Let $V_{(r,n,\tilde {m}_n,k)}^{(p)}$ and $W_{(r,n,\tilde {m}_n,k)}^{(p)}$ be the $p$-spacings of generalized order statistics based on absolutely continuous distribution functions $F$ and $G$, respectively. Imposing some conditions on $F$ and $G$ and assuming that $m_1=\cdots =m_{n-1}$, Hu and Zhuang (2006. Stochastic orderings between p-spacings of generalized order statistics from two samples. Probability in the Engineering and Informational Sciences 20: 475) established $V_{(r,n,\tilde {m}_n,k)}^{(p)} \leq _{{\rm hr}} W_{(r,n,\tilde {m}_n,k)}^{(p)}$ for $p=1$ and left the case $p\geq 2$ as an open problem. In this article, we not only resolve it but also give the result for unequal $m_i$'s. It is worth mentioning that this problem has not been proved even for ordinary order statistics so far.

Due to the importance of generalized order statistics (GOS) in many branches of Statistics, a wide interest has been shown in investigating stochastic comparisons of GOS. In this article, we study the likelihood ratio ordering of $p$-spacings of GOS, establishing some flexible and applicable results. We also settle certain unresolved related problems by providing some useful lemmas. Since we do not impose restrictions on the model parameters (as previous studies did), our findings yield new results for comparison of various useful models of ordered random variables including order statistics, sequential order statistics, $k$-record values, Pfeifer's record values, and progressive Type-II censored order statistics with arbitrary censoring plans. Some results on preservation of logconvexity properties among spacings are provided as well.

Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan–Lusztig immanants, which are indexed by permutations, involve $q=1$ specializations of Type A Kazhdan–Lusztig polynomials, and were defined by Rhoades and Skandera (2006, Journal of Algebra 304, 793–811). Using results of Haiman (1993, Journal of the American Mathematical Society 6, 569–595) and Stembridge (1991, Bulletin of the London Mathematical Society 23, 422–428), Rhoades and Skandera showed that Kazhdan–Lusztig immanants are nonnegative on matrices whose minors are nonnegative. We investigate which Kazhdan–Lusztig immanants are positive on k-positive matrices (matrices whose minors of size $k \times k$ and smaller are positive). The Kazhdan–Lusztig immanant indexed by v is positive on k-positive matrices if v avoids 1324 and 2143 and for all noninversions $i< j$ of v, either $j-i \leq k$ or $v_j-v_i \leq k$ . Our main tool is Lewis Carroll’s identity.

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

Given a sequence of polynomials $\{x_k(q)\}_{k \ges 0}$, define the transformation

$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$

$n\ges 0$

$[x_{i+j}(q)]_{i,j \ges 0}$

$\{y_n(q)\}_{n\ges 0}$

We study a class of optimal allocation problems, including the well-known bomber problem, with the following common probabilistic structure. An aircraft equipped with an amount x of ammunition is intercepted by enemy airplanes arriving according to a homogeneous Poisson process over a fixed time duration t. Upon encountering an enemy, the aircraft has the choice of spending any amount 0 ≤ y ≤ x of its ammunition, resulting in the aircraft's survival with probability equal to some known increasing function of y. Two different goals have been considered in the literature concerning the optimal amount K(x, t) of ammunition spent: (i) maximizing the probability of surviving for time t, which is the so-called bomber problem; and (ii) maximizing the number of enemy airplanes shot down during time t, which we call the fighter problem. Several authors have attempted to settle the following conjectures about the monotonicity of K(x, t): (A) K(x, t) is decreasing in t; (B) K(x, t) is increasing in x; and (C) the amount x - K(x, t) held back is increasing in x. Conjectures (A) and (C) have been shown for the bomber problem with discrete ammunition, while (B) is still an open question. In this paper we consider both time and ammunition to be continuous, and, for the bomber problem, we prove (A) and (C), while, for the fighter problem, we prove (A) and (C) for one special case and (B) and (C) for another. These proofs involve showing that the optimal survival probability and optimal number shot down are totally positive of order 2 (TP2) in the bomber and fighter problems, respectively. The TP2 property is shown by constructing convergent sequences of approximating functions through an iterative operation which preserves TP2 and other properties.

A well-known theorem states that if $f\left( z \right)$ generates a $\text{P}{{\text{F}}_{r}}$ sequence then $1/f\left( -z \right)$ generates a $\text{P}{{\text{F}}_{r}}$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f\left( z \right)$ generates a $\text{PF}$ sequence then $1/f\left( -z \right)$ generates a $\text{PF}$ sequence.

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

This study is motivated by problems of molecular sequence comparison for multiple marker arrays with correlated distributions. In this paper, the model assumes two (or more) kinds of markers, say Markers A and B, distributed along the DNA sequence. The two primary conditions of interest are (i) many of Marker B (say ≥ m) occur, and (ii) few of Marker B (say ≤ l) occur. We title these the conditional r-scan models, and inquire on the extent to which Marker A clusters or is over-dispersed in regions satisfying condition (i) or (ii). Limiting distributions for the extremal r-scan statistics from the A array satisfying conditions (i) and (ii) are derived by extending the Chen-Stein Poisson approximation method.

We consider two models for the control of a satellite–in the first, fuel is expended in a linear fashion to move a satellite following a diffusion–where the aim is to keep the satellite above a critical level for as long as possible (or indeed to reach a higher, ‘safe’ level). Under suitable assumptions for the drift and diffusion coefficients, it is shown that the stochastic maximum of the time to fall below the critical level is attained by a policy which imposes a reflecting boundary at the critical level until the fuel is exhausted and jumps the satellite directly to the safe level if this is ever possible. In the second model, there is a nonlinear response to the expenditure of fuel, and no safe level. It is shown that the optimal policy for maximizing the expected discounted time for the satellite to crash is similar, in that equal packets of fuel are used to jump the satellite upwards each time it reaches the critical level.

It is shown that totally positive order 2 (TP2) properties of the infinitesimal generator of a continuous-time Markov chain with totally ordered state space carry over to the chain's transition distribution function. For chains with such properties, failure rate characteristics of the first passage times are established. For Markov chains with partially ordered state space, it is shown that the first passage times have an IFR distribution under a multivariate total positivity condition on the transition function.

A Bayesian approach for analyzing layered defense systems is presented. This approach incorporates the dependence of penetration probabilities on the size of attackers going into any layer. A general formula is developed for computing the predictive distribution of the number of attackers surviving any layer as well as the posterior distribution of the penetration probabilities under the a priori assumptions that: (i) the probabilities are dependent and their joint distribution is Dirichlet, and (ii) the probabilities are independent. Positive dependence of the penetration probabilities as well as the number of attackers surviving the different layers is also established.

The paper investigates stochastic processes directed by a randomized time process. A new family of directing processes called Hougaard processes is introduced. Monotonicity properties preserved under subordination, and dependence among processes directed by a common randomized time are studied. Results for processes subordinated to Poisson and stable processes are presented. Potential applications to shock models and threshold models are also discussed. Only Markov processes are considered.

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

The problem of optimally allocating partially effective, defensive weapons against randomly arriving enemy aircraft so that a bomber maximizes its probability of reaching its designated target is considered in the usual continuous-time context, and in a discrete-time context. The problem becomes that of determining the optimal number of missiles K(n, t) to use against an enemy aircraft encountered at time (distance) t away from the target when n is the number of remaining weapons (missiles) in the bomber's arsenal. Various questions associated with the properties of the function K are explored including the long-standing, unproven conjecture that it is a non-decreasing function of its first variable.

Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity λ. Let Pk denote the probability that the first device will survive the first k shocks and let denote such a probability for the second device. Let and denote the survival functions of the first and the second device respectively. In this note we show that some partial orderings, namely likelihood ratio ordering, failure rate ordering, stochastic ordering, variable ordering and mean residual-life ordering between the shock survival probabilities and are preserved by the corresponding survival functions and .

Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.

Suppose Y is the strength of a system subjected to a sequence of stresses X1, X2, X3, · ··. Let N(t) be the number of stresses occurring by time t and let T be the lifetime of the system. Then the survival distribution of is given by where and . In this note we aim at determining from the above model various non-parametric classes of life distributions which are of interest in reliability theory.

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

A device is subject to damage. The damage occurs randomly in time according to a pure jump process. The device has a threshold and it fails once the damage exceeds the threshold. We show that life distribution properties of the threshold right tail probability are inherited as corresponding properties of the survival probability, under suitable conditions on the parameters of the damage process. Moreover we discuss an optimal replacement problem for such devices.