几类半参数经验似然检验问题的研究

发布时间：2018-03-30 16:51

本文选题：半参数方法　切入点：经验似然方法　出处：《哈尔滨工业大学》2016年博士论文

【摘要】：半参数经验似然方法广泛的被用于统计学中,此方法将半参数方法与经验似然方法有效的融合在一起,该方法具有如下优点:一方面,半参数方法能弥补参数方法对回归函数需要具有较强基本假设这一缺点,又能弥补非参数模型不能充分利用已知信息的缺点;另一方面,由于经验似然方法在应对复杂问题时不必受分布函数制约,因此经验似然方法能够描述一些不确定性的问题或一些无法用具体函数描述的问题。本文以多变点模型与整值时间序列模型为研究对象,对半参数经验似然检验问题进行探讨。主要研究内容有以下几个方面:1.针对相同参数权函数下含有两个变点的多变点问题,给出了半参数经验似然函数。利用Lagrange乘子方法得到了变点估计值,以及变点估计的最大似然检验统计量。并利用强大数定律获得了经验似然比检验统计量的渐近分布以及变点估计值的p-值,并证明了最大似然函数与一个连续的凸函数渐近相等,以及变点估计渐近的服从三点分布。除此之外,通过数值模拟验证了半参数经验似然方法比非参数经验似然方法能更好地检验变点估计值,最后用实际数据诊断了模型具有较好的适用性。2.分析了不同参数权函数下含有两个变点的多变点问题,该问题的数学模型是利用经验似然方法,结合Lagrange乘子,构造了半参数经验似然函数。并通过最大似然估计得到了变点的估计值以及p-值,应用强大数定律得到了经验似然比和半参数经验似然统计量的渐近分布。在数值模拟上,大量的实验表明,当变点的真实值在随机变量的相对中间位置时,半参数经验似然检验比非参数的方法相对优越,而当真实值在相对两端时半参数经验似然方法的优越性不那么明显。而实际数据依然验证了模型有很好的适用性。3.对于不同参数权函数下的含有有限个变点的多变点模型,利用半参数经验似然方法构造了经验似然函数。通过最大似然函数和强大数定律得到了变点的估计值、变点估计的p-值以及半参数经验似然统计量,建立了关于有限个变点的极大似然估计的渐近结果,并利用bootstrap方法对有限个变点的数目进行模拟估计,模拟结果显示文中所提出的经验似然方法对变点的估计是有效的。在参数估计的精度上,通过数值模拟,得出当变点在相对中间的位置时半参数经验似然有较好的效用,而当变点在相对两端时却无法判断。实例也验证了模型拟合有较好的能力。4.探讨了对含有间歇性噪声的整值时间序列INAR(k)模型,基于双似然方法建立了半参数经验似然函数。利用Lagrange乘子的方法得到了参数的估计值,并通过强大数定律和中心极限的性质证明了参数的经验似然比统计量渐进的服从自由度为k+2的χ2分布,且证明了参数的置信区间是凸集。除此之外,数值实验说明了半参数经验似然的效用,以及非间歇性噪声对模型的影响是显著的,而间歇性噪声对模型的影响是不显著的。综上所述,在多变点模型和整值时间序列模型中半参数经验似然方法都具有较高的效率。论文依次阐述了半参数经验似然方法在相同参数权、不同参数权,有限个变点的不同参数权以及整值时间序列模型中的应用。大量的数值模拟实验和实例证明了该方法的效用,体现了半参数经验似然方法的优越性。同时说明了本文所研究的方法适用于多变点和整值时间序列模型。
[Abstract]:A semiparametric empirical likelihood method is used widely in statistics, this method of semi parametric method and empirical likelihood method effectively together, this method has the following advantages: on the one hand, the semi parametric method can compensate for parameter method has a strong assumption that the disadvantages of regression function, and make up the non parametric model cannot make full use of the known information shortcomings; on the other hand, due to the empirical likelihood method without restricting the distribution function in dealing with complex problems, so the empirical likelihood method can describe some uncertain problems or some with a specific function to describe the problem. The thesis uses variable point model and integer valued time series model as the research object. Semiparametric empirical likelihood test problems are discussed. The main contents are as follows: 1. for the same power function with two parameters change point changing point to ask Questions, gives a semiparametric empirical likelihood function. By using the Lagrange method to get the change point estimation, maximum likelihood test statistic and estimate the change point. And get the empirical likelihood ratio test statistic and asymptotic distribution of point estimates of p- value using the strong law of large numbers, and proved that the maximum likelihood function and a a continuous convex function is asymptotically equal, and the change point estimation follows three asymptotic distribution. In addition, verify the semiparametric empirical likelihood method than the non parametric empirical likelihood method can better test of change point estimation through the numerical simulation, finally the actual data diagnosis model has good applicability of the.2. analysis of different parameters the right function contains two point variable problem, the mathematical model of the problem is to use the empirical likelihood method with Lagrange multipliers, constructed a semiparametric empirical likelihood function and through. The maximum likelihood estimates to get the estimation of change point value and p- value, the application of strong law of large numbers of the asymptotic distribution of empirical likelihood ratio and semiparametric empirical likelihood statistic. In numerical simulation, a large number of experiments show that when the change point in the middle position relative to the true value of random variables, semiparametric empirical likelihood test method the non parameter relative superiority, and when the true value of superiority in semiparametric empirical likelihood method is opposite less obvious. But the actual data still validate the model.3. is suitable for the variable point model a finite change point for different parameters of the weighting function containing, constructed by the likelihood function a semiparametric empirical likelihood method. The maximum likelihood function and the strong law of large numbers are estimated change point value, p- value and change point estimation semiparametric empirical likelihood statistic is established on a Co. Asymptotic results for maximum likelihood estimation of the change point, and use the bootstrap method on a number of limited change points in the simulated estimation, simulation results show that the empirical likelihood estimation method proposed in this paper on the change point is effective. The precision of parameter estimation, by numerical simulation, when the change point has a good effect in the middle of the relative position of semiparametric empirical likelihood, and when the change point in the opposite but not judgment. Results show the model has good fitting ability of.4. on the whole value of time series of INAR containing intermittent noise (k) model, semiparametric empirical likelihood function is established based on double likelihood method. By using the Lagrange obtained the estimates of the parameters, and the properties of the strong law of large numbers and central limit that the empirical likelihood ratio statistic parameters of progressive degrees of freedom for the k+2 distribution 2, and prove The confidence interval of the parameters is convex. Besides, numerical experiments show that the semiparametric empirical likelihood utility, and non intermittent noise of the model is significant, while the effect of intermittent noise on the model is not significant. In summary, the variable point model and integer valued time series model in semiparametric empirical likelihood the method has higher efficiency. This paper expounds the semiparametric empirical likelihood method in the same parameters are right, right of different parameters, different parameters of limited right to the change point and the whole value of application in time series model. Numerical simulation of a large number of quasi experiments and examples prove that the utility of this method reflects the superiority of semi parametric the empirical likelihood method. At the same time that this method is suitable for variable point and integer valued time series model.

【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O212.1

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