Geometry Method Of Estimating Functions In Nonlinear Semi-parametric Models

On the basis of differential geometry method,this paper studies estimation function of nonlinear semi-parametric model.The probability density function family of nonlinear semi-parametric statistical model is regarded as statistical manifold.Firstly,in order to research geometrical structure,nonlinear semi-parametric statistical model is regarded as a curvature exponential family model with redundant parameter by means of parametric transformation.Secondly,the error term is assumed to be caused by normal distribution and corresponding Hilbert space is established.The geometric structure of nonlinear semi-parametric statistical model and the estimation function are studied by orthogonal subspace of Hilbert space.The tangent vectors along the direction of interest parameters and redundancy parameters are scaled into corresponding tangent space through corresponding statistical manifolds of the model.Subsequently,the Hilbert space is decomposed orthogonally.This paper discusses subspace geometry of estimating function--how to select the best estimating function and look for the asymptotic property of the corresponding estimator.The nonlinearity of nonlinear function is measured by two kinds of curvature,namely inherent curvature and parameter effect curvature.The relationship between information loss and curvature of parameter estimator is further studied in this paper.A concrete example to illustrate geometric method is needed to make it intuitive so that implementation of innovations in existing theoretical method is easier.