基于变分法的一元非线性回归分析

摘要

The paper studies the analysis of two dimensional continuous random variable nonlinear regression with statistical correlation, that is, η = f(ξ) + ε , the distribution ξ is known and ξ is independent of ε, ε N(0, σ2 ) . The joint density (ξ,η) is φ(x,y) , and based on the study of the statistical properties φ(x,y) , geometrical properties, the nonlinear regression problem is considered to be attributed to functional extremum problem. By the variational method, for the different distxibutions of ξ and the dif- ferent constraint conditions of η , according to a group of sample values of (ξ, η) , the analytical method to determine the regression curvesf(x) is obtained.%研究具有统计相关关系的二维连续型随机变量（ξ，η）的非线性回归分析问题：η=f（ξ）＋ε，ξ的分布是已知的，ξ与ε独立，ε—N（0，σ2）．首先推得（ξ，η）的联合密度φ（x，y），通过对φ（x，y）统计学性质、几何性质研究，将非线性回归问题归结为泛函极值问题，应用变分法，对于ξ服从不同的分布及η满足不同的约束条件，得出了依据（ξ，η）的一组样本值确定回归曲线f（x）的解析解的方法．