Abstract
Let be an estimator obtained by integrating a kernel type density estimator based on a random sample of size n. A central limit theorem is established for the target statistic , where the underlying random vector forms an asymptotically stationary absolutely regular stochastic process, and is an estimator of a multivariate parameter ξ by using a vector of U‐statistics. The results obtained extend or generalize previous results from the stationary univariate case to the asymptotically stationary multivariate case. An example of asymptotically stationary absolutely regular multivariate ARMA process and an example of a useful estimation of F(ξ) are given in the applications.
Highlights
The purpose of this paper is to estimate the value of a multivariate distribution function, called the target distribution function, at a given point, when observing a nonstationary process
There must be a connection between the process and the target distribution
The point at which we want to estimate the target distribution is not any bona fide vector, for we will assume that it can be estimated by a vector of U-statistics
Summary
The purpose of this paper is to estimate the value of a multivariate distribution function, called the target distribution function, at a given point, when observing a nonstationary process. The point at which we want to estimate the target distribution is not any bona fide vector, for we will assume that it can be estimated by a vector of U-statistics Such a problem is clearly out of reach with that generality, and we will assume that, though nonstationary, the process exhibits an asymptotic form of stationarity and has a suitable mixing property. Even if the empirical distribution function is optimal with respect to the speed of convergence of the mean square error, it is not appropriate for not taking care of the fact that F is smooth, and of the existence of a density f It is, natural to seek an estimator of the target distribution which is smooth. For the study of some limit theorems dealing with U-statistic for processes which are uniformly mixing in both directions of time, the reader is referred to Denker and Keller
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
