【作者】 叶飞；

【导师】 杨瑛；

【作者基本信息】 清华大学 ， 统计学， 2013， 博士

【摘要】 在许多的实际应用中,例如在股价预测和寿命分析中,相对于普通误差|Y-Y|而言,人们对相对误差|(Y-Y)/Y|更感兴趣。因此,基于最小绝对相对误差和(MRE, Minimum sum of Relative Errors),以及最小相对误差平方和(RLS, the Relative Least Squares)的估计准则,很早便在不同的领域中被提了出来。最近,Chen等[2010](JASA105,1104-1112)考虑对两类相对误差|(Y-Y)/Y|+|(Y-Y)/Y|同时做极小化,提出了加速失效时间模型中参数估计的最小绝对相对误差(LARE, the least absolute relative error)准则。受此启发,本文建立起了加速失效时间(AFT. Accelerated Failure Time)模型下的相对误差估计与线性模型下的M-估计之间的联系,通过这个联系,包括LARE在内的许多相对误差估计的渐进性质都可以由已经比较完备的线性模型的M-估计理论而直接获得。但同时我们也注意到,MRE和RLS估计的渐进性质并不能由此联系而直接建立,因此本文首先在AFT模型下提出了参数估计的广义相对误差准则(GREC, the General Relative Error Criterion),然后将其转化为线性模型中的M-估计问题,并且为了得到MRE和RLS估计的渐进性质而对M-估计中此类非凸损失的情形做了推广。随机模拟的结果表明我们的方法是可行的,我们还用一个例子说明了GREC在实际中的应用。此外,我们对参数维数可变(维数随着样本量趋于无穷)的AFT模型做了研究,得到了一些具体的GREC估计的(?)n/Pn-相合性和渐近正态性,并给出了协方差矩阵的新的相合估计。模拟结果显示：我们的估计回归参数和协方差矩阵的方法是比较准确的；从模拟结果还可以看出来RLS估计对误差分布非常敏感,与之相比,最小乘积相对误差(LPARE, the Least Product Absolute Relative Errors)估计是更稳定的选择。在这些工作的基础上,我们进一步考虑了用带惩罚项的广义相对误差准则(PGREC, Penalized GREC)进行变量选择的问题。在较宽泛的正则性条件下我们下证明了,存在一组合理的阈值使得PGREC估计具有Oracle性质,即,不但能以趋于1的概率选择出正确的模型,其非零部分的参数估计还拥有真实子模型下GREC估计的渐进方差。最后,在矩阵范数的意义下,我们给出了PGREC的渐进方差和可能存在的偏倚的相合估计。更多还原

【Abstract】 Relative error|(Y-Y)/Y rather than the error Y-Y itself is of the main interest in many practical applications. Criteria based on minimizing the sum of absolute rela-tive errors (MRE), and the sum of squared relative errors (RLS) were proposed in the different areas. Motivated by Chen etc.’s recent work (JASA105,1104-1112) on the least absolute relative error (LARE) estimation minimizing two types of relative errors|(Y-Y)|Y|+|(Y-Y)|Y|for the accelerated failure time (AFT) model, in this paper we es-tablish the connection between relative error estimators and the M-estimation in the linear model. This connection allows us to deduce the asymptotic properties of many relative error estimators (e.g. LARE) by the well-developed M-estimation theories. On the other hand, the asymptotic properties of some important estimators (e.g. MRE and RLS) cannot be established directly. So in this thesis, we first propose a general relative error criteri-on (GREC) for estimating the unknown parameter in the AFT model. Then we develop the approaches to deal with the asymptotic normalities for M-estimators with non-convex and non-differentiable loss functions in the linear model. The simulation studies are con-ducted to evaluate the performance of the proposed estimates for the different scenarios, and the simulation results suggest that our method is feasible. Illustration with a real data example is also provided.Then we study the AFT model with a diverging number of parameters (which goes to infinity as the sample size goes to infinity) and obtain the-(?)n/pn-consistencies and asymptotical normalities of some GREC estimators. A new consistent estimator of the covariance matrix is also given. Results of random simulations suggest that our estimates of the regression coefficients, and the covariance matrix is practical; the RLS estimator is sensitive to the distribution of the random error, and the least product absolute relative errors (LPARE) estimator is more stable comparing with the RLS estimator.On the base of these works, we consider the variable selection problem via the pe-nalized general relative error criterion (PGREC) estimators, and prove their "Oracle" properties under certain regular conditions. The estimators select the proper model with probability tending to1, and the estimators of nonzero coefficients have the same asymp-totic distribution that they would if the zero coefficients were known in advance. At last, we propose the estimators of the asymptotic covariance matrix and the bias of the PGREC estimator, and prove their consistencies in the sense of the matrix norm.更多还原