Monotonic Research Articles

An Effective and Small Sample-Size Valid Confidence Interval for Isotonic Dose–Response Curves by Inverting a Partial Likelihood Ratio Test

A dose–response curve is essential for determining the safe dosage of a drug and is widely used in bioassay and in phase 1 clinical trials. It is generally accepted that the probability of death or the probability of dose-limiting toxicity is a nondecreasing function of the dose. This article proposes and develops an effective point-wise confidence interval for an isotonic dose–response curve, a problem without a satisfactory solution so far. We show that one subset of the observations informs the lower limit while the other subset informs the upper limit. A partial likelihood ratio test using these subsets of observations is fully investigated. The resulting confidence interval is compared to three existing methods (Morris, Meyer, and Oron) in an extensive simulation study. The new method produces tight intervals while maintaining coverage. It is therefore preferred when maintaining an actual coverage probability is crucial as often in a regulatory environment. The new method extends to one-way ANOVA of a continuous response variable in a straightforward way. An R function is provided.

Strict monotonicity of stochastic process extreme distributions

Strict monotonicity is proved for the distributions of extremes of processes consisting of series of bounded function with independent random coefficients, in particular for zero mean continuous Gaussian processes over compact metric space. These results have wide applications to global inference problems on unknown functions.

Stability of impulsive delayed switched systems with conformable fractional-order derivatives

In this paper, exponential stability is addressed for impulsive delayed switched systems with conformable derivatives by Halanay inequalities technique. First, based on the fractional-order monotonicity theorem and the definition of conformable fractional-order derivative, novel conformable fractional-order Halanay inequalities are developed that extend and improve the existing ones. Second, using the obtained Halanay inequalities and the estimate of conformable fractional-order exponential functions, globally exponential stability conditions that depend on the infimum and supremum of impulsive step sizes are proposed for the systems. Third, more relaxing stability conditions are presented that do not depend on the infimum and supremum of impulsive step sizes, which means that the conditions on impulses are weaker than those commonly used in the existing literature. Finally, examples are given to illustrate the validity of the theoretical results.

A Family of Mexican Hat Wavelet Stieltjes Transform for Unbounded Non-decreasing Functions

A Family of Mexican Hat Wavelet Stieltjes Transform for Unbounded Non-decreasing Functions

Robust non-monotonic Lyapunov based stability and stabilization methods for continuous-time systems: Applied on bilateral teleoperation system

This study introduces a Non-monotonic Lyapunov (NML) framework aimed at stability evaluation and controller design for continuous-time systems, particularly under conditions of uncertainty. Conventional Lyapunov techniques often exhibit a conservative nature, particularly in the context of uncertain systems, which necessitates the development of less conservative alternatives like NML. The NML methodology distinguishes itself by not imposing strict monotonicity requirements for demonstrating the decrease of a Lyapunov functional. Consequently, this paper derives new stability and stabilization criteria framed as matrix inequalities applicable to a specific class of uncertain systems. The practical applicability of the introduced approach is illustrated through controller design for uncertain systems, exemplified by a nonlinear bilateral teleoperation model. Assorted demonstrative examples and simulation outcomes support the findings, underscoring the NML approach’s efficaciousness.

Development of a total variation diminishing (TVD) sea ice transport scheme and its application in an ocean (SCHISM v5.11) and sea ice (Icepack v1.3.4) coupled model on unstructured grids

Abstract. As the demand for increased resolution and complexity in unstructured sea ice models is growing, higher demands are also placed on the sea ice transport scheme. In this study, we couple the Semi-implicit Cross-scale Hydro-science Integrated System Model (SCHISM, v5.11) with Icepack (v1.3.4), the column physics package of the Los Alamos sea ice model (CICE); a key step is to implement a total variation diminishing (TVD) transport scheme for the multi-class sea ice module in the coupled model. Compared with the second-order upwind scheme and the finite-element flux-corrected transport (FEM-FCT) scheme, the TVD transport scheme is overall superior when evaluated based on conservation, accuracy, efficiency (even with very high resolution), and strict monotonicity. Although it is slightly weaker than FEM-FCT in terms of accuracy alone, the TVD scheme still outperforms the other two schemes in comprehensive performance. The new coupled model outperforms the existing single-class ice model of SCHISM in the case of Lake Superior. For the Arctic Ocean case, it successfully reproduces the long-term changes in the sea ice extent, sea ice boundary, concentration observations from satellites, and thickness from in situ measurement.

Approximate utility

Approximating arbitrage or a perfect hedge in a limit implies approximation in utility (formally, expected utilities are continuous in L2), if and only if the utility function is bounded above and below by quadratics. We characterize when some special utility functions are continuous for all random consumptions with a common lower bound. Only linear, quadratic, and, in some cases, translated SAHARA utilities are continuous if there is no lower bound. We also characterize when a utility function defined on a convex subset can be extended to an L2-continuous nondecreasing and concave utility function on all of R.

Specification of the health production function and its behavioral implications.

The health production function of the canonical health-capital model is generalized to allow the state of health to affect the total and marginal products of health investment. If the total and marginal products of health investment are nonincreasing functions of the state of health, then the solution of the generalized model is locally qualitatively identical to that of the canonical model. Moreover, and in contrast to the canonical model, the generalized model is able to rationalize the cycling of the state of health and health investment observed in some individuals. The necessary conditions on the health production function for cyclical behavior are identified as well.

Submodular batch scheduling on parallel machines

AbstractThis article studies a submodular batch scheduling problem motivated by the vacuum heat treatment. The batch processing time is formulated by a monotone nondecreasing submodular function characterized by decreasing marginal gain property. The objective is to minimize the makespan. We show the NP‐hardness of the problem on a single machine and of finding a polynomial‐time approximation algorithm with the worst‐case performance ratio strictly less than for the problem on parallel machines. We introduce a bounded interval to model the batch processing time using two parameters, that is, the total curvature and the quantization indicator. Based on the decreasing marginal gain property and the two parameters, we make a systematic analysis of the full batch longest processing time algorithm and the longest processing time greedy algorithm, and propose the instances with the bound of batch capacity for these two algorithms for the submodular batch scheduling problem. Moreover, we prove the submodularity of batch processing time function of the existing batch models including the parallel batch, serial batch, and mixed batch models. We compare the worst‐case performance ratios in the existing batch models with those deduced from our work in the submodular batch model. In most situations, the worst‐case performance ratios deduced from our work are comparable to the best‐known worst‐case performance ratios with tailored examinations.

On indication, strict monotonicity, and efficiency of projections in a general class of path-based data envelopment analysis models

Data envelopment analysis (DEA) theory formulates a number of desirable properties that DEA models should satisfy. Among these, indication, strict monotonicity, and strong efficiency of projections tend to be grouped together in the sense that, in individual models, typically, either all three are satisfied or all three fail at the same time. Specifically, in slacks-based graph models, the three properties are always met; in path-based models, such as radial models, directional distance function models, and the hyperbolic function model, the three properties, with some minor exceptions, typically all fail.Motivated by this observation, the article examines relationships among indication, strict monotonicity, and strong efficiency of projections in the class of path-based models over variable returns-to-scale technology sets. Under mild assumptions, it is shown that the property of strict monotonicity and strong efficiency of projections are equivalent, and that both properties imply indication. This paper also characterises a narrow class of technology sets and path directions for which the three properties hold in path-based models.

State Estimation of Switched Time-Delay Complex Networks With Strict Decreasing LKF.

State estimation issue is investigated for a switched complex network (CN) with time delay and external disturbances. The considered model is general with a one-sided Lipschitz (OSL) nonlinear term, which is less conservative than Lipschitz one and has wide applications. Adaptive mode-dependent nonidentical event-triggered control (ETC) mechanisms for only partial nodes are proposed for state estimators, which are not only more practical and flexible but also reduce the conservatism of the results. By using dwell-time (DT) segmentation and convex combination methods, a novel discretized Lyapunov-Krasovskii functional (LKF) is developed such that the value of LKF at switching instants is strict monotone decreasing, which makes it easy for nonweighted L2 -gain analysis without additional conservative transformation. The main results are given in the form of linear matrix inequalities (LMIs), by which the control gains of the state estimator are designed. A numerical example is given to illustrate the advantages of the novel analytical method.

A novel probabilistic model: Simulation and modeling the time duration in musical education and engineering

Statistical analysis of the time-to-event scenarios commonly involves the use of the exponential and Rayleigh distributions. However, the exponential distribution is constrained by its constant failure rate, while the Rayleigh distribution exhibits a growing (or increasing) failure rate. The linear failure rate distribution provides a broader perspective by encompassing both of these distributions, including the possibility of a non-increasing hazard function. In this paper, a new distribution is introduced, which could be considered as one of the major modifications to the linear failure rate distribution, known as the weighted linear failure rate distribution. The model proposed illustrates both monotonic and a known non-monotonic failure rate. This paper extensively explores the characteristics of the proposed distribution. It presents the maximum likelihood estimates for the unknown parameters and includes a simulation study. The analysis focuses on three real data sets obtained from the fields of musical education and engineering. By employing three decisive tools, it is observed that the current distribution provides a superior fit in comparison to several other widely recognized distributions.

A general law of the iterated logarithm for non-additive probabilities

Motivated by some interesting problems in mathematical economics, quantum mechanics and finance, non-additive probabilities have been used to describe the phenomena which are generally non-additive. In this paper, we further study the law of the iterated logarithm (LIL) for non-additive probabilities, based on existing results. Under the framework of sublinear expectation initiated by Peng, we give two convergence results of Vn≔∑i=1nXinϕ(n) under some reasonable assumptions, where {Xi}i=1∞ is a sequence of random variables and ϕ is a positive nondecreasing function. From these, a general LIL for non-additive probabilities is proved for negatively dependent and non-identically distributed random variables. It turns out that our result is a natural extension of the Kolmogorov LIL and the Hartman–Wintner LIL. Theorem 1 and Theorem 2 in this paper can be seen an extension of Theorem 1 in Chen and Hu (2014).

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

Strict monotonicity for first passage percolation on graphs of polynomial growth and quasi-trees

Extrapolation results on variable exponent grand Lebesgue space with Bp(·)$B_{p(\cdot)}$ weights

AbstractIn this paper, we study Rubio de Francia extrapolation theorems in the framework of the variable grand Lebesgue spaces with weights. As an application of the extrapolation theorems, we prove the boundedness of the Hardy averaging operator and the fractional Riemann Liouville transform for nonnegative and nonincreasing measurable functions. Some structural properties of the weighted grand Lebesgue spaces with variable exponent are also investigated.

Convexity of shock polars

Abstract We show that the shock polars of compressible full potential flow are strictly convex if the enthalpy per mass is a convex function of volume per mass, in particular when the sound speed is a non-decreasing function of density. Counterexamples are given for some cases that violate the condition on enthalpy. For the full Euler equations with convex equation of state satisfying the ideal gas law, polars are strictly convex if heat capacity is constant, but counterexamples are given in variable cases, showing no useful generalizations are possible.

Fast Exact Simulation of the First Passage of a Tempered Stable Subordinator Across a Non-Increasing Function

Fast Exact Simulation of the First Passage of a Tempered Stable Subordinator Across a Non-Increasing Function

Massless Lifshitz field theory for arbitrary z

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z-dependent radius scale for the Lifshitz background. This relation is consistent with the z-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

Geometric random intersection graphs with general connection probabilities

Abstract Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$ . A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points (v, u) with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$ , where g is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$ . This gives rise to a random intersection graph on $\mathbb{R}^d$ . Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function g. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether g has bounded or unbounded support.

Strictly monotone sequences of lower and upper bounds on Perron values and their combinatorial applications

In this paper, we present monotone sequences of lower and upper bounds on the Perron value of a nonnegative matrix, and we study their strict monotonicity. Using those sequences, we provide two combinatorial applications. One is to improve bounds on Perron values of rooted trees in combinatorial settings, in order to find characteristic sets of trees. The other is to generate log-concave and log-convex sequences through the monotone sequences.