Monotonicity Assumption Research Articles

Wegner estimate for discrete Schrödinger operators with Gaussian random potentials

Abstract We prove a Wegner estimate for discrete Schrödinger operators with a potential given by a Gaussian random process. The only assumption is that the covariance function decays exponentially; no monotonicity assumption is required. This improves earlier results where abstract conditions on the conditional distribution, compactly supported and non-negative, or compactly supported covariance functions with positive mean are considered.

Bayes Factors for Evaluating Latent Monotonicity in Polytomous Item Response Theory Models

The assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score. Three forms of latent monotonicity can be distinguished: monotonicity of the cumulative probabilities, of the continuation ratios, and of the adjacent-category ratios. Observable consequences of these different forms of latent monotonicity are derived, and Bayes factor methods for testing these consequences are proposed. These methods allow for the quantification of the evidence both in favor and against the tested property. Both item-level and category-level Bayes factors are considered, and their performance is evaluated using a simulation study. The methods are applied to an empirical example consisting of a 10-item Likert scale to investigate whether a polytomous item scoring rule results in item scores that are of ordinal level measurement.

A Comparison of Two Quantile Models With Endogeneity

This article studies the relationship between the two most-used quantile models with endogeneity: the instrumental variable quantile regression (IVQR) model (Chernozhukov and Hansen 2005) and the local quantile treatment effects (LQTE) model (Abadie, Angrist, and Imbens 2002). The key condition of the IVQR model is the rank similarity assumption, a restriction on the evolution of individual ranks across treatment states, under which population quantile treatment effects (QTE) are identified. By contrast, the LQTE model achieves identification through a monotonicity assumption on the selection equation but only identifies QTE for the subpopulation of compliers. This article shows that, despite these differences, there is a close connection between both models: (i) the IVQR estimands correspond to QTE for the compliers at transformed quantile levels and (ii) the IVQR estimand of the average treatment effect is equal to a convex combination of the local average treatment effect and a weighted average of integrated QTE for the compliers. These results do not rely on the rank similarity assumption and therefore provide a characterization of IVQR in settings where this key condition is violated. Underpinning the analysis are novel closed-form representations of the IVQR estimands. I illustrate the theoretical results with two empirical applications.

Revisiting the cavity-method threshold for random 3-SAT.

A detailed Monte Carlo study of the satisfiability threshold for random 3-SAT has been undertaken. In combination with a monotonicity assumption we find that the threshold for random 3-SAT satisfies α_{3}≤4.262. If the assumption is correct, this means that the actual threshold value for k=3 is lower than that given by the cavity method. In contrast the latter has recently been shown to give the correct value for large k. Our result thus indicate that there are distinct behaviors for k above and below some critical k_{c}, and the cavity method may provide a correct mean-field picture for the range above k_{c}.

Consistency Theory for the General Nonparametric Classification Method.

Parametric likelihood estimation is the prevailing method for fitting cognitive diagnosis models-also called diagnostic classification models (DCMs). Nonparametric concepts and methods that do not rely on a parametric statistical model have been proposed for cognitive diagnosis. These methods are particularly useful when sample sizes are small. The general nonparametric classification (GNPC) method for assigning examinees to proficiency classes can accommodate assessment data conforming to any diagnostic classification model that describes the probability of a correct item response as an increasing function of the number of required attributes mastered by an examinee (known as the "monotonicity assumption"). Hence, the GNPC method can be used with any model that can be represented as a general DCM. However, the statistical properties of the estimator of examinees' proficiency class are currently unknown. In this article, the consistency theory of the GNPC proficiency-class estimator is developed and its statistical consistency is proven.

Measurement equivalence of PROMIS depression in Spain and the United States.

In this study, we assessed the psychometric properties of the Spanish Patient-Reported Outcomes Measurement Information System (PROMIS) Depression in an adult population-representative sample from Spain (n = 1,503). We tested unidimensionality and local independence item response theory (IRT) assumptions with confirmatory factor and bifactor models under the exploratory structural equations modeling framework. We evaluated item monotonicity assumption with Mokken scaling analysis. We calibrated the items with an IRT-graded response model and assessed score reliability and test information, and evidence of validity with regard to scores on external measures. To examine differential item functioning by age, sex, education, and country (United States vs. Spain, N = 2,271), we used ordinal logistic regression. Results support compliance with IRT assumptions. We found few signs of differential item functioning: Only one item showed country differential functioning between the United States (n = 768) and Spain, with minimal impact on the overall score. Information values were equivalent to reliabilities over 0.90 from -1 (low depression) to +4 SD (high depression) around the population score mean. Evidence of validity in relation to concurrent measures was supported by the expected correlation pattern with external variables of depression, but higher than expected correlations with anxiety were found. Results indicate that the Spanish version of PROMIS Depression is adequate for assessing and monitoring depression levels in the general population and that PROMIS Depression is especially suitable for cross-national comparisons. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

Safeguarding against bad luck when attempting to discredit a state-trace model

A state-trace model for a natural phenomenon proposes that there is a causal “bottleneck” that makes the joint causal effects of two independent variables look like the effect of a single independent variable. Imagine that a state-trace model has been formulated, but the model seems implausible and we wish to find empirical evidence that it is wrong. How can we do that? Given reasonable monotonicity assumptions, a state-trace model is wrong if a point on one state trace is delta-discordant with a point on another state trace. [We say that two points (x,y) and (x′,y′) in the plane are delta-discordant if Δx=x′−x and Δy=y′−y are nonzero and have opposite signs.] So, to show the model wrong, we need to find a pair of delta-discordant points lying on different state traces. One method for finding two such points is to observe multiple points on each of two state traces with the hope that, among those points, there will be a delta-discordant pair. But this approach can have bad luck and fail to find delta-discordant pairs even though they exist. A new approach to “locating” a delta-discordant pair of points, an approach that is much less dependent on luck, is formulated in this paper. In addition, the paper describes a statistical test for evaluating whether the “located” points are really delta-discordant.

Strong approximation of monotone stochastic partial differential equations driven by white noise

Abstract We establish an optimal strong convergence rate of a fully discrete numerical scheme for second-order parabolic stochastic partial differential equations with monotone drifts, including the stochastic Allen–Cahn equation, driven by an additive space-time white noise. Our first step is to transform the original stochastic equation into an equivalent random equation whose solution possesses more regularity than the original one. Then we use the backward Euler in time and spectral Galerkin in space to fully discretise this random equation. By the monotonicity assumption, in combination with the factorisation method and stochastic calculus in martingale-type 2 Banach spaces, we derive a uniform maximum norm estimation and a Hölder-type regularity for both stochastic and random equations. Finally, the strong convergence rate of the proposed fully discrete scheme is obtained. Several numerical experiments are carried out to verify the theoretical result.

Experiment-inspired comments on John Roemer’s theory of cooperation

We report on a nonsocial experiment where we find that all participants choose the dominant strategy in the experimental payoff, and compare it with the payoff-isomorphic, but social, Prisoner’s Dilemma treatment presented in a recent paper by Bosch-Domènech and Silvestre where 28% choose cooperation instead of the dominant strategy. The contrast reinforces Roemer’s emphasis on human cooperation. Next, we argue that Roemer’s Simple Kantian Equilibrium works well as a theory of cooperation under the assumption of monotonicity (positive or negative externalities), but not when efficient cooperation requires the division of labor by coordinating dissimilar tasks.

Improved one-sided deviation inequalities under regularity assumptions for product measures

This note is concerned with lower tail estimates for product measures. Some improved deviation inequalities are obtained for functions satisfying some regularity and monotonicity assumptions. The arguments are based on semigroup interpolation together with Harris’s negative association inequality and hypercontractive estimates.

Sharp Bounds on the MTE With Sample Selection

I propose a Generalized Roy Model with sample selection that can be used to analyze treatment effects in a variety of empirical problems. First, I decompose, under a monotonicity assumption on the sample selection indicator, the MTR function for the observable outcome when treated as a weighted average of (i) the MTR on the outcome of interest for the always-observed sub-population and (ii) the MTE on the observable outcome for the observed-only-when-treated sub-population, and show that such decomposition can provide point-wise sharp bounds on the MTE of interest for the always-observed sub-population. Moreover, I impose an extra mean dominance assumption and tighten the previous bounds. I, then, show how to point-identify those bounds when the support of the propensity score is continuous. After that, I show how to (partially) identify the MTE of interest when the support of the propensity score is discrete. At the end, I estimate bounds on the MTE of the Job Corps Training Program on hourly wages for the always-employed sub-population and find that it is decreasing with the likelihood of attending the program for the Non-Hispanic group. For example, I find that the ATT is between $.38 and $1.17 while the ATU is between $.73 and $3.14.

Convergence Analysis of Sample Average Approximation of Two-Stage Stochastic Generalized Equations

A solution of two-stage stochastic generalized equations is a pair: a first stage solution which is independent of realization of the random data and a second stage solution which is a function of random variables.This paper studies convergence of the sample average approximation of two-stage stochastic nonlinear generalized equations. In particular an exponential rate of the convergence is shown by using the perturbed partial linearization of functions. Moreover, sufficient conditions for the existence, uniqueness, continuity and regularity of solutions of two-stage stochastic generalized equations are presented under an assumption of monotonicity of the involved functions. These theoretical results are given without assuming relatively complete recourse, and are illustrated by two-stage stochastic non-cooperative games of two players.

Mathematical analysis of boundary layers in two-dimensional incompressible viscous heat conducting flows

This paper is to review our recent study on the well-posedness and blowup of the boundary layer equations in small viscosity and heat conductivity limit for the two-dimensionalincompressible viscous heat conducting flows near a physical boundary. In the case that the viscosity and heat conductivity have the same scale, first we derive the boundary layer equations of the viscous layer and thermal layer for the incompressible Navier-Stokes-Fourier equations by multi-scale analysis,and then we shall review a well-posedness result established under the monotonicity condition of tangential velocity by using the Crocco transformation and the energy method.After that, when the tangential velocity does not satisfy the monotonicity assumption, we shall present a well-posedness result when the data are analytic with respect to the tangential variable,by using the Littlewood-Paley theory. We also present a blowup result in a finite time by introducing a Lyapunov functional, when the monotonicity condition is violated for the initial velocity. This shows that the analytic solution which we obtained exists in a finite time only in general.

Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation

We show the convergence of finite state symmetric N-player differential games, where players control their transition rates from state to state, to a limiting dynamics given by a finite state Mean Field Game system made of two coupled forward–backward ODEs. We exploit the so-called Master Equation, which in this finite-dimensional framework is a first order PDE in the simplex of probability measures, obtaining the convergence of the feedback Nash equilibria, the value functions and the optimal trajectories. The convergence argument requires only the regularity of a solution to the Master Equation. Moreover, we employ the convergence results to prove a Central Limit Theorem and a Large Deviation Principle for the evolution of the N-player empirical measures. The well-posedness and regularity of solution to the Master Equation are also studied, under monotonicity assumptions.

A forward–backward penalty scheme with inertial effects for monotone inclusions. Applications to convex bilevel programming

ABSTRACTWe investigate a forward–backward splitting algorithm of penalty type with inertial effects for finding the zeros of the sum of a maximally monotone operator and a cocoercive one and the convex normal cone to the set of zeroes of an another cocoercive operator. Weak ergodic convergence is obtained for the iterates, provided that a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions, strong convergence for the sequence of generated iterates is proved. As a particular instance we consider a convex bilevel minimization problem including the sum of a non-smooth and a smooth function in the upper level and another smooth function in the lower level. We show that in this context weak non-ergodic and strong convergence can be also achieved under inf-compactness assumptions for the involved functions.

Qualitative Effects of Hydraulic Conductivity Distribution on Groundwater Flow in Heterogeneous Soils

One of the most significant difficulties in subsurface hydrology is the considerable uncertainty in hydraulic conductivity values in the medium. This stimulates qualitative analysis of the effect of conductivity distribution on the solutions or on some components of the solutions of groundwater flow equations. This work is an attempt to develop a rigorous basis for deciding whether the solutions are monotonous with respect to hydraulic conductivity. Such monotonicity is analogous to the well-known comparison principles with respect to variations of initial data or external supplies. Some example problems are given in this paper, including a problem with a free boundary, in which the monotonous dependence of the solution on the conductivity distribution is proved rigorously. Examples are also given, in which monotonicity assumptions, despite being apparently obvious, are proved to be invalid.

Specific Properties of Groundwater Flows in Heterogeneous Media

The paper deals with justification of estimates for solutions of groundwater flow equations in the case of incomplete information on soil permeability. In engineering applications, this problem is treated by means of mathematical modeling and implies a search through numerous alternate distributions of unknown parameters. Usually, these approaches involve, explicitly or implicitly, the monotonicity assumption, which provides an opportunity to reduce the number of alternative calculations. The paper presents some counterexamples to show that the monotonicity assumption, even if it is in agreement with engineering intuition, can be invalid.

Continuity of Solutions for Parametric Set Optimization Problems via Scalarization Methods

The aim of this paper is to investigate the continuity of solution mappings for parametric set optimization problems with upper and lower set less order relations by scalarization methods. First, we recall some linear and nonlinear scalarization properties used to characterize the set order relations. Subsequently, we introduce new monotonicity concepts of the set-valued mapping by linear and nonlinear scalarization methods. Finally, we obtain the semicontinuity and closedness of solution mappings for parametric set optimization problems (both convex and nonconvex cases) under the monotonicity assumption and other suitable conditions. The results achieved do not impose the continuity of the set-valued objective mapping, which are obviously different from the related ones in the literature.

On the uniqueness of maximal immediate extensions of valued differential fields

So far there exist just a few results about the uniqueness of maximal immediate valued differential field extensions and about the relationship between differential-algebraic maximality and differential-henselianity; see [1, Chapter 7]. We remove here the assumption of monotonicity in these results but replace it with the assumption that the value group is the union of its convex subgroups of finite (archimedean) rank. We also show the existence and uniqueness of differential-henselizations of asymptotic fields with such a value group.

확률적 단조성 하에서 실패까지 걸린 시간에 대한 유용한 모형들 : 모형 설정 및 통계적 추론

We propose some useful models for hitting time under stochastic monotonicity assumption. Depending on whether the underlying shock process is governed by a Poisson process or a Brownian motion, on whether each shock has a deterministic impact size or a random impact size, and on whether failure is defined by a threshold crossing or by a hazard rate, we develop various “failure” time models. We then explain how to utilize information contained in the failure time to make inferences on the underlying model parameters which characterizes shock processes and the impact sizes. Real life applications are briefly mentioned.