Positive Definite Nonparametric Regression with Application to Covariance Function Estimation

Abstract

We propose a new nonparametric regression method subject to the positive definiteness constraint for estimating the covariance function of a stationary stochastic process. Our method ensures that the estimator can satisfy positive definiteness, as well as both isotropy and monotonicity. Also, unlike previous methods, our estimators can be computed automatically with cross validation. To achieve these advantages, we define our estimators by taking the integral transform of kernel distribution estimators and suggest universal estimation algorithm based on the framework of Estimation of Distribution Algorithms (EDAs). A small simulation study is performed that demonstrates the usefulness of our method to estimate the covariance function more robustly than other typical nonparametric regression estimators. An application to the sic.100 dataset is also presented.