Spectral and Probability Density Estimation From Irregularly Observed Data

Abstract

Let X = {X(t), − ∞ < t < ∞} be a stationary stochastic process with univariate probability density f(x), covariance function R(t), and spectral density φ(λ). The nonparametric estimation of f, R, and φ on the basis of irregularly-spaced observations $\left\{ {{\text{X}}\left( {{\text{t}}_{\text{k}} } \right){\text{,t}}_{\text{k}} } \right\}_{{\text{k = 1}}}^{\text{n}}$ is considered. The sampling schemes {t_k} which allow the consistent estimation of f, R, and φ, as the sample size n → ∞, are identified and the asymptotic statistical properties of the corresponding estimates ${\hat f_n}\left( x \right),{\hat f_n}\left( t \right)$, and $\hat \varphi _{\text{n}} \left( \lambda \right)$ are presented.