具有限制條件的部分線性模型的經驗似然推斷（英文）

劉常勝+李永獻

摘 要 本文將經驗似然方法應用到具有限制假設條件的部分線性模型中. 為了檢驗假設條件， 構造基于零假設和對立假設條件下的極大經驗對數似然比估計值的差值統計量. 而且在零假設下證明該統計量的極限分布為標準的χ2分布. 數值模擬表明所提出的檢驗統計量的優勢.

關鍵詞 經驗似然； 限制條件； 部分線性模型； 假設檢驗； 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.

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