【作者】 王慧；

【导师】 方兴；

【作者基本信息】 武汉大学 ， 大地测量学与测量工程， 2020， 硕士

【摘要】 统计学家指出粗差的出现总是无法完全消除,稳健估计可以尽量地减小或者消除粗差对于参数估计的影响,因此稳健估计的研究十分重要。目前常用的稳健估计准则包括极大似然估计(M估计)、排序线性估计(L估计)、秩估计(R估计)。本文以M估计为出发点,研究随机等式约束和固定等式约束下的M估计。具体实施将利用等价权原理把M估计转化为抗差最小二乘估计。本文主要工作与贡献如下:(1)推导一般M估计参数估计值表达式以及观测值影响函数的表达式。通过一维M估计准则以及一维M估计影响曲线的定义,详细推导了多维M估计的参数估计值以及影响函数。并以间接平差模型为例,从独立观测值开始推导观测值的影响函数,并进一步推导相关观测值的影响函数。(2)推导等式约束M估计中参数估计值表达式。将等式约束分为随机等式约束与固定等式约束。随机等式约束下,首先推导无污染分布时,最小二乘准则下参数的估计值以及对应观测值和先验信息的影响函数。接着根据不同的污染分布,将稳健估计准则根据污染分布的出现情况分为M-LS估计(观测值服从污染分布,参数先验信息服从正态分布)、LS-M估计(观测值服从正态分布,参数先验信息服从污染分布)和M-M估计(观测值和参数先验信息均服从污染分布)。推导三种估计准则下参数的抗差估计值。在固定等式约束下,推导观测值含污染分布时参数的抗差估计值。(3)推导等式约束稳健估计中影响函数的表达式。随机等式约束下,推导MLS、LS-M以及M-M估计模型中观测值以及先验信息的影响函数。固定等式约束下,推导观测值含污染分布时观测值的影响函数表达式。(4)随机等式约束与固定等式约束实例计算。计算随机等式约束下无污染分布时,最小二乘估计准则的计算结果。在不同的污染分布下,对比最小二乘准则和稳健估计准则参数估计值与影响函数的估计值。固定约束下,计算无污染分布时最小二乘估计准则的计算结果。观测值受污染分布时,对比最小二乘估计准则与M估计准则的参数估计值与影响函数估计值。更多还原

【Abstract】 Statisticians point out that the occurrence of gross error can never be completely eliminated,and robust estimation can reduce or eliminate the impact of gross error on parameter estimation as far as possible,so the study of robust estimation is very important.Currently,the commonly used robust estimation criteria include Maximum Likelihood estimation(M estimation),Ordered Linear estimation(L estimation)and Rank estimation(R estimation).Based on M estimation,this paper studies M estimation under stochastic equality constraint and fixed equality constraint.The principle of equivalent weight is used to transform M estimation into Robust Least Square estimation.The main work and contributions of this paper are as follows:(1)Deduce the expression of the estimated value of the general M estimation parameter and the expression of the influence function of the observed value.Based on the one-dimensional M estimation criterion and the definition of the onedimensional M estimation influence curve,the M estimation and the influence function of multi-dimensional parameter estimation are derived in detail.Taking the indirect adjustment model as an example,the influence functions are derived from the independent observations,and the influence functions of the relevant observations are further derived.(2)Deduce the expression of parameter estimation value in M estimation with equality constraint.The equality constraint is divided into random equality constraint and fixed equality constraint.Under the stochastic equality constraint without pollution distribution,the estimated values of parameters and the influence functions of the corresponding observations and prior information under the Least Square criterion are derived firstly.Then,according to different pollution distribution situation,robust estimation criterion is divided into M-LS estimation(the observations follow pollution distribution,the priori information follows normal distribution),LS-M estimation(the observations follow normal distribution,the priori information follows pollution distribution)and M-M estimation(both the observations and the priori information follows pollution distribution).The robust estimation of the parameters under three estimation criteria are derived.Under the constraint of fixed equation,the robust estimation of the parameters under the pollution distribution of the observations values are derived.(3)Deduce the expression of the influence function in the robust estimation of equality constraints.Under the constraint of random equation,the expressions of the influence functions of the observations and the prior information in the M-LS,LS-M and M-M estimation models are derived.Under the constraint of fixed equation,the expression of the influence function of the observations with pollution distribution is derived.(4)Random equality constraint and fixed equality constraint example calculation.The results of Least Squares estimation are obtained when no pollution distribution is calculated under stochastic equality constraints.Under different pollution distributions,the estimated parameters of the Least Square criterion and the robust estimation criterion are compared with the estimated values of the influence function.The results of the Least Squares estimation when the pollution distribution is calculated in the fixed constraint.When the observations are contaminated,the estimated values of the parameters and the estimated values of the influence function are compared between the Least Square estimation and the M estimation.更多还原