Journal of Statistical Planning and Inference 22 (1989) 213-222 North-Holland 213 SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS FOR QUANTILES IN THE PRESENCE OF CENSORING I. GIJBELS and N. VERAVERBEKE Limburgs Universitair Centrum, B-3610 Diepenbeek, Belgium Received 25 January 1988; revised manuscript received 23 May 1988 Recommended by R.N. Bhattacharya Abstruct: This paper studies the asymptotic properties of sequential fixed-width confidence inter- vals for quantiles of the survival distribution function in the random censorship model. The inter- val is formed by a pair of quantiles of the product-limit estimator. The set up requires new results on the almost sure behaviour of such empirical quantiles. AMS Subject Classification: Primary 60F15; secondary 62LlO. Key words and phrases: Bahadur representation; product-limit estimator; quantiles; sequential fixed-width confidence intervals. 1. Introduction Let Xi, . . . . X,, be nonnegative survival times and Y,, . . . , Y, nonnegative censor- ing times. The X, are independent and identically distributed (i.i.d.) with contin- uous distribution function zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG F, the Y, are i.i.d. with continuous distribution function G and for each i, X, and I$ are independent. In the right random censorship model the observations are the pairs (7;, 6,), i= 1, . . . , n, where 7; = min(X,, Y) and 6;= 1(X, 5 Y). It follows that the 7; are i.i.d. with continuous distribution function H= 1 -(l -F)(l -G). A common estimator for 1 -F(t), based on (7;,6,), i= 1, . . . . n, is the product- limit (PL) estimator 1 -F,,(t) defined by (see Kaplan and Meier (1958)): 0 if t 2 Ten,, where T,,,< ... I Ten, are the order statistics of the 7; and 6(i), . . . , ~3~~) are the corre- sponding 6,. 0378-3758/89/$3.50 CCC 1989, Elsevier Science Publishers B.V. (North-Holland)