Uniform Strong Consistent Estimation of an Ifra Distribution Function

Let F̄n be an estimator of an IFRA survival function F̄ and let A be such that 0 < F̄(A) < 1. The main result constructs an IFRA estimator by splicing the smallest increasing failure rate on the average majorant and greatest increasing failure rate on the average minorant of the restrictions of F̄n to the intervals [0, A] and [A, ∞), respectively. The resulting etı̃mator F̃n has the property that supx |F̃n − F̄| ≤ k supx |F̄n − F̄ where k ≥ 2, and k = 2 if and only if A is the median of F. As a consequence, if F̄n represents the empirical survival function, or the Kaplan-Meier estimator, the estimator F̃n inherits the strong and uniform convergence properties, as well as the optimal rates of convergence of the empirical survival function and Kaplan-Meier estimator respectively. Simulations show a substantial improvement in mean-squared error when comparing F̃n to those IFRA estimators available in the literature. Under suitable conditions, asymptotic confidence intervals for F̄(t0) are also provided.