Trust Region Method And Statistical Properties Of M-estimate In The Nonlinear Regression Model Of Implicit Function

With the rapid development of science and technology, nonlinear statistical models are introduced in all kinds of regions such as biology, medicine, economics and engineering etc. It is impossible to linearize models in most conditions. Nonlinear models are generalizations of linear models, so their theoretical researches draw more and more statisticians' attentions.In this paper, we consider a nonlinear regression model of implicit function:f(θ)+ ε = 0, where θ is parameter and s is random error. The M-estimate theoriesand algorithm are researched systematically, and the applications of geometrical concepts and methods are stressed.There are five chapters in the paper. Chapters 1, the theories of linear and nonlinear models are introduced, and we also discuss prepared knowledge which include M-estimate, nonlinear least square estimate, curvature array and trust region method. Chapter 2, the trust region method of M-estimate for parameter θ is given, and the assumptions, algorithm and convergence properties are investigated. Chapter 3, themodel of linear constraints is considered. α_i~Tθ=b_i i∈E , a_j~Tθ≥>b_j,j ∈ I , where a_i,a_j are given vectors, b_i,b_j are given numbers, E is an index set of equality, andI is a index set of inequality. Chapter 4, we get the statistical properties of M-estimate that is derived from trust region algorithm. The properties cover asymptotic normality, biased estimate and non-least variance. Chapter 5, an example is given by stochastic simulation method, which is the application of least absolute deviation estimate in location by GPS.In the end of this paper, we draw the conclusion that M-estimate is better than least square estimate under disturbed conditions.