Quantum mechanical view of mathematical statistics

Y. Bogdanov
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引用次数: 3

Abstract

Multiparametric statistical model providing stable reconstruction of parameters by observations is considered. The only general method of this kind is the root model based on the representation of the probability density as a squared absolute value of a certain function, which is referred to as a psi function in analogy with quantum mechanics. The psi function is represented by an expansion in terms of an orthonormal set of functions. It is shown that the introduction of the psi function allows one to represent the Fisher information matrix as well as statistical properties of the estimator of the state vector (state estimator) in simple analytical forms. A new statistical characteristic, a confidence cone, is introduced instead of a standard confidence interval. The chi-square test is considered to test the hypotheses that the estimated vector converges to the state vector of a general population and that both samples are homogeneous. The expansion coefficients are estimated by the maximum likelihood method. An iteration algorithm for solving the likelihood equation is presented. The stability and rate of convergence of the solution are studied. A special iteration parameter is introduced: its optimal value is chosen on the basis of the maximin strategy. Numerical simulation is performed using the set of the Chebyshev-Hermite functions as a basis.
数理统计的量子力学观点
考虑了多参数统计模型,该模型可通过观测值稳定地重建参数。这类方法的唯一通用方法是基于概率密度表示为某个函数的平方绝对值的根模型,这个函数在量子力学中被称为psi函数。函数是用一个标准正交函数集的展开式来表示的。结果表明,psi函数的引入使得人们可以用简单的解析形式表示Fisher信息矩阵以及状态向量估计量(状态估计量)的统计性质。引入了一种新的统计特征——置信锥来代替标准置信区间。卡方检验被认为是用来检验估计向量收敛于一般总体的状态向量和两个样本是齐次的假设。用极大似然法估计了膨胀系数。提出了一种求解似然方程的迭代算法。研究了该解的稳定性和收敛速度。引入了一种特殊的迭代参数,根据极大值策略选择其最优值。以切比雪夫-埃尔米特函数集为基础进行了数值模拟。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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