ISSN:
1588-2829

Keywords:
Primary 60F15
;
62G05
;
Strong consistency
;
product limit
;
censored data

Source:
Springer Online Journal Archives 1860-2000

Topics:
Mathematics

Notes:
Abstract In reliability and survival-time studies one frequently encounters the followingrandom censorship model:X 1,Y 1,X 2,Y 2,… is an independent sequence of nonnegative rv's, theX n' s having common distributionF and theY n' s having common distributionG, Z n =min{X n ,Y n },T n =I[X n 〈-Y n ]; ifX n represents the (potential) time to death of then-th individual in the sample andY n is his (potential) censoring time thenZ n represents the actual observation time andT n represents the type of observation (T n =O is a censoring,T n =1 is a death). One way to estimateF from the observationsZ 1.T 1,Z 2,T 2, … (and without recourse to theX n' s) is by means of theproduct limit estimator $$\hat F_n $$ (Kaplan andMeier [6]). It is shown that $$\left| {\hat F_N (x) - F(x)} \right| \to 0$$ a.s., uniformly on [0,T] ifH(T −)〈1 wherel−H=(l−F) (l−G), uniformly onR if $$G(T_{\bar F} )〈 1$$ whereT F =sup {x:F(x)〈1}; rates of convergence are also established. These results are used in Part II of this study to establish strong consistency of some density and failure rate estimators based on $$\hat F_N $$ .

Type of Medium:
Electronic Resource

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