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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.
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference