensionality. This finding reveals a theoretical advantage in the power of tests based on weighted residual empirical process without martingale transformations over their martingale-transformed counterparts, challenging the conventional wisdom of existing asymptotically distribution-free tests based on martingale transformations. To validate our approach, we conduct simulation studies and apply the proposed tests to a real-world dataset, demonstrating their practical effectiveness.

Figures (2)

• Figure 1: (a) Scatter plot of $Y$ versus $\hat{\beta}_1^{\top}X$ with $\hat{\beta}_1$ obtained from a linear model. (b) Scatter plot of $residuals$ from a linear model versus the fitted value $\hat{\beta}_1^{\top}X$.
• Figure 2: (a) Scatter plot of $Y$ versus $\hat{\beta}_2^{\top}X$ with $\hat{\beta}_2$ obtained from model (\ref{['quadratic']}). (b) Scatter plot of $residuals$ from the quadratic model (\ref{['quadratic']}) versus the fitted value $\hat{Y}$.

Theorems & Definitions (15)

• Proposition 1
• Remark 1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Proposition 2
• Theorem 3.1
• Theorem 3.2
• Theorem 4.1