Wavelet estimates of density by observations from mixtures. (Ukrainian, English) Zbl 1071.62033

Pokhyl’ko, D.

Teor. Jmovirn. Mat. Stat. 70, 121-130 (2004); translation in Theory Probab. Math. Stat. 70, 135-145 (2005).

Projection estimates for a PDF are constructed by a sample \(\Xi_N\) from a mixture with varying concentrations. I.e., it is assumed that \(\Xi_N=(\xi_1,\dots,\xi_N)\), where \(\xi_j\) are independent and the PDF of \(\xi_j\) is \(p_{t_j}(x)=\sum_{k=1}^M w_k(t_j)p_k(x)\), \(p_k\) being the PDF of the \(k\)-th component in the mixture, \(w_k(t_j)\), \(k=1,\dots,M\), are the mixing probabilities at the moment \(t_j\) of the \(j\)-th observation. The author is interested in the case where a projection is made on some wavelet system. Linear and hard thresholded estimates are considered. Rates of convergence are derived.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

62G07	Density estimation
42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

projection estimate; adaptive estimate; mixture with varying concentrations; hard threshold