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Asymptotics of a kernel estimatefor distribution density constructed from observations of a mixture with varying concentration. (English. Ukrainian original) Zbl 0953.62036
Theory Probab. Math. Stat. 59, 161-171 (1999); translation from Teor. Jmovirn. Mat. Stat. 59, 156-166 (1998).
The problem of probability density estimation is considered for the case where the training sample contains observations from $$M$$ different subpopulations. It is unknown to which subpopulation an element belongs. The statistician knows only the probabilities $$w_j^k$$ to obtain an element of the $$k$$-th subpopulation in the $$j$$-th observation. If $$H_k$$ denotes the distribution of observations from the $$k$$-th subpopulation, then the distribution of the $$j$$-th observation is $\Pr\{\xi_j<x\}=\sum_{k=1}^M w_j^k H_k(x),\quad j=1,\dots, N.$ The author considers a weighted kernel estimate for the density of $$H_k$$: $\hat p_N^k(x)=(Nh_N)^{-1}\sum_{j=1}^N a_j^N K\left((x-\xi_j)/ h_N)\right),$ where $$a_j^k$$ are some nonrandom weights, $$K$$ is a kernel and $$h_N$$ is a bandwidth.
An asymptotic normality result and a convergence rate in $$L_2$$ are obtained for this estimate. An adaptive bandwidth selection algorithm is constructed for which the optimal $$L_2$$ convergence rate is attained.
##### MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference