劉常勝+李永獻(xiàn)

摘 要 本文將經(jīng)驗(yàn)似然方法應(yīng)用到具有限制假設(shè)條件的部分線性模型中. 為了檢驗(yàn)假設(shè)條件， 構(gòu)造基于零假設(shè)和對(duì)立假設(shè)條件下的極大經(jīng)驗(yàn)對(duì)數(shù)似然比估計(jì)值的差值統(tǒng)計(jì)量. 而且在零假設(shè)下證明該統(tǒng)計(jì)量的極限分布為標(biāo)準(zhǔn)的χ2分布. 數(shù)值模擬表明所提出的檢驗(yàn)統(tǒng)計(jì)量的優(yōu)勢(shì).

關(guān)鍵詞 經(jīng)驗(yàn)似然； 限制條件； 部分線性模型； 假設(shè)檢驗(yàn)； χ2分布

It is well known that there are some striking advantages with the empirical likelihood （EL） method proposed by Owen[1-2] in the construction of confidence intervals for unknown parameter， e.g.， it turns out that by using the empirical likelihood method， one does not have to explicitly estimate the asymptotic variance of the quantity. The EL method only uses the data to determine the shape and orientation of confidence regions. Note that using empirical likelihood for confidence interval is a well known procedure in nonparametric statistics. The EL method has been applied to partially linear regression model by Shi and Lau[3]， Wang and Jing[4-5].

For a long time， however， its main theory was about the parametric regressions. The aforementioned articles mainly discussed the estimation of the parametric component rather than testing. In practice， after solving the problems of identification and fitting of the model （1）， one of the further inference issues is to evaluate if certain explanatory variables in the parametric components influence the response significantly. The study of this issue is also one of the important aspects in regression analysis because it is not only useful in variable selection but also in explanatory power.

Motivated by the above， we shall apply EL method to the partially linear model with restricted condition. More generally， we consider the following linear hypothesis：

The rest of this paper is organized as follows. In Section 2， applying the profile empirical likelihood procedure， we construct the empirical likelihood estimator on parameter with linear restriction and without any restriction. In Section 3， we construct the test statistic and study its asymptotic distribution. In Section 4， simulations are conducted to examine the performance of the proposed approaches. The proofs of the main results are given in Section 5.

1 Empirical likelihood estimation on parameter

For the need of constructing the test statistic， we first develop estimating approach for model （1） under the null hypothesis in this section. That is， we estimate the unknown quantities in model （1） with the restricted condition Aβ=b. Then model （1） can be written as

We summarize our findings as follows. When the null hypothesis is true （that is， c=0）， the rejection frequencies （estimated sizes） of both our proposed test based Tn and the restricted least-squares approach test based Wn are quite good and close to their nominal levels 0.05 under different error distributions. Under the alternative hypothesis， the rejection rate seems very robust to the variation of the type of error distribution. With the increasing of c， the test power of our proposed test is slightly better than the test based on the residual sum of squares.

References：

[1] OWEN A B. Empirical likelihood ratio confidence intervals for a single functional[J]. Biometrika，1988，75（2）：237-249.

[2] OWEN A B. Empirical likelihood ratio confidence regions[J]. Ann Stat， 1990，18（1）：90-120.

[3] SHI J， LAU T S. Empirical likelihood for partially linear models[J]. J Multiv Anal， 2000，72（1）：132-148.

[4] WANG Q H， JING B Y. Empirical likelihood for partial linear models with fixed designs[J]. Stat Prob Lett， 1999，41（4）：425-433.

[5] WANG Q H， JING B Y. Empirical likelihood for partially linear models[J]. Ann Inst Stat Math， 2003，55（3）：585-595.

[6] FAN J. Local linear regression smoothers and their minimax efficiencies[J]. Ann Stat， 1993，21（1）：196-216.

[7] FAN J， GIJBELS I. Local polynomial modelling and its applications[M]. New Tork： Chapman & Hall Press， 1996.

[8] WEI C， WANG Q. Statistical inference on restricted partially linear additive errors-in-variables models[J]. Test， 2012，21（4）：757-774.

[9] LIANG H， HRDLE W， CARROLL R J. Estimation in a semiparametric partially linear errors-in-variables model[J]. Ann Stat， 1999，27（5）：1519-1535.

[10] LIANG H， THURSTON S W， RUPPERT D， et al. Additive partial linear models with measurement errors[J].Biometrika， 2008，95（3）：667-678.

[11] LIANG H Y， JING B Y. Asymptotic normality in partial linear models based on dependent errors[J].J Stat Plan Infer， 2009，139（4）：1357-1371.

[12] 洪圣巖. 一類(lèi)半?yún)?shù)回歸模型的估計(jì)理論[J]. 中國(guó)科學(xué)：A 輯， 1991，34（12）：1258-1272.

[13] 孫耀東. 分歧泊松自回歸模型的馬爾可夫性[J]. 湖南師范大學(xué)自然科學(xué)學(xué)報(bào)， 2011，34（4）：18-20.

[14] WU C. Some algorithmic aspects of the empirical likelihood method in survey sampling[J]. Stat Sin， 2004，14（4）：1057-1068.

[15] XUE L G， ZHU L X. Empirical likelihood for a varying coefficient model with longitudinal data[J]. J Am Stat Assoc， 2007，102（478）：642-654.

[16] ZHU L， XUE L. Empirical likelihood confidence regions in a partially linear single-index model[J].J Royal Stat Soc： Ser B， 2006，68（3）：549-570.