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Estimating a distribution function with truncated data. (English) Zbl 0574.62040
Let X and Y be independent positive r.v.’s with unknown distribution functions F and G and let $(X\sb 1,Y\sb 1),...,(X\sb N,Y\sb N)$ be i.i.d. as (X,Y). Here the number N is assumed to be unknown. Suppose that one observes only those pairs $(X\sb i,Y\sb i)$ for which $i\le N$ and $Y\sb i\le X\sb i$. Denote $n=\#\{i\le N:$ $Y\sb i\le X\sb i\}$. The estimators $\hat F\sb n$, $\hat G\sb n$ and $\hat N\sb n$ for F, G and N, respectively, are described. Under some conditions on the d.f.’s F and G consistency results for these estimators are obtained. The convergence in distribution of the processes $\sqrt{n}[\hat F\sb n-F]$ to some Gaussian process is established.
Reviewer: R.Mnatsakanov

##### MSC:
 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics
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