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Wavelet estimation of a density in a GARCH-type model. (English) Zbl 1298.62061

Summary: We consider the GARCH-type model: \(S=\sigma^2Z\), where \(\sigma^2\) and \(Z\) are independent random variables. The density of \(\sigma^2\) is unknown whereas the one of \(Z\) is known. We want to estimate the density of \(\sigma^2\) from \(n\) observations of \(S\) under some dependence assumption (the exponentially strongly mixing dependence). Adopting the wavelet methodology, we construct a nonadaptive estimator based on projections and an adaptive estimator based on the hard thresholding rule. Taking the mean integrated squared error over Besov balls, we prove that the adaptive one attains a sharp rate of convergence.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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