# zbMATH — the first resource for mathematics

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

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