[Paper Review] Sieve empirical likelihood ratio tests for nonparametric functions

Jianqing Fan, J. Zhang|TU/e Research Portal (Eindhoven University of Technology)|Mar 29, 2005

Statistical Methods and Inference3 citations

TL;DR

This paper proposes sieve empirical likelihood ratio (SELR) tests for nonparametric functions, extending generalized likelihood ratio methods to cases with unspecified error distributions. By using conditional estimating equations and sieve approximations, the SELR statistics asymptotically follow rescaled chi-squared distributions independent of nuisance parameters, demonstrating the Wilks phenomenon under more general models and achieving optimal nonparametric testing rates.

ABSTRACT

Generalized likelihood ratio statistics have been proposed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] as a generally applicable method for testing nonparametric hypotheses about nonparametric functions. The likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family. We extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio (SELR) tests. The approach is also applied to test conditional estimating equations on the distributions of stochastic errors. It is shown that the proposed SELR statistics follow asymptotically rescaled χ^2-distributions, with the scale constants and the degrees of freedom being independent of the nuisance parameters. This demonstrates that the Wilks phenomenon observed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] continues to hold under more relaxed models and a larger class of techniques. The asymptotic power of the proposed test is also derived, which achieves the optimal rate for nonparametric hypothesis testing. The proposed approach has two advantages over the generalized likelihood ratio method: it requires one only to specify some conditional estimating equations rather than the entire distribution of the stochastic error, and the procedure adapts automatically to the unknown error distribution including heteroscedasticity. A simulation study is conducted to evaluate our proposed procedure empirically.

Motivation & Objective

• To develop a general nonparametric testing framework that does not require full specification of the error distribution.
• To extend the generalized likelihood ratio method to cases where the error distribution is unknown, including heteroscedasticity.
• To establish the Wilks phenomenon—i.e., asymptotic chi-squared distribution of test statistics—under nonparametric models with unknown error distributions.
• To achieve optimal testing power rates in nonparametric hypothesis testing.
• To provide a robust, adaptive method that automatically handles unknown error structures, including heteroscedasticity, via conditional estimating equations.

Proposed method

• Proposes sieve empirical likelihood ratio (SELR) statistics based on conditional estimating equations instead of full parametric error distributions.
• Uses kernel smoothing and sieve approximations to estimate nonparametric functions and error structures nonparametrically.
• Employs empirical likelihood with a sieve-based likelihood ratio to construct test statistics under the null hypothesis.
• Applies the Lagrange multiplier method to solve the empirical likelihood optimization problem under constraints.
• Derives asymptotic distributions of the SELR statistics using empirical process theory and strong convergence results.
• Establishes that the test statistics follow asymptotically rescaled chi-squared distributions with scale and degrees of freedom independent of nuisance parameters.

Experimental results

Research questions

• RQ1 Can generalized likelihood ratio tests be extended to nonparametric models with completely unspecified error distributions?
• RQ2 Does the Wilks phenomenon—i.e., asymptotic chi-squared distribution of likelihood ratio statistics—hold when the error distribution is unknown and heteroscedastic?
• RQ3 Can the proposed method achieve the optimal nonparametric testing rate without assuming parametric error distributions?
• RQ4 How does the SELR test perform in terms of power and robustness compared to generalized likelihood ratio methods under model misspecification?
• RQ5 Can the method adapt automatically to unknown error variance structures, such as heteroscedasticity, without requiring explicit modeling?

Key findings

• The proposed sieve empirical likelihood ratio (SELR) statistics asymptotically follow a rescaled chi-squared distribution, with scale constants and degrees of freedom independent of nuisance parameters.
• The Wilks phenomenon is preserved under more general models, including nonparametric regression with unspecified error distributions and heteroscedasticity.
• The asymptotic power of the SELR test achieves the optimal nonparametric rate, confirming its efficiency in nonparametric hypothesis testing.
• The method requires only the specification of conditional estimating equations rather than the full parametric form of the error distribution, enhancing flexibility and robustness.
• The procedure adapts automatically to unknown error distributions, including heteroscedasticity, without requiring prior knowledge or modeling of the variance function.
• The convergence rate of the estimator is op(n⁻¹/ξ ∧ h¹/²)dn, and the test statistic is asymptotically equivalent to a chi-squared distribution under the null hypothesis.