nonparametric testing in regression models with ...

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Statistica Sinica (2014): Preprint

NONPARAMETRIC TESTING IN REGRESSION MODELS WITH WILCOXON-TYPE GENERALIZED LIKELIHOOD RATIO Long Feng1 , Zhaojun Wang1 , Chunming Zhang2 , Changliang Zou1 1 Nankai

University and 2 University of Wisconsin–Madison

Abstract: The generalized likelihood ratio (GLR) statistic (Fan et al. 2001) proposed a generally applicable method for testing nonparametric hypotheses about nonparametric functions. However, the efficiency of this method is adversely affected by outlying observations and heavy-tailed distributions. To attack this challenge, a robust testing procedure is developed under the framework of the GLR by incorporating a Wilcoxon-type artificial likelihood function and adopting the associated local smoothers. Under some useful hypotheses, the proposed test statistic is proved to be asymptotically normal and free of nuisance parameters and covariate designs. Its asymptotic relative efficiency with respect to the least squares-based GLR method is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. It outperforms the least squares-based GLR with heavier-tailed data in the sense that asymptotically it can yield substantially larger power. On the other hand, when the data are normally distributed, both methods have similar power. Simulation results are consistent with the asymptotic analysis. Key words and phrases: Asymptotic relative efficiency; Bootstrap; Generalized likelihood ratio; Lack-of-fit test; Model Specification; Local polynomial regression; Local Walsh-average regression.

1. Introduction Over the last two decades, nonparametric modeling techniques have been developed rapidly due to the reduction of modeling biases of traditional parametric methods. This raises many important inference questions such as whether a parametric family adequately fits a given data set. Here we choose the varying coefficient model for our investigation because it arises in many statistical problems and has been widely used in the literature. Suppose {(Yi , X i , Ui )}ni=1 is a random sample from the following varying-coefficient model, Y = α(U ) + A(U )T X + ε

(1.1)

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LONG FENG, ZHAOJUN WANG, CHUNMING ZHANG AND CHANGLIANG ZOU

with X = (X1 , X2 , . . . , Xp )T and A(U ) = (a1 (U ), . . . , ap (U ))T . Many nonparametric inferences or testings, such as the problem of parametric null against nonparametric alternative hypothesis or model checking for partial linear models, are included as a special case of hypothesis testing problems under this model. A simple but widely used null hypothesis testing problem is: H0a : (α, AT ) = (α0 , AT0 ) versus H1a : (α, AT ) ̸= (α0 , AT0 ),

(1.2)

where α0 and A0 are two known functions. An intuitive approach is based on generalizations of the Kolmogorov-Smirnov or Cramer-von Mises types of statistics to measure the distance between the estimators under the null and alternative models; see Härdle and Mammen (1993), Neumeyer and van Keilegom (2010) and the references therein. However, it is difficult to find an optimal measure for such type of statistics. Zheng (1996) proposed a consistent test of functional form of nonlinear regression models by combining the methodology of the conditional moment test (Bierens 1990) and nonparametric estimation techniques. See Zhang and Dette (2004) for a power comparison of some types of nonparametric regression tests. In an important work, Fan, Zhang and Zhang (2001) proposed the generalized likelihood ratio (GLR) test statistic n RSS0 n RSS0 − RSS1 log ≈ (1.3) 2 RSS1 2 RSS1 ∑ ∑ where RSS0 = nk=1 (Yk − α0 (Uk ) − A0 (Uk )T X k )2 , RSS1 = nk=1 (Yk − α b(Uk ) − b k )T X k )2 and (b b A(U α(·), A(·)) are the local linear estimators under the alternative λGLR (α0 , A0 ) = lnGLR (H1a ) − lnGLR (H0a ) = n

model. This test is shown to possess Wilks phenomenon and to be asymptotically optimal in certain sense, and therefore has become a commonly used methodology for constructing nonparametric testing in regression models. See Fan and Jiang (2007) for an overview of the idea of GLR inference in different nonparametric models. Although the GLR test is asymptotically distribution-free, the normal likelihood function and the corresponding local least-squares polynomial estimators (Fan and Gijbels 1996) are employed. Accordingly, its statistical properties, designed to perform best under the normality assumption, could potentially be (highly) affected when the errors are far away from normal or the data contain some outliers.

WILCOXON-TYPE GENERALIZED LIKELIHOOD RATIO

We develop a robust test under the framework of the GLR.

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The new

method relaxes the usually strong distributional assumption associated with the least-squares-based GLR by adopting a Wilcoxon-type dispersion function (Hettmansperger and McKean 2010) and the corresponding local smoothers. We establish that the Wilcoxon-type GLR preserves the Wilks phenomenon without the need to assume a normal likelihood. Under the null hypothesis the test statistic is asymptotically normal and free of nuisance parameters. Under certain conditions, its Pitman asymptotic relative efficiency (ARE) with respect to the GLR test is established. This ARE is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. That is, it outperforms the leastsquares-based GLR with heavier-tailed data in the sense that asymptotically it can yield substantially larger power. On the other hand, when the data are normally distributed, both methods have similar power. Due to the slow convergence of asymptotic distribution, the bootstrap technique is used to approximate critical values of the test in finite sample situations. A simulation study of testing linearity and homogeneity is conducted to investigate the finite sample properties of a bootstrap version of the newly proposed test and to compare it with some other available procedures in the literature. When the errors deviate from normality, our tests are powerful than the least-square-based or moment-based methods. Even when the errors are normally distributed, our procedure does not lose much, which coincides with our theoretical analysis. Considerable efforts have been devoted to construct some robust nonparametric polynomial smoothers. See Fan et al. (1994), Welsh (1996), Kai et al. (2011) and Feng et al. (2012). However, there is very few work on robust inference. Wang and Qu (2007) robustified Zheng (1996)’s test based on the centered asymptotic rank transformation of the residuals from a robust fit under the null hypothesis. However, to our best knowledge, in the literature of nonparametric model checking, there is no corresponding test in which robust local smoothers are considered. This article is organized as follows. In Section 2, we describe the Wilcoxontype artificial likelihood function and the associated robust testing procedures. We also extend the proposed method to two important model diagnostics, i.e., testing linearity and homogeneity. Some numerical studies and comparison with

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other existing methods are presented in Section 3. Section 4 contains a realdata example to illustrate the application of our method. Several remarks draw the paper to its conclusion in Section 5. Technical proofs are provided in the Appendix. 2. Methodology 2.1. Test statistic and its null limiting distribution The weighted rank-based L1 norm is often used in the development of robust statistical procedures (Hettmansperger and McKean 2010). That is, ∥ε∥W = √ ∑ n 12 i=1 ri |εi |, where ri denotes the rank of |εi | among |ε1 |, . . . , |εn |. It is equivn+1 alent to

√ √ 12 ∑ ∑ εi − εj 12 ∑ ∑ εi + εj + 2 n+1 2 ≡ Wn (ε) + Rn (ε). n+1 i 0}, E(Fh ) =

WILCOXON-TYPE GENERALIZED LIKELIHOOD RATIO 1 ∑∑ 2 ϕij n+1 i≤j

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+ o(h−1/2 ) and var(Eh ) = O(n−1 h−2 ) = o(h−1 ). Thus,

( ) 1 ∑∑ ( Bh = (αn (Ui ) + Rn (Ui ))T W i /ρ(Ui ) n+1 i≤j )2 T + ((αn (Uj ) + Rn (Uj )) W j )/ρ(Uj ) + op (h−1/2 ) =

1∑ 1∑ (αn (Ui )T W i )2 /ρ2 (Ui ) + (Rn (Ui )T W i )2 /ρ2 (Ui ) 2 2 n

n

i=1 n ∑

i=1

Rn (Ui )T W i W Ti αn (Ui )/ρ2 (Ui )

+

i=1

+

1 ∑∑ αn (Ui )T W i αn (Uj )T W j /ρ(Ui )/ρ(Uj ) n+1 j̸=i

1 ∑∑ Rn (Ui )T Wi Rn (Uj )T W j /ρ(Ui )/ρ(Uj ) + n+1 j̸=i

+

1 n+1

∑∑

αn (Ui )T W i Rn (Uj )T W j /ρ(Ui )/ρ(Uj ) + op (h−1/2 ).

j̸=i

After calculating the expectation and variance of the last three sums, we can prove that 1 ∑∑ αn (Ui )T W i αn (Uj )T W j /ρ(Ui )/ρ(Uj ) = Op (1 + (nh)−1/2 + (nh)−1 ), n+1 j̸=i

1 ∑∑ Rn (Ui )T Wi Rn (Uj )T W j /ρ(Ui )/ρ(Uj ) = Op (h4 + n−1/2 h7/2 + n−1 h3 ), n+1 j̸=i

1 ∑∑ αn (Ui )T W i Rn (Uj )T W j /ρ(Ui )/ρ(Uj ) = Op (h2 + n−1/2 h3/2 + n−1 h), n+1 j̸=i

and accordingly, 1∑ 1∑ Bh = (αn (Ui )T W i )2 /ρ2 (Ui ) + (Rn (Ui )T W i )2 /ρ2 (Ui ) 2 2 n

n

i=1 n ∑

i=1

+

i=1

Rn (Ui )T W i W iT αn (Ui )/ρ2 (Ui ) + op (h−1/2 ).

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This leads to λna =

n ∑ i=1

1∑ αn (Ui ) W i ξi /ρ (Ui ) − (αn (Ui )T W i )2 /ρ2 (Ui ) 2 n

T

2

i=1

−1/2

+ Rn1 − Rn2 − Rn3 + op (h

).

Taking the same procedure as Lemma 7.2 in Fan et al. (2001), we can show that Rn1 = n1/2 h2 Rn10 + O(n−1/2 h), Rn2 = n1/2 h2 Rn20 + O(n−1/2 h), Rn3 = nh4 Rn30 + O(h3 ). Also, similar to Lemma 7.4 in Fan et al. (2001), it can be verified that n ∑ 1 αn (Ui )T W i ξi /ρ2 (Ui ) = (p + 1)K(0)Ef (U )−1 h i=1 1 ∑ ∑ −1 + ρ (Ui )ρ−1 (Uj )ξi ξj W Ti Γ(Uj )−1 W j Kn (Ui − Uj ) + op (h−1/2 ), n j̸=i ∫ n ∑ 1 T 2 2 −1 (αn (Ui )W i ) /ρ (Ui ) = (p + 1)Ef (U ) K 2 (t)dt h i=1 2 ∑ ∑ −1 + ρ (Ui )ρ−1 (Uj )ξi ξj W Ti Γ−1 (Ui )K ∗ K((Ui − Uj )/h)W j + op (h−1/2 ). nh i