半参数模型的估计理论及其应用
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摘要
半参数模型是二十世纪八十年代发展起来的一种重要的统计模型,它引入了表示模型误差或其它系统误差的非参数分量,从而使这种模型既含有参数分量,又含有非参数分量,兼顾了参数模型和非参数模型的优点,较单纯的参数模型或非参数模型有更大的适应性,并具有更强的解释能力。
在许多实际问题中,我们遇到的系数是非随机设计点列,即固定设计点列的情况。因而本论文主要研究在固定设计情况下,半参数模型中参数分量和非参数分量的估计量的构造、估计结果的大样本性质及其应用。
将参数模型和非参数模型估计理论中的参数估计扩展到半参数模型,初步建立半参数模型最小二乘估计理论是本文所做的主要工作。将测量数据处理中影响观测值的因素分为两个部分:一部分为线性部分,另一部分为某种干扰因素,它同观测量的关系是完全未知的,没有理由将其归入误差项,可以将其看成半参数模型中的非参数分量,即用非参数分量表达参数模型表达不完善的部分。因此,半参数模型可以克服参数模型在表达客观模型方面的局限性。一方面使数学模型与客观实际更接近,另一方面能从误差中分离出系统误差和偶然误差,提供更丰富的解算结果。从而,半参数模型可以概括和描述众多实际问题,更接近于真实,因而引起了广泛的重视,研究日益成熟,本文的研究具有理论意义和实用价值。
本论文将结合数学界的理论研究工作与测绘界的实际需要,系统地研究半参数模型的各种估计方法(补偿最小二乘法、两步估计法、二阶段估计法、小波估计法、迭代法等)及其在测量数据处理中的应用,具体地说,主要研究了如下内容:
在第二章,基于最小二乘极值问题的求解,提出了补偿最小二乘准则。在该准则下,得到了正规化矩阵正定、半正定情况下模型中参数分量、非参数分量的估计值及其观测值的改正值的表达式。较为系统地讨论了平滑因子及正规化矩阵的选取方法。利用补偿最小二乘原理构造加权补偿平方和,得到了半参数模型中正规化矩阵正定时参数和非参数的估计量。从偶然误差的统计特征出发,详细讨论了这种平差方法得到的参数估计值的有偏性、误差大小等统计性质,并对半参数平差与最小二乘法的参数估计值进行了比较。理论分析表明,通过选取合适的平滑因子,半参数平差方法优于最小二乘法。另外从数理统计的角度对平滑因子的选取进行了分析,得到了平滑因子的取值范围。在均方误差准则下,对半参数模型和参数模型的估计的准确度进行了比较,给出了参数分量为O的T统计检验的实用统计量的构造公式和检验方法。这对于上述估计方法的应用有实际意义。
采用模拟数据对补偿最小二乘法进行了算例验证,与忽略系统误差采用参数模型在最小二乘准则下的估计结果进行了比较,证明采用半参数模型,可以估计出系
A semiparametric model have been an important statistics model since 1980s,this kind of model includes not only a parametric component ,but also a nonparametric component. So it has the advantages of the parametric model and the nonparametric model. It has the more implements and stronger explanations than the pure parametric or nonparametric model.In many practical problems, unknown function usually is some nonrandom design points, that is, fixed design points. So, the purpose of this paper concentrates on the semiparametric model's large sample property and application research.Parametric and nonparametric models have been studied deeply and a set of basic theory has been set up. The focus of this paper is to expand the estimating technique of nonparametric to semiparametric models under Least-squares principle. If factor of impacting observed values can be divided into two parts: main part is linear relation, another part is a certain interference factor, relation to observation values is complete unknown, it also fall under error item without any reason, too many information will be lost if the non-parametric model (though it has bigger flexibility )is used, imitated result is bad if the linear model is adopted. Semiparametric not only contain the parametric Weight (described known composition of function relation in observation values),but also contain the non-parametric weight (exclusively show the model deviation unknown in function relation),can generalize and describe numerous actual problems, and it even near to true thing. As a result, the model is extensively thought, its research is increasingly mature.In the paper, combining the theoretical research work of mathematics field with practical requirement of surveying and mapping field, estimation methods of the semiparametric model are systematically investigated, including penalized least squares method, two steps estimate method, two stages estimate methods, wavelet estimate method etc. Their applications in the surveying data processing are studied. Say in a specific way, the main researched contents are as follows:In chapter 2,the penalized least squares method of the semiparametric model is clarified. In order to get only one minimal solution and smooth the curve of non-parametric estimate, the penalized least squares principle is put forwards. Under the principle, the parametric and nonparametric estimators and correct values of observed values are got. The choices of smooth factor and regular matrix are systematically discussed. The nonparametric and parametric estimator is attained; some statistical properties of some estimators are also discussed.This paper gives a method for getting the values of the unknown parameters of the semiparametric model under the principle of penalized least squares with a positive definite regularity matrix. Based on the statistic characteristic of random errors, the mathematical expectation, variance and mean square error of the parameters estimator getting from this method are discussed in detail. The difference between the parameters
values that were given by semiparametric models and by general least squares is compared. It is shown clearly by the theoretical analysis and the simulating computation that the method of semiparametric adjustment is better than that of least squares if the smoothing parameter takes a suitable value. Our study shows how to choose a reasonable value of the smoothing parameter and it's influence to the precision of the mathematic model is also given. By some simulating examples and actual applications, for example: GPS position, gravity measuring etc., success and validity of the method are illuminated. In chapter 3, we deal with the two stages estimation method of semiparametric model. And define the first stage estimators of parametric and nonparametric by using least square and two-stage estimation under the additivity of the model. And then the paper proves the estimators are the consistency. Obtain the second stage estimators of parametric and nonparametric with the first stage estimators by using generated least square and nearer kernel weight function. The second stage estimators are better than the first stage ones concerning variance, and meantime the paper proves estimator is the approximate normal distribution. We first give the two estimators of variance with the estimators obtained before, and then define their Bootstraps by using method of resampling, and prove the Bootstraps are the consistency and the approximate normal distribution, and obtain some useful conclusion. In addition, these methods and results are still verified and explained with some examples.In chapter 4, the theory of p-norm semiparametric model is introduced. Using the moment method of estimation, We obtain the parameter estimation of/?-norm distribution, Which is effective to decide the distribution of random errors. The calculating formulas of the monadic/?-norm maximum likelihood adjustment are going into particulars. Under the assumption that the distribution of observations is unimodal and symmetry, this method can give the estimated values of parametric, but the calculating time is too long. The moment method of estimation and the maximum likelihood adjustment are combining to calculate parameter. Two examples are presented finally, the new method presented in this paper shows an effective way of solving the problem, and the estimated values are nearer to their theoretical ones than the moment method of estimation or the maximum likelihood adjustment. Using the kernel weight function, we obtain the parameter and nonparametric estimation of /?-norm distribution in semiparametric model, which is effective to decide the distribution of random errors. Under the assumption that the distribution of observations is unimodal and symmetry, this method can give the estimates of parametric, nonparametric and variance, two simulated adjustment problem are constructed to explain this method. The new method presented in this paper shows an effective way of solving the problem, the estimated values are nearer to their theoretical ones than those by least squares adjustment.In chapter 5, the theory of heteroscedasticity semiparametric model is introduced. Since the semiparametric model is usually studied when the error sequence has the same variance, this chapter deals with a special heteroscedasticity situation. Namely, et = aiei,
where a] = /(w;), {￡,} is an i.i.d. random error sequence with mean 0 and variance 1.Under above consumption, the kernel-weighted estimators of parametric and nonparametric are defined. Moreover, their weak consistency, strong consistency and asymptotic normality are proved. The effect of some techniques is tested with numerical examples.In chapter 6, the estimating theory of semiparametric model under random censorship is researched. When the observation variable are case one randomly censored and the censoring distribution function is known, parametric and nonparametric estimate is obtained using the least squares and usual nonparametric weighted method. Under some natural and reasonable conditions, we have the strong consistency and p order mean consistency of the estimators, the asymptotic normality of parametric estimate, the optimal convergence rate of the nonparametric. The effect of some techniques is tested with numerical examples.In chapter 7, by using the least square method and the weighted function method of nonparametric estimation tighter, defines the wavelet estimators of parametric and nonparametric. But the weighted function here is not in the common weighted functions, which include kernel weighted function and nearest neighbour weighted function, it is a kind of weighted function deduced by wavelet method, so we call it wavelet weighted function here. This chapter established some new research and discussion and obtained satisfactory results.In addition, principal research work and creativeness of this paper are summarized; at the same time, some problems of deserving discussion are put forward for further studying the semiparametric model.
在许多实际问题中,我们遇到的系数是非随机设计点列,即固定设计点列的情况。因而本论文主要研究在固定设计情况下,半参数模型中参数分量和非参数分量的估计量的构造、估计结果的大样本性质及其应用。
将参数模型和非参数模型估计理论中的参数估计扩展到半参数模型,初步建立半参数模型最小二乘估计理论是本文所做的主要工作。将测量数据处理中影响观测值的因素分为两个部分:一部分为线性部分,另一部分为某种干扰因素,它同观测量的关系是完全未知的,没有理由将其归入误差项,可以将其看成半参数模型中的非参数分量,即用非参数分量表达参数模型表达不完善的部分。因此,半参数模型可以克服参数模型在表达客观模型方面的局限性。一方面使数学模型与客观实际更接近,另一方面能从误差中分离出系统误差和偶然误差,提供更丰富的解算结果。从而,半参数模型可以概括和描述众多实际问题,更接近于真实,因而引起了广泛的重视,研究日益成熟,本文的研究具有理论意义和实用价值。
本论文将结合数学界的理论研究工作与测绘界的实际需要,系统地研究半参数模型的各种估计方法(补偿最小二乘法、两步估计法、二阶段估计法、小波估计法、迭代法等)及其在测量数据处理中的应用,具体地说,主要研究了如下内容:
在第二章,基于最小二乘极值问题的求解,提出了补偿最小二乘准则。在该准则下,得到了正规化矩阵正定、半正定情况下模型中参数分量、非参数分量的估计值及其观测值的改正值的表达式。较为系统地讨论了平滑因子及正规化矩阵的选取方法。利用补偿最小二乘原理构造加权补偿平方和,得到了半参数模型中正规化矩阵正定时参数和非参数的估计量。从偶然误差的统计特征出发,详细讨论了这种平差方法得到的参数估计值的有偏性、误差大小等统计性质,并对半参数平差与最小二乘法的参数估计值进行了比较。理论分析表明,通过选取合适的平滑因子,半参数平差方法优于最小二乘法。另外从数理统计的角度对平滑因子的选取进行了分析,得到了平滑因子的取值范围。在均方误差准则下,对半参数模型和参数模型的估计的准确度进行了比较,给出了参数分量为O的T统计检验的实用统计量的构造公式和检验方法。这对于上述估计方法的应用有实际意义。
采用模拟数据对补偿最小二乘法进行了算例验证,与忽略系统误差采用参数模型在最小二乘准则下的估计结果进行了比较,证明采用半参数模型,可以估计出系
A semiparametric model have been an important statistics model since 1980s,this kind of model includes not only a parametric component ,but also a nonparametric component. So it has the advantages of the parametric model and the nonparametric model. It has the more implements and stronger explanations than the pure parametric or nonparametric model.In many practical problems, unknown function usually is some nonrandom design points, that is, fixed design points. So, the purpose of this paper concentrates on the semiparametric model's large sample property and application research.Parametric and nonparametric models have been studied deeply and a set of basic theory has been set up. The focus of this paper is to expand the estimating technique of nonparametric to semiparametric models under Least-squares principle. If factor of impacting observed values can be divided into two parts: main part is linear relation, another part is a certain interference factor, relation to observation values is complete unknown, it also fall under error item without any reason, too many information will be lost if the non-parametric model (though it has bigger flexibility )is used, imitated result is bad if the linear model is adopted. Semiparametric not only contain the parametric Weight (described known composition of function relation in observation values),but also contain the non-parametric weight (exclusively show the model deviation unknown in function relation),can generalize and describe numerous actual problems, and it even near to true thing. As a result, the model is extensively thought, its research is increasingly mature.In the paper, combining the theoretical research work of mathematics field with practical requirement of surveying and mapping field, estimation methods of the semiparametric model are systematically investigated, including penalized least squares method, two steps estimate method, two stages estimate methods, wavelet estimate method etc. Their applications in the surveying data processing are studied. Say in a specific way, the main researched contents are as follows:In chapter 2,the penalized least squares method of the semiparametric model is clarified. In order to get only one minimal solution and smooth the curve of non-parametric estimate, the penalized least squares principle is put forwards. Under the principle, the parametric and nonparametric estimators and correct values of observed values are got. The choices of smooth factor and regular matrix are systematically discussed. The nonparametric and parametric estimator is attained; some statistical properties of some estimators are also discussed.This paper gives a method for getting the values of the unknown parameters of the semiparametric model under the principle of penalized least squares with a positive definite regularity matrix. Based on the statistic characteristic of random errors, the mathematical expectation, variance and mean square error of the parameters estimator getting from this method are discussed in detail. The difference between the parameters
values that were given by semiparametric models and by general least squares is compared. It is shown clearly by the theoretical analysis and the simulating computation that the method of semiparametric adjustment is better than that of least squares if the smoothing parameter takes a suitable value. Our study shows how to choose a reasonable value of the smoothing parameter and it's influence to the precision of the mathematic model is also given. By some simulating examples and actual applications, for example: GPS position, gravity measuring etc., success and validity of the method are illuminated. In chapter 3, we deal with the two stages estimation method of semiparametric model. And define the first stage estimators of parametric and nonparametric by using least square and two-stage estimation under the additivity of the model. And then the paper proves the estimators are the consistency. Obtain the second stage estimators of parametric and nonparametric with the first stage estimators by using generated least square and nearer kernel weight function. The second stage estimators are better than the first stage ones concerning variance, and meantime the paper proves estimator is the approximate normal distribution. We first give the two estimators of variance with the estimators obtained before, and then define their Bootstraps by using method of resampling, and prove the Bootstraps are the consistency and the approximate normal distribution, and obtain some useful conclusion. In addition, these methods and results are still verified and explained with some examples.In chapter 4, the theory of p-norm semiparametric model is introduced. Using the moment method of estimation, We obtain the parameter estimation of/?-norm distribution, Which is effective to decide the distribution of random errors. The calculating formulas of the monadic/?-norm maximum likelihood adjustment are going into particulars. Under the assumption that the distribution of observations is unimodal and symmetry, this method can give the estimated values of parametric, but the calculating time is too long. The moment method of estimation and the maximum likelihood adjustment are combining to calculate parameter. Two examples are presented finally, the new method presented in this paper shows an effective way of solving the problem, and the estimated values are nearer to their theoretical ones than the moment method of estimation or the maximum likelihood adjustment. Using the kernel weight function, we obtain the parameter and nonparametric estimation of /?-norm distribution in semiparametric model, which is effective to decide the distribution of random errors. Under the assumption that the distribution of observations is unimodal and symmetry, this method can give the estimates of parametric, nonparametric and variance, two simulated adjustment problem are constructed to explain this method. The new method presented in this paper shows an effective way of solving the problem, the estimated values are nearer to their theoretical ones than those by least squares adjustment.In chapter 5, the theory of heteroscedasticity semiparametric model is introduced. Since the semiparametric model is usually studied when the error sequence has the same variance, this chapter deals with a special heteroscedasticity situation. Namely, et = aiei,
where a] = /(w;), {￡,} is an i.i.d. random error sequence with mean 0 and variance 1.Under above consumption, the kernel-weighted estimators of parametric and nonparametric are defined. Moreover, their weak consistency, strong consistency and asymptotic normality are proved. The effect of some techniques is tested with numerical examples.In chapter 6, the estimating theory of semiparametric model under random censorship is researched. When the observation variable are case one randomly censored and the censoring distribution function is known, parametric and nonparametric estimate is obtained using the least squares and usual nonparametric weighted method. Under some natural and reasonable conditions, we have the strong consistency and p order mean consistency of the estimators, the asymptotic normality of parametric estimate, the optimal convergence rate of the nonparametric. The effect of some techniques is tested with numerical examples.In chapter 7, by using the least square method and the weighted function method of nonparametric estimation tighter, defines the wavelet estimators of parametric and nonparametric. But the weighted function here is not in the common weighted functions, which include kernel weighted function and nearest neighbour weighted function, it is a kind of weighted function deduced by wavelet method, so we call it wavelet weighted function here. This chapter established some new research and discussion and obtained satisfactory results.In addition, principal research work and creativeness of this paper are summarized; at the same time, some problems of deserving discussion are put forward for further studying the semiparametric model.
引文
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