Non parametric estimation of smooth stationary covariance functions by interpolation methods. (English) Zbl 1204.62148
Summary: We introduce a nonparametric approach for the estimation of the covariance function of a stationary stochastic process \(X_t\) indexed by \(t \in {\mathbb{R}}^+\). The data consist of a finite number of observations of the process at irregularly spaced time points and the aim is to estimate the covariance at any lag point without parametric assumptions, and in such a way that it is a positive definite function. After interpolating the process, we use the estimator designed by E. Parzen [Technometrics 3, 167–190 (1961; Zbl 0101.12306)] for continuous-time data. Our estimator is shown to be consistent under smoothness assumptions on the covariance. Its performance is evaluated by simulations.
MSC:
| 62M09 | Non-Markovian processes: estimation |
| 62G05 | Nonparametric estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 65C60 | Computational problems in statistics (MSC2010) |
Keywords:
covariance estimation; geostatistics; positive definite; spline interpolation; stationary processesCitations:
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