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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 \((X_ 1,Y_ 1),...,(X_ N,Y_ N)\) be i.i.d. as (X,Y). Here the number N is assumed to be unknown. Suppose that one observes only those pairs \((X_ i,Y_ i)\) for which \(i\leq N\) and \(Y_ i\leq X_ i\). Denote \(n=\#\{i\leq N:\) \(Y_ i\leq X_ i\}\). The estimators \(\hat F_ n\), \(\hat G_ n\) and \(\hat N_ n\) 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 \(\sqrt{n}[\hat F_ n-F]\) to some Gaussian process is established.
Reviewer: R.Mnatsakanov

62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
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