On the moments of gamma order statistics

A recurrence relation between the moments of order statistics from the gamma distribution having an integer parameter r is obtained. It is shown that if the negative moments of orders -(r?1), …, ?1 of the smallest order statistic in random samples of size n are known, then one can obtain all the moments. Tables of negative moments for r = 2 (1) 5 are also given.     

The asymptotic distribution of order statistics

For each n., X₁(n), X₂(n), …, X_n(n) are IID, with common pdf f_n(x). y₁(n) < … < Y_n (n) are the ordered values of X₁ (n), …, X_n(n). K_n is a positive integer, with lim K_n = ∞. Under certain conditions on K_n and f_n (x), it was shown in an earlier paper that the joint distribution of a special set of K_n + 1 of the variables Y₁ (n), …, Y_n (n) can be assumed to be normal for all asymptotic probability calculations. In another paper, it was shown that if f_n (x) approaches the pdf which is uniform over (0, 1) at a certain rate as n increases, then the conditional distribution of the order statistics not in the special set can be assumed to be uniform for all asymptotic probability calculations. The present paper shows that even if f_n (x) does not approach the uniform distribution as n increases, the distribution of the order statistics contained between order statistics in the special set can be assumed to be the distribution of a quadratic function of uniform random variables, for all asymptotic probability calculations. Applications to statistical inference are given.     

Tests of independence of continuous random variables which guard against special alternatives

For the problem of testing the independence of a set of continuous random variables, where the test is to be based on n independent and identically distributed observations, tests are constructed that guard against all alternatives but are particularly effective against special alternatives of interest. The tests are based on the numbers of observations falling in the cells of a grid constructed using order statistics.     

Exact first and second order moments of order statistics from the truncated exponential distribution

Exact expressions for the first and second order moments of order statistics from the truncated exponential distribution, when the proportion 1–P of truncation is known in advance, are presented in this paper. Tables of expected values and variances-covariances are given for P = 0.5 (0.1) 0.9 and n = 1 (1) 10.     

A sequential scheduling problem with impatient jobs

There are n customers that need to be served. Customer i will only wait in queue for an exponentially distributed time with rate λ_i before departing the system. The service time of customer i has distribution F_i, and on completion of service of customer i a positive reward r_i is earned. There is a single server and the problem is to choose, after each service completion, which currently in queue customer to serve next so as to maximize the expected total return. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 659–663, 2015     

Characterizations of the exponential distribution by higher-order gap

Suppose X₁,X₂, ?,X_n is a random sample of size n from a continuous distribution function F(x) and let X_1,n, ≦ X_2,n ≦ ? ≦ X_n,n be the corresponding order statistics. We define the jth-order gap g_i,j as g_i,j = X_i+j,n ? X_i,n, 1 ≦ i < n, 1 ≦ j ≦ n ? i. In this article characterizations of the exponential distribution are given by considering the distributional properties of g_k,n-k, 1 ≦ k ≦ n.     

Probability inequalities via negative dependence for random variables conditioned on order statistics

Distributions are studied which arise by considering independent and identically distributed random variables conditioned on events involving order statistics. It is shown that these distributions are negatively dependent in a very strong sense. Furthermore, bounds are found on the distribution functions. The conditioning events considered occur naturally in reliability theory as the time to system failure for k-out-of-n systems. An application to systems formed with “second-hand” components is given.     

Relaxation method for the solution of the minimax location-allocation problem in euclidean space

A method previously devised for the solution of the p-center problem on a network has now been extended to solve the analogous minimax location-allocation problem in continuous space. The essence of the method is that we choose a subset of the n points to be served and consider the circles based on one, two, or three points. Using a set-covering algorithm we find a set of p such circles which cover the points in the relaxed problem (the one with m < n points). If this is possible, we check whether the n original points are covered by the solution; if so, we have a feasible solution to the problem. We now delete the largest circle with radius r_p (which is currently an upper limit to the optimal solution) and try to find a better feasible solution. If we have a feasible solution to the relaxed problem which is not feasible to the original, we augment the relaxed problem by adding a point, preferably the one which is farthest from its nearest center. If we have a feasible solution to the original problem and we delete the largest circle and find that the relaxed problem cannot be covered by p circles, we conclude that the latest feasible solution to the original problem is optimal. An example of the solution of a problem with ten demand points and two and three service points is given in some detail. Computational data for problems of 30 demand points and 1–30 service points, and 100, 200, and 300 demand points and 1–3 service points are reported.     

Reliability of linear multistate multiple sliding window systems

This paper proposes a new model that generalizes the linear consecutive k‐out‐of‐r‐from‐n:F system to multistate case with multiple failure criteria. In this model (named linear multistate multiple sliding window system) the system consists of n linearly ordered multistate elements (MEs). Each ME can have different states: from complete failure up to perfect functioning. A performance rate is associated with each state. Several functions are defined for a set of integer numbers ρ in such a way that for each r ∈ ρ corresponding function f_r produces negative values if the combination of performance rates of r consecutive MEs corresponds to the unacceptable state of the system. The system fails if at least one of functions f_r for any r consecutive MEs for r ∈ ρ produces a negative value. An algorithm for system reliability evaluation is suggested which is based on an extended universal moment generating function. Examples of system reliability evaluation are presented. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Signature based analysis of m‐Consecutive‐k‐out‐of‐n: F systems with exchangeable components

In this article, we study reliability properties of m‐consecutive‐k‐out‐of‐n: F systems with exchangeable components. We deduce exact formulae and recurrence relations for the signature of the system. Closed form expressions for the survival function and the lifetime distribution as a mixture of the distribution of order statistics are established as well. These representations facilitate the computation of several reliability characteristics of the system for a given exchangeable joint distribution or survival function. Finally, we provide signature‐based stochastic ordering results for the system's lifetime and investigate the IFR preservation property under the formulation of m‐consecutive‐k‐out‐of‐n: F systems. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Tail hazard rate ordering properties of order statistics and coherent systems

We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k‐out‐of‐n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Evaluating methods for the reliability of a large 2‐dimensional rectangular k‐within‐consecutive‐(r,s)‐out‐of‐(m,n):F system

A 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system consists of m × n components, and fails if and only if k or more components fail in an r × s submatrix. This system can be treated as a reliability model for TFT liquid crystal displays, wireless communication networks, etc. Although an effective method has been developed for evaluating the exact system reliability of small or medium‐sized systems, that method needs extremely high computing time and memory capacity when applied to larger systems. Therefore, developing upper and lower bounds and accurate approximations for system reliability is useful for large systems. In this paper, first, we propose new upper and lower bounds for the reliability of a 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system. Secondly, we propose two limit theorems for that system. With these theorems we can obtain accurate approximations for system reliabilities when the system is large and component reliabilities are close to one. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Distribution free bounds for service constrained (Q,r) inventory systems

A classical and important problem in stochastic inventory theory is to determine the order quantity (Q) and the reorder level (r) to minimize inventory holding and backorder costs subject to a service constraint that the fill rate, i.e., the fraction of demand satisfied by inventory in stock, is at least equal to a desired value. This problem is often hard to solve because the fill rate constraint is not convex in (Q, r) unless additional assumptions are made about the distribution of demand during the lead‐time. As a consequence, there are no known algorithms, other than exhaustive search, that are available for solving this problem in its full generality. Our paper derives the first known bounds to the fill‐rate constrained (Q, r) inventory problem. We derive upper and lower bounds for the optimal values of the order quantity and the reorder level for this problem that are independent of the distribution of demand during the lead time and its variance. We show that the classical economic order quantity is a lower bound on the optimal ordering quantity. We present an efficient solution procedure that exploits these bounds and has a guaranteed bound on the error. When the Lagrangian of the fill rate constraint is convex or when the fill rate constraint does not exist, our bounds can be used to enhance the efficiency of existing algorithms. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 635–656, 2000     

Reliability of a 2‐dimensional k‐within‐consecutive‐r × s‐out‐of‐m × n:F system

A 2‐dimensional rectangular (cylindrical) k‐within‐consecutive‐r × s‐out‐of‐m × n:F system is the rectangular (cylindrical) m × n‐system if the system fails whenever k components in a r × s‐submatrix fail. This paper proposes a recursive algorithm for the reliability of the 2‐dimensional k‐within‐consecutive‐r × s‐out‐m × n:F system, in the rectangular case and the cylindrical case. This algorithm requires min ( O (mk^r(n?s)), O (nk^s(m?r))), and O (mk^rn) computing time in the rectangular case and the cylindrical case, respectively. The proposed algorithm will be demonstrated and some numerical examples will be shown. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 625–637, 2001.     

Cyclical job sequencing on multiple sets of identical machines

The problem posed in this paper is to sequence or route n jobs, each originating at a particular location or machine, undergoing r?1 operations or repairs, and terminating at the location or machine from which it originated. The problem is formulated as a 0-1 integer program, with block diagonal structure, comprised of r assignment subproblems; and a joint set of constraints to insure cyclical squences. To obtain integer results the solutions to each subproblem are ranked as required and combinations thereof are implicitly enumerated. The procedure may be terminated at any step to obtain an approximate solution. Some limited computational results are presented.     

Probabilistic analysis of an asymptotically optimal solution for the completion time variance problem

Scheduling a set of n jobs on a single machine so as to minimize the completion time variance is a well‐known NP‐hard problem. In this paper, we propose a sequence, which can be constructed in O(n log n) time, as a solution for the problem. Our primary concern is to establish the asymptotical optimality of the sequence within the framework of probabilistic analysis. Our main result is that, when the processing times are randomly and independently drawn from the same uniform distribution, the sequence is asymptotically optimal in the sense that its relative error converges to zero in probability as n increases. Other theoretical results are also derived, including: (i) When the processing times follow a symmetric structure, the problem has 2^{⌊(n−1)/2⌋} optimal sequences, which include our proposed sequence and other heuristic sequences suggested in the literature; and (ii) when these 2^{⌊(n−1)/2⌋} sequences are used as approximate solutions for a general problem, our proposed sequence yields the best approximation (in an average sense) while another sequence, which is commonly believed to be a good approximation in the literature, is interestingly the worst. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 373–398, 1999     

Estimating a survival curve when new is better than used in expectation

Let X be a positive random variable. The distribution F of X is said to be “new better than used in expectation,” or “NBUE,” if E(X) ⩾ E(X – t|X > t) for all t ⩾ 0. Suppose X₁, …, X_n, is a random sample from an NBUE distribution F. The problem of estimating F by a distribution which is itself NBUE is considered. The estimator G_n, defined as the NBUE distribution supported on the sample which minimizes the (sup norm) distance between the NBUE class and the empirical distribution function, is studied. The strong uniform consistency of G_n, is proven, and a numerical algorithm for obtaining G_n, is given. Our approach is applied to provide an estimate of the distribution of lifetime following the diagnosis of chronic granulocytic leukemia based on data from a National Cancer Institute study.     

Isotonic median regression for orders representable by rooted trees

The isotonic median regression problem arising in statistics is as follows. We are given m observations falling into n sets, the ith set containing m_i observations. The problem requires the determination of n real numbers, the ith being the value “fitted” to each observation in the ith set. These n numbers chosen must satisfy certain (total or partial) order requirements and minimize the distance between the vectors of observed and fitted values in the l₁ norm. We present a simple algorithm, of time complexity O(mn), for calculating isotonic median regression for orders representable by rooted trees. We believe that this algorithm is the best currently available for this problem. The algorithm is validated by a linear programming approach which provides additional insight.     

Confidence bounds and propagation of uncertainties in system availability and reliability computations

This paper develops bounds on the uncertainties in system availabilities or reliabilities which have been computed from structural (series, parallel, etc.) relations among uncertain subsystem availabilities or reliabilities. It is assumed that the highly available (reliable) subsystems have been tested or simulated to determine their unavailabilities (unreliabilities) to within some small percentages of uncertainty. It is shown that series, parallel and r out of n structures which are nominally highly available will have unavailability uncertainties whose percentages errors are of the same order as the subsystem uncertainties. Thus overall system analysis errors, even for large systems, are of the same order of magnitude as the uncertainties in the component probabilities. Both systematic (bias type) uncertainties and independent random uncertainties are considered.     

Test for the extreme value and weibull distributions based on normalized spacings

Discussed in this article are tests for the extreme-value distribution, or, equivalently, for the two-parameter Weibull distribution when parameters are unknown and the sample may be censored. The three tests investigated are based on the median, the mean, and the Anderson-Darling A² statistic calculated from a set zⁱ of values derived from the spacings of the sample. The median and the mean have previously been discussed by Mann, Scheuer, and Fertig [10] and by Tiku and Singh [14]. Asymptotic distributions and points are given for the test statistics, based on recently developed theory, and power studies are conducted to compare them with each other and with two other statistics suitable for the test. Of the normalized spacings tests, A² is recommended overall; the mean also gives good power in many situations, but can be nonconsistent.