Estimating a survival curve when new is better than used with respect to a distinguished set

Hollander, Park, and Proschan define a survival function S of a positive random variable X to be new better than used at age t₀ (NBU-{t₀}) if S satisfies $ \begin{array}{*{20}c} {\frac{{S(x + t_0)}}{{S\left({t_0} \right)}} \le S\left(x \right),} & {{\rm for}\,{\rm all}\,x\, \ge \,0,} \\ \end{array}$ where S(x) = P(X > x). The NBU-{t₀} class is a special case of the NBU-A family of survival distributions, where A is a subset of [0, ∞). These families introduce a variety of modeling possibilities for use in reliability studies. We treat problems of nonparametric estimation of survival functions from these classes by estimators which are themselves members of the classes of interest. For a number of such classes, a recursive estimation technique is shown to produce closed-form estimators which are strongly consistent and converge to the true survival distribution at optimal rates. For other classes, additional assumptions are required to guarantee the consistency of recursive estimators. As an example of the latter case, we demonstrate the consistency of a recursive estimator for S ∈ NBU-[t₀, ∞) based on lifetime data from items surviving a preliminary “burn-in” test. The relative precision of the empirical survival curve and several recursive estimators of S are investigated via simulation; the results provide support for the claim that recursive estimators are superior to the empirical survival curve in restricted nonparametric estimation problems of the type studied here.     

Vector processing approach to constrained network problems

Modification of algorithms designed for scalar computing, to take advantage of vector processing, raises several challenges. This article presents the vectorization of the primal simplex based network algorithm and results in a 50% improvement in computational time. One of the major contributors to this improvement is the matching of the size of the pricing candidate list to the vector register size. The side constraints are relaxed into a single surrogate constraint. The single constraint network algorithm is vectorized and used as the basis for solving large-scale constrained network problems. Computational experiments are presented which illustrate the vectorization of the network code as well as the ability of the surrogate constraint approach to deal with large constrained network problems.     

Estimation of the survival function of a repairable standby series system

This article deals with the statistical analysis of an N-component series system supported by an active standby and one repair facility. Assuming that the life and repair times of the components are independent exponential random variables, the probability distribution of the first passage to the system failure time is shown to be a convolution of two independent exponential distributions. Three observation schemes are considered to obtain the maximum likelihood estimates of the survival function. Information matrices are supplied. Numerical results based on Monte Carlo simulation are presented. It is noted that component level information (failure rate, repair rate) is not necessary for estimating the survival function of the system.     

An algorithm and new penalties for concave integer minimization over a polyhedron

We present a branch-and-bound algorithm for globally minimizing a concave function over linear constraints and integer variables. Concave cost functions and integer variables arise in many applications, such as production planning, engineering design, and capacity expansion. To reduce the number of subproblems solved during the branch-and-bound search, we also develop a framework for computing new and existing penalties. Computational testing indicates that penalties based on the Tuy cutting plane provide large decreases in solution time for some problems. A combination of Driebeek-Tomlin and Tuy penalties can provide further decreases in solution time. © 1994 John Wiley & Sons, Inc.     

Localization of optimal strategies in certain games

Assume the payoffs of a matrix game are concave in the index of the maximizing player. That player is shown to have an optimal strategy which uses at most two consecutive pure strategies, identifiable through approximate solution of a related continuous game. Generalizations are given, and the results are applied to a motivating hidden-target model due to Shapley. © 1994 John Wiley & Sons, Inc.     

A note on the optimality conditions for the bilevel programming problem

A counterexample is given to demonstrate that previously proposed necessary conditions for the bilevel programming problem are not correct. An interpretation of the difficulty is given by appealing to a “theorem of alternative” result presented in the original work.     

Stochastic linear fractional programming with the ratio of independent Cauchy variates

In this article is studied a stochastic linear fractional programming problem, in which the parameters of both the numerator and the denominator are assumed to be mutually independent Cauchy variates. The deterministic equivalent of the problem is obtained and is shown to be a linear fractional program. A numerical example is also added for illustration.     

Estimation of the common scale parameter of two pareto distributions in censored samples

The problem of estimation of the common scale parameter of two Pareto distributions with unknown and unequal shape parameters in censored samples is considered. The uniformly minimum variance unbiased estimator (UMVUE) is given along with the UMVUE of its variance.     

An extremal problem in subconnectivity

The connectivity of a subgraph of a graph can exceed the connectivity of the graph. We call the largest of the connectivities of all subgraphs the subconnectivity. We then give the exact solution to the extremal problem of determining the maximum number of lines in a p-point graph of subconnectivity two.