Using Dirichlet Kernel to Analyze Long Memory Time Series

Abstract

In this paper, we propose a new spectral density estimator based on Dirichlet -Kernel homogeneity in the cycle diagram. Our methodology is nonparametric and can be applied to processes that do not necessarily follow a normal distribution. Dirichlet kernel is an asymmetric density function of a shape that varies according to the frequency at which the spectrum is estimated. Dirichlet kernel smoothing was introduced because Dirichlet kernel diverges at zero when its bandwidth shrinks; it becomes smoother and more attractive than the cycle diagram when the process is a long memory. It automatically adjusts to a time series range. If the process is a short memory, the resulting estimation of the spectral density is automatically constrained, while the estimator diverges at the origin when applied to the certified long-term data. Kernel smoothing or kernel density estimation is a well-known methodology for non-parametric characterization of the probability density function of a random variable or random vector. It can be considered as a proxy of packages for histograms, and is particularly useful in multivariate cases; density estimation for non-parametric alternatives for regression and classification can be used to represent how the conditional probabilities of a categorical variable depend on quantitative variables. Our main goal in this paper is to find a non-parametric estimation for a long memory time series, to reconsider the Dirichlet Kernel estimator and to study its asymptotic properties in detail. Our main contribution to this paper is to find asymptotic expressions for point bias, point variance, mean squared error (MSE) and mean integrated square error (MISE). These results generalize to those of the beta kernel. Choosing the optimal bandwidth for parameter b (bandwidth parameters b).