Sieve empirical likelihood ratio tests for nonparametric functions

By Jianqing Fan, Jian Zhang. 2004

76 citations recorded by the source corpus.

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 norparametic 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 resealed chi(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.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b9941173a7a4ed48f8be4809277097bfd12545fa

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Citations in the corpus

An empirical likelihood goodness‐of‐fit test for time series (2003, methodology): “For independent data, EL has been employed for testing conditional moment restrictions (Tripathi and Kitamura, 2000) and for nonparametric functions based on a local approximation (Fan and Zhang, 2000).”
Generalized likelihood ratio statistics and Wilks phenomenon (2001, background): “See also Owen (1990), Hall and Owen (1993), Chen and Qin (1993), Li, Hollander, McKeague and Yang (1996) for applications of the empirical likelihood.”
A Test for model specification of diffusion processes (2007, background): “Kernel–based tests have been shown to be able to effectively test for discrete time series models as demonstrated in Robinson (1989), Hjellvik and Tjøstheim (1995), Fan and Li (1996), Hjellvik and Yao and Tjøstheim (1998), Kreiss, Neumann and Yao (1998), Li (1999), Aı̈t-Sahalia, Bickel and Stoker (2”
Local and global asymptotic inference in smoothing spline models (2012, background): “Because of the technical complexity, they focused on the local polynomial fitting; see [10] for a sieve version.”
A difference based approach to the semiparametric partial linear model (2011, methodology): “The test of significance of partial linear model was discussed in [15, 23, 35].”
A review on empirical likelihood methods for regression (2009, background): “Fan and Zhang (2004) propose a sieve EL test for testing a varying-coefficient regression model that extends the generalized likelihood ratio test of Fan, Zhang and Zhang (2001).”