Monotonicity Restrictions Research Articles

Synthesis of Optimal Correction Functions in the Class of Disjunctive Normal Forms

The paper proposes to consider individual heuristics as unreliably operating parts of the information processing system. In a separate case, several different heuristics are adopted to solve the same problem, and the results obtained are adjusted in a certain way. In this case, problems arise that are close in methodology to the problems of synthesizing reliable circuits from unreliable elements or making a collective expert decision. The work solves the problem of constructing an optimal correction function based on control material; classes of functions of k-valued logic under monotonicity restrictions are studied. A theorem on the completeness of the class of monotonic functions of k-valued logic for arbitrary k is proved, and a basis in the given class is proved and constructed. The problem of constructing an optimal corrector in the class of disjunctive normal forms of k-valued functions is solved.

Planar Schrödinger-Poisson system with zero mass potential and critical exponential growth

This paper is concerned with the following planar Schrödinger-Poisson system with zero mass potential{−Δu+ϕu=Q(|x|)f(u),x∈R2,Δϕ=u2,x∈R2, where Q∈C((0,∞),(0,∞)) may be singular or unbound and f∈C(R,R) is of critical exponential growth and there is no monotonicity restriction on f(u)/u3. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality within the working space associated with the energy functional relevant to the aforementioned problem, which is different from Albuquerque et al. (2021) [5]. By combining the variational methods and delicate estimates, we prove the existence of a non-trivial mountain-pass solution to the above system under mild assumptions on Q and f.

Identification of mixtures of dynamic discrete choices

This paper provides new identification results for finite mixtures of Markov processes. Our arguments yield identification from knowledge of the cross-sectional distribution of three (or more) effective time-series observations under simple conditions. We explain how our approach and results are different from those in previous work by Kasahara and Shimotsu (2009) and Hu and Shum (2012). Most notably, outside information, such as monotonicity restrictions that link conditional distributions to latent types, is not needed.

Judging Judge Fixed Effects

We propose a nonparametric test for the exclusion and monotonicity assumptions invoked in instrumental variable (IV) designs based on the random assignment of cases to judges. We show its asymptotic validity and demonstrate its finite-sample performance in simulations. We apply our test in an empirical setting from the literature examining the effects of pretrial detention on defendant outcomes in New York. When the assumptions are not satisfied, we propose weaker versions of the usual exclusion and monotonicity restrictions under which the IV estimator still converges to a proper weighted average of treatment effects. (JEL H76, K41)

Critical Schrödinger equations with Stein-Weiss convolution parts in [formula omitted

We study the following planar Schrödinger equations with Stein-Weiss convolution parts−Δu+u=1|x|β(∫R2F(u)|x−y|μ|y|βdy)f(u),x∈R2, where β≥0, 0<μ<2 with 0<2β+μ<2 and F is the primitive of f that fulfills a critical exponential growth in the Trudinger-Moser sense. Via establishing a general version of Pohoz̆aev identity, we shall exploit the general minimax principle to investigate the existence of mountain-pass type solutions for the equation. Although there is no any monotonicity restriction on the nonlinearity f, we can show that the mountain-pass value equals to the least energy level. Our results extend the ones in, e.g. Yang-Rădulescu-Zhou (2022) [44] and Du-Gao-Yang (2022) [15], to the lower dimension.

Using monotonicity restrictions to identify models with partially latent covariates

This paper develops a new method for identifying econometric models with partially latent covariates. Such data structures arise in industrial organization and labor economics settings where data are collected using an input-based sampling strategy, e.g., if the sampling unit is one of multiple labor input factors. We show that the latent covariates can be nonparametrically identified, if they are functions of a common shock satisfying some plausible monotonicity assumptions. With the latent covariates identified, semiparametric estimation of the outcome equation proceeds within a standard IV framework that accounts for the endogeneity of the covariates. We illustrate the usefulness of our method using a new application that focuses on the production functions of pharmacies. We find that differences in technology between chains and independent pharmacies may partially explain the observed transformation of the industry structure.

SIMPLE SEMIPARAMETRIC ESTIMATION OF ORDERED RESPONSE MODELS

We propose two simple semiparametric estimation methods for ordered response models with an unknown error distribution. The proposed methods do not require users to choose any tuning parameters, and they automatically incorporate the monotonicity restriction of the unknown distribution function. Fixing finite-dimensional parameters in the model, we construct nonparametric maximum likelihood estimates for the error distribution based on the related binary choice data or the entire ordered response data. We then obtain estimates for finite-dimensional parameters based on moment conditions given the estimated distribution function. Our semiparametric approaches deliver root-n consistent and asymptotically normal estimators of the regression coefficient and threshold parameter. We also develop valid bootstrap procedures for inference. The advantages of our methods are borne out in simulation studies and a real data application.

Planar Schrödinger-Poisson system with critical exponential growth in the zero mass case

In this paper, we develop new proof techniques and analytical methods to prove the existence of ground state solutions for the following planar Schrödinger-Poisson system with zero mass{−Δu+ϕu=f(u),x∈R2,Δϕ=2πu2,x∈R2, where f∈C(R,R) has the critical exponential growth at infinity and there is no monotonicity restriction on f(u)/u3. In particular, by using delicate estimates we obtain a desired upper bound for the Mountain Pass level just with the optimal asymptotic condition κ=liminf|t|→∞t2F(t)eα0t2>0 to restore the compactness in the presence of critical exponential growth, which significantly improves analogous assumptions on asymptotic behavior of t2F(t)eα0t2 or tf(t)eα0t2 at infinity in the previous works. Moreover, we use a different approach from the one of Du and Weth (2017) [21] dealing with the power nonlinearities to establish the Pohozaev type identity, which not only allows critical exponential growth nonlinearities, but also deals with the non-autonomous case containing a linear term V(x)u in the first equation, both of which are not covered in the existing literature. To our knowledge, there has not been any work in the literature on the subject, even for the simpler equation: −Δu=f(u) in R2.

Exploratory Restricted Latent Class Models with Monotonicity Requirements under PÒLYA-GAMMA Data Augmentation.

Restricted latent class models (RLCMs) provide an important framework for supporting diagnostic research in education and psychology. Recent research proposed fully exploratory methods for inferring the latent structure. However, prior research is limited by the use of restrictive monotonicity condition or prior formulations that are unable to incorporate prior information about the latent structure to validate expert knowledge. We develop new methods that relax existing monotonicity restrictions and provide greater insight about the latent structure. Furthermore, existing Bayesian methods only use a probit link function and we provide a new formulation for using the exploratory RLCM with a logit link function that has an additional advantage of being computationally more efficient for larger sample sizes. We present four new Bayesian formulations that employ different link functions (i.e., the logit using the Pòlya-gamma data augmentation versus the probit) and priors for inducing sparsity in the latent structure. We report Monte Carlo simulation studies to demonstrate accurate parameter recovery. Furthermore, we report results from an application to the Last Series of the Standard Progressive Matrices to illustrate our new methods.

Random generation of capacities and its application in comprehensive decision aiding

The capacities and the Choquet integral are powerful tools to represent decision problems with dependencies and aggregate correlated decision criteria. Random generation of suitable capacities is a vital and challenging task in this decision model because of an exponential number of the involved parameters as well as the associated monotonicity restrictions. In this paper we present various approaches to representing decision makers’ preferences on aggregation through linear constraints, random generation of capacities from the selected polytope, testing the uniformity of the resulting distribution, and constructing the dominance relations between the alternatives, which are subsequently used to get the most credible ranking of the alternatives.

Using Monotonicity Restrictions to Identify Models with Partially Latent Covariates

This paper develops a new method for identifying econometric models with partially latent covariates. Such data structures arise naturally in industrial organization and labor economics settings where data are collected using an “input-based sampling” strategy, e.g., if the sampling unit is one of multiple labor input factors. We show that the latent covariates can be nonparametrically identified, if they are functions of a common shock satisfying some plausible monotonicity assumptions. With the latent covariates identified, semiparametric estimation of the outcome equation proceeds within a standard IV framework that accounts for the endogeneity of the covariates. We illustrate the usefulness of our method using two applications. The first focuses on pharmacies: we find that production function differences between chains and independent pharmacies may partially explain the observed transformation of the industry structure. Our second application investigates education achievement functions and illustrates important differences in child investments between married and divorced couples. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Using Monotonicity Restrictions to Identify Models with Partially Latent Covariates

This paper develops a new method for identifying econometric models with partially latent covariates. Such data structures arise in industrial organization and labor economics settings where data are collected using an input-based sampling strategy, e.g., if the sampling unit is one of multiple labor input factors. We show that the latent covariates can be nonparametrically identified, if they are functions of a common shock satisfying some plausible monotonicity assumptions. With the latent covariates identified, semiparametric estimation of the outcome equation proceeds within a standard IV framework that accounts for the endogeneity of the covariates. We illustrate the usefulness of our method using a new application that focuses on the production functions of pharmacies. We find that differences in technology between chains and independent pharmacies may partially explain the observed transformation of the industry structure.

Using Monotonicity Restrictions to Identify Models with Partially Latent Covariates

This paper develops a new method for identifying and estimating production functions with partially latent inputs. Such data structures arise naturally when data are collected using an ``input-based strategy, e.g., if the sampling unit is one of multiple labor input factors. We show that the latent inputs can be nonparametrically identified, if they are strictly monotone functions of a scalar shock a la Olley & Pakes (1996). With the latent inputs identified, semiparametric estimation of the production function proceeds within an IV framework that accounts for the endogeneity of the covariates. We illustrate the usefulness of our method using two applications. The first focuses on pharmacies: we find that production function differences between chains and independent pharmacies may partially explain the observed transformation of the industry structure. Our second application investigates skill production functions and illustrates important differences in child investments between married and divorced couples.

Calculation of time to complete a task by a group of employees using fuzzy sets for the sustainable development of the enterprise

The article considers the use of fuzzy sets in determining the execution time of tasks. An approach is proposed that makes it possible to obtain an expert estimate of the time taken to complete a task by a specific employee in the form of a fuzzy set. A procedure is proposed for generalizing fuzzy sets defined on different bearing sets. To apply the proposed procedure, fuzzy sets are subject to continuity and monotonicity restrictions. The generalization procedure is based on the process of finding the value of a fuzzy set for a certain value of its membership function. For piecewise linear functions, the operation of calculating the points of the resulting function is defined. Piecewise nonlinear functions are considered and an algorithm for searching for individual values of the generalized membership function is described. In addition to generalizing fuzzy sets, an approach to calculating the interaction time of workers is proposed. Various approaches to defuzzification of fuzzy sets of task execution time are considered. The proposed approach will improve the work planning for sustainability of enterprise.

Kripke Semantics for Intuitionistic \u0141ukasiewicz Logic

This paper proposes a generalization of the Kripke semantics of intuitionistic logic IL appropriate for intuitionistic Łukasiewicz logicIŁL —a logic in the intersection between IL and (classical) Łukasiewicz logic. This generalised Kripke semantics is based on the poset sum construction, used in Bova and Montagna (Theoret Comput Sci 410(12):1143–1158, 2009). to show the decidability (and PSPACE completeness) of the quasiequational theory of commutative, integral and bounded GBL algebras. The main idea is that w Vdash psi —which for ILis a relation between worlds w and formulas psi , and can be seen as a function taking values in the booleans (w Vdash psi ) in {{mathbb {B}}}—becomes a function taking values in the unit interval (w Vdash psi ) in [0,1]. An appropriate monotonicity restriction (which we call sloping functions) needs to be put on such functions in order to ensure soundness and completeness of the semantics.

Estimation and inference for non-crossing multiple-index quantile regression

Though the theoretical properties of quantile regression have been extensively studied in the past three decades,in practice it is not unusual to obtain crossing surfaces by estimating regression functions at different quantile levels with regular approaches. The crossing quantile surfaces are intrinsically uninterpretable. To address this issue, we consider the semiparametric multi-index quantile regression subject to the monotonicity restriction at different quantile levels. We first connect the semiparametric multi-index quantile regression with a dimension-reducible model. Such a relationship allows us to estimate the index coefficients consistently. The B-splines are then used to approximate the nonparametric function under the monotonicity restriction, which numerically corresponds to a constrained linear programming problem. To further improve the efficiency, we estimate the B-spline coefficients based on the dual of the constrained optimization problem. We assess the finite-sample performance of our proposed method through a comprehensive simulation study, and compare the prediction performance of different methods through a real case study.

Survival estimation through the cumulative hazard with monotone natural cubic splines using convex optimization-the HCNS approach

Background and objectivesIn survival analysis both the Kaplan-Meier estimate and the Cox model enjoy a broad acceptance. We present an improved spline-based survival estimate and offer a fully automated software for its implementation. We explore the use of natural cubic splines that are constrained to be monotone. Apart from its superiority over the Kaplan Meier estimator our approach overcomes limitations of other known smoothing approaches and can accommodate covariates. Unlike other spline methods, concerns of computational problems and issues of overfitting are resolved since no attempt is made to maximize a likelihood once the Kaplan-Meier estimator is obtained. An application to laryngeal cancer data, a simulation study and illustrations of the broad application of the method and its software are provided. In addition to presenting our approaches, this work contributes to bridging a communication gap between clinicians and statisticians that is often apparent in the medical literature. MethodsWe employ a two-stage approach: first obtain the stepwise cumulative hazard and then consider a natural cubic spline to smooth its steps under restrictions of monotonicity between any consecutive knots. The underlying region of monotonicity corresponds to a non-linear region that encompasses the full family of monotone third-degree polynomials. We approximate it linearly and reduce the problem to a restricted least squares one under linear restrictions. This ensures convexity. We evaluate our method through simulations against competitive traditional approaches. ResultsOur method is compared to the popular Kaplan Meier estimate both in terms of mean squared error and in terms of coverage. Over-fitting is avoided by construction, as our spline attempts to approximate the empirical estimate of the cumulative hazard itself, and is not fitted directly on the data. ConclusionsThe proposed approach will enable clinical researchers to obtain improved survival estimates and valid confidence intervals over the full spectrum of the range of the survival data. Our methods outperform conventional approaches and can be readily utilized in settings beyond survival analysis such as diagnostic testing.

Learning bipartite Bayesian networks under monotonicity restrictions

ABSTRACTLearning parameters of a probabilistic model is a necessary step in machine learning tasks. We present a method to improve learning from small datasets by using monotonicity conditions. Monotonicity simplifies the learning and it is often required by users. We present an algorithm for Bayesian Networks parameter learning. The algorithm and monotonicity conditions are described, and it is shown that with the monotonicity conditions we can better fit underlying data. Our algorithm is tested on artificial and empiric datasets. We use different methods satisfying monotonicity conditions: the proposed gradient descent, isotonic regression EM, and non-linear optimization. We also provide results of unrestricted EM and gradient descent methods. Learned models are compared with respect to their ability to fit data in terms of log-likelihood and their fit of parameters of the generating model. Our proposed method outperforms other methods for small sets, and provides better or comparable results for larger sets.

Simple Semiparametric Estimation of Ordered Response Models: With an Application to the Interdependent Durations Model

We propose two simple semiparametric estimation methods for ordered response models with an unknown error distribution. The proposed methods do not require users to choose any tuning parameter and they automatically incorporate the monotonicity restriction of the unknown distribution function. Fixing finite dimensional parameters in the model, we construct nonparametric maximum likelihood estimates (NPMLE) for the error distribution based on the related binary choice data or the entire ordered response data. We then obtain estimates for finite dimensional parameters based on moment conditions given the estimated distribution function. Our semiparametric approaches deliver root-n consistent and asymptotically normal estimators of the regression coefficient and threshold parameter. We also develop valid bootstrap procedures for inference. We apply our methods to the interdependent durations model in Honore and de Paula (2010), where the social interaction effect is directly related to the threshold parameter in the corresponding ordered response model. The advantages of our methods are borne out in simulation studies and a real data application to the joint retirement decision of married couples.

Monotonic classification: An overview on algorithms, performance measures and data sets

Currently, knowledge discovery in databases is an essential first step when identifying valid, novel and useful patterns for decision making. There are many real-world scenarios, such as bankruptcy prediction, option pricing or medical diagnosis, where the classification models to be learned need to fulfill restrictions of monotonicity (i.e. the target class label should not decrease when input attributes values increase). For instance, it is rational to assume that a higher debt ratio of a company should never result in a lower level of bankruptcy risk. Consequently, there is a growing interest from the data mining research community concerning monotonic predictive models. This paper aims to present an overview of the literature in the field, analyzing existing techniques and proposing a taxonomy of the algorithms based on the type of model generated. For each method, we review the quality metrics considered in the evaluation and the different data sets and monotonic problems used in the analysis. In this way, this paper serves as an overview of monotonic classification research in specialized literature and can be used as a functional guide for the field.