## Improved estimates for moments by observations from mixtures.(Ukrainian, English)Zbl 1070.62017

Teor. Jmovirn. Mat. Stat. 70, 74-81 (2004); translation in Theory Probab. Math. Stat. 70, 83-92 (2005).
A sample $$\Xi_N=(\xi_1,\dots,\xi_N)$$ from a mixture with varying concentrations is considered, i.e., $$\xi_j$$ are independent r.v.s and $\Pr\{\xi_j<x\}=\sum_{m=1}^M w_j^m H_m(x),$ where $$w_j^m$$ are mixing probabilities for the $$j$$-th observation (known), and $$H_m$$ are the unknown distributions of the mixture components. The weighted empirical CDF (weCDF) $$\hat H_m(x)= N^{-1} \sum_{j=1}^N a_j^m\mathbf{1}_{\{\xi_j<x\}}$$ can be used for estimation of $$H_m$$, where $$a_j^m$$ are nonrandom weights defined by $$(w_j^m)_{j,m}$$. The functional moment of the $$m$$-th component $$\bar g_m=\int g(t)H_m(dt)$$ can be estimated as $\hat g_m=\int g(t)\hat H_m(dt)=N^{-1}\sum_{j=1}^N a_j^m g(\xi_j).$ Since the optimal weights $$a_j^m$$ for estimation of $$H_m$$ usually are negative for some $$j$$, the estimates $$\hat g_m$$ can have some bad properties, e.g., the estimate of the second moment can be negative. The authors consider some improvements of $$\hat H_m$$ (e.g., the simplest one is $$H^{+}(x)=\sup_{y<x}\hat H_m(y)$$) which makes it a true CDF. The improved moment estimator is just an integral with respect to the improved estimate of the CDF. It is shown that the asymptotic distribution of the improved moment estimates is the same as that of $$\hat g_m$$.

### MSC:

 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference