Jee Soo Kim’s research while affiliated with Florida State University and other places

Let λ=(λ1,…,λn), λ1 ⩽…⩽λn, and . A function is said to be arrangement increasing (AI) if (i) ƒ is permutation invariant in both arguments λ and , and (ii) ) ⩾ whenever x and x′ differ in two coordinates only, say i and j, (xi-xj)(i–j)⩾0, and xi′=xj, xj′=xi. This paper reviews concepts and many of the basic properties of AI functions, their preservation properties under mixtures, compositions and integral transformations. The AI class of functions includes as special cases other well-known classes of functions such as Schur functions, totally positive functions of order two and positive set functions. We present a number of applications of AI functions to problems in probability, statistics, reliability theory and mathematics. A multivariate extension of the arrangement ordering is also reviewed.

Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say is Laplace-smaller than for all s > 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/ 1 queues, time series, coherent systems, shock models and cumulative damage models.

Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say F ¯ is Laplace-smaller than G ¯ if ∫ 0 ∞ e -st F ¯(t)dt≤∫ 0 ∞ e -st G ¯(t)dt for all s>0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.

This paper reviews recent developments in the stochastic comparison of order statistics. The results discussed are basically: (l) Stochastic comparisons of linear combinations of order statistics from distributions F and G where G−1 F is convex or starshaped. (2) Stochastic comparisons of individual order statistics and of vectors of order statistics from underlying heterogeneous distributions by the use of majorization and Schur function theory. (3) Stochastic comparison of random processes. Applications to reliability problems are presented illustrating the use and value of the theoretical results described

The be k exponential populations. The problem of subset selection for these k populations is formulated in order to accommodate randomly censored observations. A selection procedure is proposed, based on the maximum likelihood estimators of θi i = 1, 2, … , k. It is shown that the is independent of the meanlife ξ of the censoring distribution and that the specification of P - value depends on k. This dependency is one of the distinct features that is inherent to the selection procedure under random censoring. Some desirable properties and the condition under which Gupta's (1963) multiplicative constants can be used in the proposed selection procedure are discussed. In the context of reliability, one can employ the same procedure to select the best population with respect to the hazard rate or system reliability.