Nonparametric inference for additive models estimated via simplified smooth backfitting

TL;DR

This paper develops hypothesis testing methods for nonparametric additive models estimated via simplified smooth backfitting, demonstrating their asymptotic properties, optimality, and practical effectiveness through simulations and real data analysis.

Contribution

It introduces a generalized likelihood ratio and a loss function testing framework with proven asymptotic properties for additive models estimated by simplified smooth backfitting, including the Wilks phenomenon.

Findings

Both tests have asymptotic chi-squared distributions under the null hypothesis.

The loss function test is asymptotically more powerful than the GLR test.

Simulations and real data illustrate the tests' effectiveness and the Wilks phenomenon.

Abstract

We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, \textbf{28(2)}, 386--400, 2019). Simplified smooth backfitting achieves oracle properties under regularity conditions and provides closed-form expressions of the estimators that are useful for deriving asymptotic properties. We develop a generalized likelihood ratio (GLR) and a loss function (LF) based testing framework for inference. Under the null hypothesis, both the GLR and LF tests have asymptotically rescaled chi-squared distributions, and both exhibit the Wilks phenomenon, which means the scaling constants and degrees of freedom are independent of nuisance parameters. These tests are asymptotically optimal in terms of rates of convergence for nonparametric hypothesis testing. Additionally, the bandwidths that are well-suited for model estimation may be useful for testing. We show that in additive models, the LF test is asymptotically more powerful than the GLR test. We use simulations to demonstrate the Wilks phenomenon and the power of these proposed GLR and LF tests, and a real example to illustrate their usefulness.