广义线性模型的若干估计及比较

发布时间：2018-08-28 10:04

【摘要】：广义线性模型是线性模型的推广,其适应范围更加广泛.它既适应于因变量为连续型的随机变量,又适应于因变量为离散型的随机变量.这就决定了其有更加广泛的应用价值.广义线性模型可以应用于保险,金融,医学,社会统计等众多领域中.而参数估计是统计学中重要内容.它是研究其他统计性质的基础.目前广义线性模型中参数估计方法中应用最多的就是极大似然估计,极大似然估计可以通过迭代的方法估计出感兴趣的参数.在广义线性模型中,极大似然估计通常情况下表现良好,是一种不错的参数估计方法.然而,广义线性模型中的参数的极大似然估计方法在有些情况下并不是最理想的参数估计方法.例如,有的时候被估计的参数会受到部分的约束,如果不考虑这些约束而对参数进行估计,那么所得到的估计值与真实值应该是有偏差的.所以在本文我们介绍了广义线性模型的压缩估计,试图改善这种情况.并讨论了压缩估计的渐近性质,给出了压缩估计的渐近风险函数并与极大似然估计的渐近风险函数做了比较,接着又给出了压缩估计的数值模拟.此外,当模型存在复共线性时,极大似然估计的均方误差将会变的很大,此时极大似然估计并不是一种很好的参数估计方法.本文将讨论一些克服模型复共线性的估计方法,并通过数值实例对这些估计方法进行比较.另外,本文对广义线性模型的岭估计进行了改进,提出了广义线性模型的广义岭估计的概念,并证明了它的一些相关性质,说明了相对于广义线性模型的岭估计,广义线性模型的广义岭估计确实有更加优良性质.
[Abstract]:Generalized linear model is a generalization of linear model, and its adaptive range is more extensive. It is not only suitable for random variables of continuous type, but also suitable for random variables of discrete type. This determines that it has more extensive application value. Generalized linear model can be used in many fields such as insurance, finance, medicine, social statistics and so on. Parameter estimation is an important part of statistics. It is the basis for the study of other statistical properties. At present, the most widely used parameter estimation methods in generalized linear models are maximum likelihood estimation, which can estimate the parameters of interest by iterative method. In the generalized linear model, the maximum likelihood estimation (MLE) usually performs well and is a good parameter estimation method. However, the maximum likelihood method for parameter estimation in generalized linear models is not the most ideal in some cases. For example, sometimes the estimated parameters will be partially constrained. If the parameters are estimated without these constraints, the estimated value should be deviated from the real value. So in this paper, we introduce the contractive estimation of the generalized linear model and try to improve the situation. The asymptotic properties of the contractive estimator are discussed, the asymptotic risk function of the contractive estimate is given and compared with the asymptotic risk function of the maximum likelihood estimator, and the numerical simulation of the contractive estimate is given. In addition, when the model has complex collinearity, the mean square error of maximum likelihood estimation will become very large. At this time, maximum likelihood estimation is not a good parameter estimation method. In this paper, we discuss some estimation methods to overcome the complex collinearity of the model, and compare them with numerical examples. In addition, in this paper, the ridge estimation of generalized linear model is improved, the concept of generalized ridge estimate of generalized linear model is proposed, and some related properties are proved, and the ridge estimate relative to the generalized linear model is proved. The generalized ridge estimators of generalized linear models do have better properties.
【学位授予单位】：扬州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212

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