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A class of statistics based on the integrated weighted difference in Kaplan-Meier estimators is introduced for the two-sample censored data problem. With positive weight functions these statistics are intuitive for and sensitive against the alternative of stochastic ordering. The standard weighted log-rank statistics are not always sensitive against this alternative, particularly if the hazard functions cross. Qualitative comparisons are made between the weighted log-rank statistics and these weighted Kaplan-Meier (WKM) statistics. A statement of null asymptotic distribution theory is given and the choice of weight function is discussed in some detail. Results from small-sample simulation studies indicate that these statistics compare favorably with the log-rank procedure even under the proportional hazards alternative, and may perform better than it under the crossing hazards alternative.
Pepe et al. (1989) studied this question.