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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]