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A life distribution function F is said to have an increasing failure rate average if H (x) ∕ x is nondecreasing where H (x) is the corresponding cumulative hazard function. In this paper we provide a uniformly strongly consistent estimator of F and derive the convergence in distribution of the estimator at a point where H (x) ∕ x is increasing using the arg max theorem. We also show using simulations that, unlike other estimators of the past, this new estimator outperforms the empirical distribution in terms of mean square error at all quantiles. An example is also discussed to illustrate the theoretical results • Uniformly consistent estimation of an IFRA distribution function. • Convergence in distribution of an estimate of an IFRA distribution. • Use of the arg max theorem for asymptotic distribution. [ABSTRACT FROM AUTHOR]

In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t for a particular type of failure in the presence of other risks. Its estimation and asymptotic distribution theory have been studied by many. In some cases there are reasons to believe that the CIFs due to two types of failure are order restricted. Several procedures have appeared in the literature for testing for such orders. In this paper we initiate the study of estimation of two CIFs subject to a type of stochastic ordering, both when there are just two causes of failure and when there are more than two causes of failure, treating those other than the two of interest as a censoring mechanism. We do not assume independence of the two types of failure of interest; however, these are assumed to be independent of the other causes in the censored case. Weak convergence results for the estimators have been derived. It is shown that when the order restriction is strict, the asymptotic distributions are the same as those for the empirical estimators without the order restriction. Thus we get the restricted estimators “free of charge”, at least in the asymptotic sense. When the two CIFs are equal, the asymptotic MSE is reduced by using the order restriction. For finite sample sizes simulations seem to indicate that the restricted estimators have uniformly smaller MSEs than the unrestricted ones in all cases. [Copyright &y& Elsevier]

If X is a life distribution with finite mean then its mean residual life function (MRLF) is defined by M(x)=E[X−x|X>x]. It has been found to be a very intuitive way of describing the aging process. Suppose that M1 and M2 are two MRLFs, e.g., those corresponding to the control and the experimental groups in a clinical trial. It may be reasonable to assume that the remaining life expectancy for the experimental group is higher than that of the control group at all times in the future, i.e., M1(x)⩽M2(x) for all x. Randomness of data will frequently show reversals of this order restriction in the empirical observations. In this paper we propose estimators of M1 and M2 subject to this order restriction. They are shown to be strongly uniformly consistent and asymptotically unbiased. We have also developed the weak convergence theory for these estimators. Simulations seem to indicate that, even when M1=M2, both of the restricted estimators improve on the empirical (unrestricted) estimators in terms of mean squared error, uniformly at all quantiles, and for a variety of distributions. [Copyright &y& Elsevier]