IfFandGare cumulative distribution functions on [0, ∞) governing the lifetimes of specific systems under study, and ifFandGare their corresponding survival functions, thenFis said to be uniformly stochastically smaller thanG, denoted byF<(+)G, if and only if the ratiol(x) =G(x)/F(x) is nondecreasing forx∈ [0, sup{t: [0, sup {(t:F(t) > 0}). WhenFandGare absolutely continuous,F(+)Gis equivalent to the assumption that the corresponding failure rates are ordered. The applicability of the notion of uniform stochastic ordering in reliability and life testing is discussed. Given that a random sampleX1, …,Xnof lifetimes has been obtained fromF, whereFis assumed to satisfy the uniform stochastic ordering constraintF<(+)G(or alternatively,F>(+)G), whereGis fixed and known, the problem of estimatingFis addressed. It has been shown elsewhere that the method of nonparametric maximum likelihood estimation fails to provide consistent estimators in this type of problem. Here, a recursive approach is shown to provide estimators that converge uniformly toFwith probability 1 and are as close or closer toF, in the sup norm, than is the empirical distribution function. This leads to a proof of the inadmissibility of the empirical distribution function, relative to the sup norm loss criterion, when estimatingF<(+)G(orF>(+)<G) withGcontinuous. The two-sample estimation problem is also discussed briefly.