Estimation Methods And Their Applications In Semiparametric Model

Along with scientific and technological development of surveying and mapping, science of surveying and mapping oneself and other related science put forward the higher request to the modern surveying data processing. However, the current theories of data processing can't resolve the some new problems met in surveying practice, so the technical development and application of surveying and mapping are limited and tied up. Hence, the theories of data processing must be further studied and improved, and new theories and method are put forward and developed.In the course of surveying data processing, many researchers use the parametric model because its construction is simple and apt to be processed. Further more, under a majority of situations (for instance, kinds of static problems of conventional geodetic surveying) , it is in accordance with objective fact and can satisfy practical needs because a majority of system errors are be compensated and cleared up and can be expressed in the parameter model before data processing. However, under some situations (for instance, some dynamic issues of geodetic surveying) , since observed values include system errors which can't be cleared up and parametrized, it brings on non-neglectable difference between the parametric model and objective practicality.On the other hand, it is not always very scientific that the system errors are always tried to eliminate or compensate as the harmful composition. In fact, the system errors contain lots information that influence observed values, if they can be identified and withdrawn rightly, not only the accuracy of parameter estimate can be increased, but also the data can be provided for the research of the other subjects.In addition, if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. Here, 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.Whereas the above problems, other data processing models need consider, that is the smeiparametric modelIt is a kind of important statistical model developed in 1980's. Because it not only contain the parameter weight (described known composition of function relation in observation values), but also contain the non-parameter 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.Generally speaking, the problem of surveying data processing is ultimately come down to the one of parametric or non-parametric estimate. Up to the present, the research of the smeiparametric model has existed a lot of estimate methods, such as earlier period parametrization thought, two stages estimate (including neighbor estimate, weighted estimate, kernel estimate, wavelet estimate etc.), two stages estimate, robust estimate, penalized least squares method etc.. However, in the mathematics etc. theoretical field, its research is almost theoretical estimates and big sample properties, thus it is very difficult to apply them. Moreover, at surveying and mapping etc. applied field, the majority of research results in semiparametric model exist some shortagessuch as no thorough theories research and single method etc.. The writer try to set up a bridge at the two field such that their shortages are offset.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, wavelet estimate method, universal least squares method, robust estimate method, iterative method, accumulation method, two stages 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 forward as followsUnder the principle, the parametric and nonparameteric estimators and correct values of observed values are got. Using cubic splines interpolation, the nonparameteric estimator is attained. Some statistical properties are studied. The choices of smooth factor a and regular matrix R are systematically discussed. By some simulating examples and actual applications (for example, coordinate systems transformation, GPS position, gravity measuring etc.), success and validity of the method are illuminated. The popular cubic nature splines method is regarded as one of penalized least squares method, thus the former is researched as special case of the latter.In chapter 3, wavelet estimation method is taken as an example of the two steps estimate method. The thought of two steps estimate is: First, the nonparametric estimator is defined according to the assumption parameter known; Then parametric estimator is attained by using least squares method, thereby the non-parametric estimator is attained. When random error sequences are martingale difference sequences, the semiparametric model is studied. Not only the parametric and nonparametric estimators are attained, but also their asymptotic normality and strong consistency and moment consistency are studied. And an application of wavelet analysis in gross errors processing of the surveying data is preliminarily studied.In chapter 4, the universal least squares method is proposed for the very first time, its standard is given byThe method is an expansion of some methods, such as penalized least squares method, rank-deficient (weighting) minimum norm, ridge estimation. The research of the universal least squares method is begun from a linear model. Combined penalized least squares method and two steps estimation, the universal least squares method of the semiparametric model is researched. At the same time, after the universal least squares method is compared with the penalized least squares method and two steps method and ridge estimate method, the method is better than past estimate methods under square mean meaning and some appropriate conditions. In addition, these methods and results are still verified and explained with some examples.In chapter 5, the writer use accumulation estimates of the parameter model for investigating the semiparametric model. Combining with the penalized least squares method and two steps estimate method, the pilot study is carried through in semiparametric model, and the parametric and non parametric estimators are attained. The method is showed by example.In chapter 6, the robust estimation of the semiparametric model is researched. Using equivalent weight principle, some expressions are got, such as robust two steps estimate, robust penalized least squares estimation, robust universal least squares estimation etc. Their influence functions are derived, which explain that two steps estimate and penalized least squares estimation and universal least squares estimation are not robustness. The robust penalized least squares estimation and the robust universal least squares estimation have robustness by imitated example. M-estimation based on two steps estimation is studied, the linear representation is attained.In chapter 7, the iterative method of semiparametric model is systematically studied, and iterative equations are constructed as follows:Under the certain conditions, the iterative equations converge to the parameters and non-parameters of the semiparametric model, rigorous and theoretical foundation of iterative method are provided. The difference estimate and its statistical characteristics are studied, so theoretical foundation is provided for the actual application. Two stages estimation of the semiparametric model is studied in brief. The thought of robust two stages estimation is putting forward, and the free level net is calculated by using it. The parametrization of the semiparametric model is described in brief.In addition, principal research work and creativeness of the paper are summarized; at the same time, some problems of deserving discussion are put forward for further study the semiparametric model.