Nonparametric Estimation of a Distribution Subject to a Stochastic Precedence Constraint

Abstract

For any two random variables X and Y with distributions F and G de ned on 601ˆ5, X is said to stochastically precede Y if P4X µ Y 5 ¶ 1=2. For independent X and Y , stochastic precedence (denoted by X µ sp Y ) is equivalent to E6G4Xƒ57 µ 1=2. The applicability of stochastic precedence in various statistical contexts, including reliability modeling, tests for distributional equality versus various alternatives, and the relative performance of comparable tolerance bounds, is discussed. The problem of estimating the underlying distribution(s) of experimental data under the assumption that they obey a stochastic precedence (sp) constraint is treated in detail. Two estimation approaches, one based on data shrinkage and the other involving data translation, are used to construct estimators that conform to the sp constraint, and each is shown to lead to a root n-consistent estimator of the underlying distribution. The asymptotic behavior of each of the estimators is fully characterized. Conditions are given under which each estimator is asymptotically equivalent to the corresponding empirical distribution function or, in the case of right censoring, the Kaplan-Meier estimator. In the complementary cases, evidence is presented, both analytically and via simulation, demonstrating that the new estimators tend to outperform the empirical distribution function when sample sizes are suf ciently large.

Hypothesis tests with precedence probabilities and precedence‐type tests

“…Arcones et al 42 argued that stochastic ordering is a too strong assumption to be used practically since it assumes uniform domination of one distribution over another. They introduce the concept of precedence ordering.…”

Bayesian semiparametric modeling for stochastic precedence, with applications in epidemiology and survival analysis

“…Arcones et al (2002) introduced stochastic precedence, a weaker constraint than stochastic order, which builds the restriction through Pr(X 1 ≤ X 2 ) for two independent random variables X 1 and X 2 with distribution functions F 1 and F 2 , respectively. Specifically, X 1 is said to stochastically precede X 2 if Pr(X 1 ≤ X 2 ) ≥ 0.5, denoted by X 1 ≤ sp X 2 or, equivalently, F 1 ≤ sp F 2 .…”

“…For example, the normal distribution is stochastic precedence constrained with respect to its mean, in particular, N(θ 1 , σ 2 1 ) ≤ sp N(θ 2 , σ 2 2 ) if and only if θ 1 ≤ θ 2 , with no further restriction needed on the variances σ 2 1 and σ 2 2 . Arcones et al (2002) discussed potential applications for stochastic precedence, and developed classical nonparametric stochastic precedence constrained estimators for F 1 (with F 2 assumed known) and for both F 1 and F 2 .…”

Relations Between Stochastic Orderings and Generalized Stochastic Precedence

“…The interest of this concept for applications has been pointed out several times in the literature (see in particular [1], [3] and [9]). We recall the reader's attention on the fact that stochastic precedence does not define a stochastic order in that, for instance, it is not transitive.…”

“…When X 1 , X 2 are independent the implication X 1 st X 2 ⇒ X 1 sp X 2 holds (see [1]). It is also easy to find several other examples of bivariate probability models where the same implication holds.…”