Abstract
This paper is dedicated to the study of local polynomial estimations of time-varying coefficients for a local stationary diffusion model. Based on local polynomial fitting, the estimations of drift parametric functions are obtained by using local weighted least squares method. By applying the forward Kolmogorov equation, the estimation of the diffusion coefficient is proposed. The consistency, asymptotic normality and uniform convergence of the estimations that we proposed are established.
Highlights
1 Introduction The theory of local stationary process is introduced by Dahlhaus [, ]
Speaking, a process is locally stationary if the process behaves like a stationary diffusion process in a neighborhood of a chosen time point
We study local polynomial estimations of time-varying coefficients for local stationary diffusion models
Summary
The theory of local stationary process is introduced by Dahlhaus [ , ]. We study local polynomial estimations of time-varying coefficients for local stationary diffusion models. In Section , the local polynomial estimations of the drift parameters are proposed, and the consistency, asymptotic normality and uniform convergence rate of the estimations for the drift functions are established. By using p-polynomial fitting for α(u) and β(u) (see Fan and Gijbels [ ]), we can obtain the local estimations of time-varying parameters α(u) and β(u) by minimizing the following objective function,. We establish the asymptotic properties of the local polynomial estimations of the drift parameters as the following. The following Theorem and Theorem show the weak consistency and asymptotic normality of our proposed local estimations, respectively. Theorem shows the consistence and asymptotic normality of the estimation of the diffusion coefficient, and Theorem gives the uniform convergence rate of our estimation
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