Strong Regularity Condition Research Articles

Bootstrapping ARMA time series models after model selection

Inference after model selection is a very important problem. This article derives the asymptotic distribution of some model selection estimators for autoregressive moving average time series models. Under strong regularity conditions, the model selection estimators are asymptotically normal, but generally the asymptotic distribution is a non normal mixture distribution. Hence bootstrap confidence regions that can handle this complicated distribution were used for hypothesis testing. A bootstrap technique to eliminate selection bias is to fit the model selection estimator β ̂ M S ∗ to a bootstrap sample to find a submodel, then draw another bootstrap sample, and fit the same submodel to get the bootstrap estimator β ̂ M I X ∗ .

Stochastic Approximation for Multi-period Simulation Optimization with Streaming Input Data

We consider a continuous-valued simulation optimization (SO) problem, where a simulator is built to optimize an expected performance measure of a real-world system while parameters of the simulator are estimated from streaming data collected periodically from the system. At each period, a new batch of data is combined with the cumulative data and the parameters are re-estimated with higher precision. The system requires the decision variable to be selected in all periods. Therefore, it is sensible for the decision-maker to update the decision variable at each period by solving a more precise SO problem with the updated parameter estimate to reduce the performance loss with respect to the target system. We define this decision-making process as the multi-period SO problem and introduce a multi-period stochastic approximation (SA) framework that generates a sequence of solutions. Two algorithms are proposed: Re-start SA ( ReSA ) reinitializes the stepsize sequence in each period, whereas Warm-start SA ( WaSA ) carefully tunes the stepsizes, taking both fewer and shorter gradient-descent steps in later periods as parameter estimates become increasingly more precise. We show that under suitable strong convexity and regularity conditions, ReSA and WaSA achieve the best possible convergence rate in expected sub-optimality either when an unbiased or a simultaneous perturbation gradient estimator is employed, while WaSA accrues significantly lower computational cost as the number of periods increases. In addition, we present the regularized ReSA which obviates the need to know the strong convexity constant and achieves the same convergence rate at the expense of additional computation.

A new fast predictor-corrector method for nonlinear time-fractional reaction-diffusion equation with nonhomogeneous terms

Time-fractional reaction-diffusion (TFRD) equation is an important fractional parabolic equation, and the research of its numerical solution has scientific significance and engineering application value. In this paper, a new fast predictor-corrector (FP-C) scheme is constructed based on fast L1 approximation of Caputo fractional derivative for solving nonlinear TFRD equation with nonhomogeneous terms. The linearized implicit difference scheme is used for the predictor step and Crank-Nicolson(C-N) scheme is used for the corrector step. Theoretical analysis proves that FP-C scheme is convergent and stable unconditionally for nonlinear TFRD equation. Numerical analysis and experiments show that the computational accuracy of FP-C scheme is $ O(\tau^{2-\alpha}+h^2) $ under the strong regularity condition and $ O(\tau^{\alpha}+h^2) $ under the weak regularity condition. Compared with the classical predictor-corrector (P-C) scheme based on standard L1 approximation, the FP-C scheme improves the computational efficiency without losing the computational accuracy. It is shown that the new FP-C scheme is an efficient method to solve nonlinear TFRD equation with nonhomogeneous terms.

Optimal multiple stopping problem under nonlinear expectation

AbstractIn this paper, we study the optimal multiple stopping problem under the filtration-consistent nonlinear expectations. The reward is given by a set of random variables satisfying some appropriate assumptions, rather than a process that is right-continuous with left limits. We first construct the optimal stopping time for the single stopping problem, which is no longer given by the first hitting time of processes. We then prove by induction that the value function of the multiple stopping problem can be interpreted as the one for the single stopping problem associated with a new reward family, which allows us to construct the optimal multiple stopping times. If the reward family satisfies some strong regularity conditions, we show that the reward family and the value functions can be aggregated by some progressive processes. Hence, the optimal stopping times can be represented as hitting times.

Understanding the dual formulation for the hedging of path-dependent options with price impact

We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (SIAM J. Control Optim. 57 (2019) 4125–4149), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under strong regularity conditions. Using only probabilistic arguments, we prove, in a path-dependent setting and under weak regularity conditions, that any solution to this dual problem actually allows one to construct explicitly a perfect hedging portfolio. From a pure probabilistic point of view, our approach also allows one to exhibit solutions to a specific class of second order forward backward stochastic differential equations, in the sense of Cheridito et al. (Comm. Pure Appl. Math. 60 (2007) 1081–1110). Existence of a solution to the dual optimal control problem is also addressed in particular settings. As a by-product of our arguments, we prove a version of Itô’s lemma for path-dependent functionals that are only C0,1 in the sense of Dupire.

Bootstrapping some GLM and survival regression variable selection estimators

Inference after variable selection is a very important problem. This paper derives the asymptotic distribution of many variable selection estimators, such as forward selection and backward elimination, when the number of predictors is fixed. Under strong regularity conditions, the variable selection estimators are asymptotically normal, but generally the asymptotic distribution is a nonnormal mixture distribution. The theory shows that the lasso variable selection and elastic net variable selection estimators are consistent estimators of when lasso and elastic net are consistent estimators of A bootstrap technique to eliminate selection bias is to fit the variable selection estimator to a bootstrap sample to find a submodel, then draw another bootstrap sample and fit the same submodel to get the bootstrap estimator Bootstrap confidence regions were used for hypothesis testing.

Strong convergence order for slow–fast McKean–Vlasov stochastic differential equations

In this paper, we consider the averaging principle for a class of McKean–Vlasov stochastic differential equations with slow and fast time-scales. Under some proper assumptions on the coefficients, we first prove that the slow component strongly converges to the solution of the corresponding averaged equation with convergence order 1/3 using the approach of time discretization. Furthermore, under stronger regularity conditions on the coefficients, we use the technique of Poisson equation to improve the order to 1/2, which is the optimal order of strong convergence in general.

Two-sample Behrens–Fisher problems for high-dimensional data: A normal reference approach

High-dimensional data are frequently encountered with the development of modern data collection techniques. Testing the equality of the mean vectors of two high-dimensional samples with possibly different covariance matrices is usually referred to as a high-dimensional two-sample Behrens–Fisher (BF) problem. In the high-dimensional setting, the classical BF solutions are expected to perform poorly or become inapplicable due to the singularity of the sample covariance matrices. Several approaches have been proposed in the literature to address this challenging issue but they all require strong regularity conditions on the underlying covariance matrices to guarantee that their test statistics are asymptotically normally distributed. To overcome this difficulty, an L2-norm-based test is proposed and studied in this article. It is shown that under some regularity conditions and the null hypothesis, the test statistic and a chi-square-type mixture have the same normal or non-normal limiting distribution. It is then natural to approximate the null distribution of the proposed test using that of the chi-square-type mixture, which is actually obtained from the proposed test statistic when the two high-dimensional samples are normally distributed. The resulting test is then referred to as a normal reference test. The distribution of the chi-square-type mixture can then be well approximated by the Welch–Satterthwaite χ2-approximation with the approximation parameters consistently estimated from the data. The asymptotic power of the proposed test is established. Good performance of the proposed test against several existing competitors is demonstrated via several simulation studies and illustrated by a real data example.

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand.

Time evolutions of copulas and foreign exchange markets

Time evolution of copulas is well visible in such a dynamical market as foreign exchange market (ForeX, FX, or currency market). We first show how several families of copulas evolving in time for EURO-JPY and CHF-JPY at ForeX market. Black-Scholes paradigm suggest to apply evolution of copulas with respect to heat equation. Stationary limit of such an evolution is proven to be an independence copula under strong regularity conditions. However, empirical observations of ForeX stock confirm that reality can be more delicate, because of the ForeX market violations.The manuscript shows that under slight changes of topology, the limiting object is not a copula, because the 1-Lipschitzianity continuity is violated. The authors study these kinds of convergences with respect to FEMA (Foreign Exchange Management Act) violations.

Weak differentiability of Wiener functionals and occupation times

In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Leão, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t. Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite p-variation (p≥2) regularity is also discussed. Under stronger regularity conditions in the sense of finite (p,q)-variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established.

Robin problems with indefinite and unbounded potential, resonant at $-\infty$, superlinear at $+\infty$

We consider a semilinear Robin problem with an indefinite and unbounded potential and a reaction which exhibits asymmetric behavior as $x\rightarrow\pm\infty$. More precisely it is sublinear near $-\infty$ with possible resonance with respect to the principal eigenvalue of the negative Robin Laplacian and it is superlinear at $+\infty$. Resonance is also allowed at zero with respect to any nonprincipal eigenvalue. We prove two multiplicity results. In the first one, we obtain two nontrivial solutions and in the second, under stronger regularity conditions on the reaction, we produce three nontrivial solutions. Our work generalizes the recent one by Recova-Rumbos (Nonlin. Anal. 112 (2015), 181--198).

Simulation of multibody systems with servo constraints through optimal control

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path.Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques.By means of the well-known n-car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.

Bilevel Optimization for Calibrating Point Spread Functions in Blind Deconvolution

Blind deconvolution problems arise in many imaging modalities, where both the underlying point spread function, which parameterizes the convolution operator, and the source image need to be identified. In this work, a novel bilevel optimization approach to blind deconvolution is proposed. The lower-level problem refers to the minimization of a total-variation model, as is typically done in non-blind image deconvolution. The upper-level objective takes into account additional statistical information depending on the particular imaging modality. Bilevel problems of such type are investigated systematically. Analytical properties of the lower-level solution mapping are established based on Robinson's strong regularity condition. Furthermore, several stationarity conditions are derived from the variational geometry induced by the lower-level problem. Numerically, a projected-gradient-type method is employed to obtain a Clarke-type stationary point and its convergence properties are analyzed. We also implement an efficient version of the proposed algorithm and test it through the experiments on point spread function calibration and multiframe blind deconvolution.

Investigation of regularity conditions in optimal control problems with geometric mixed constraints

We investigate regularity conditions in optimal control problems with mixed constraints of a general geometric type, in which a closed non-convex constraint set appears. A closely related question to this issue concerns the derivation of necessary optimality conditions under some regularity conditions on the constraints. By imposing strong and weak regularity condition on the constraints, we provide necessary optimality conditions in the form of Pontryagin maximum principle for the control problem with mixed constraints. The optimality conditions obtained here turn out to be more general than earlier results even in the case when the constraint set is convex. The proofs of our main results are based on a series of technical lemmas which are gathered in the Appendix.

The asymptotic behavior of the order parameter for the infinite-N Kuramoto model

The Kuramoto model, first proposed in 1975, consists of a population of sinusoidally coupled oscillators with random natural frequencies. It has served as an idealized model for coupled oscillator systems in physics, chemistry, and biology. This paper addresses a long-standing problem about the infinite-N Kuramoto model, which is to describe the asymptotic behavior of the order parameter for this system. For coupling below a critical level, Kuramoto predicted that the order parameter would decay to 0. We use Fourier transform methods to prove that for general initial conditions, this decay is not exponential; in fact, exponential decay to 0 can only occur if the initial condition satisfies a fairly strong regularity condition that we describe. Our theorem is a partial converse to the recent results of Ott and Antonsen, who proved that for a special class of initial conditions, the order parameter does converge exponentially to its limiting value. Consequently, our result shows that the Ott-Antonsen ansatz does not completely capture all the possible asymptotic behavior in the full Kuramoto system.

Revisited mixed-value method via symmetric BEM in the substructuring approach

Abstract Within the Symmetric Boundary Element Method, the mixed-value analysis is re-formulated. This analysis method contemplates the subdivision of the body into substructures having interface kinematical and mechanical quantities. For each substructure an elasticity equation, connecting weighted displacements and tractions to nodal displacements and forces of the same interface boundary and to external action vector, is introduced. The assembly of the substructures is performed through both the strong and weak regularity conditions of the displacements and tractions. We obtain the solving equations where the compatibility and the equilibrium are guaranteed in the domain Ω for the use of the fundamental solution and at the interface nodes for the strong regularity conditions imposed, whereas the previous quantities are respected in weighted form along the interface boundaries. The mixed-value method leads to a better solution than those obtained through the displacement method of the Symmetric Boundary Element Method, if compared with the analytical solution. By using the Karnak.sGbem program, developed with other researchers and updated through the implementation of the present method, some examples are made which show the advantages related to the computational aspects and to the convergence of the numerical response.

Advertising Budget And Sales Paths Under The Dynamics Of The Student Work Control Problem And Regularity Requirements

Consider a firm promoting a product in a fast expanding industry by using advertising as its single promotional tool. The firms objective is to minimize the overall cost of advertising necessary for reaching certain target sales of the product by the end of a given planning period. We adopt the Student Work Control problem (SWC) framework for modeling this marketing context, in general, and the advertising-sales response function, in particular. We compare the SWCs optimal control budgeting principle with the solutions of equally effective, alternative advertising budgeting principles, which require strong regularity conditions on the path of either the advertising outlays or sales. In contrast to the other principles, SWCs optimal sales path is highly convex to the point that the firm may deliberately accept decreasing sales at the earliest periods. However, its optimal solution requires the firm to advertise in every period and to continue to accelerate its advertising outlays. The resulting Advertising Sales Response function, too, may therefore have a convex section with declining sales, a finding contributing to an optimization-driven explanation of threshold in advertising effect on sales.

A note on regularity and positive definiteness of interval matrices

Abstract We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval matrices. Comparing it with Beeck’s strong regularity condition, we show by counterexamples that none of the two conditions is more general than the other one.

Revisiting Tests for Neglected Nonlinearity Using Artificial Neural Networks

Tests for regression neglected nonlinearity based on artificial neural networks (ANNs) have so far been studied by separately analyzing the two ways in which the null of regression linearity can hold. This implies that the asymptotic behavior of general ANN-based tests for neglected nonlinearity is still an open question. Here we analyze a convenient ANN-based quasi-likelihood ratio statistic for testing neglected nonlinearity, paying careful attention to both components of the null. We derive the asymptotic null distribution under each component separately and analyze their interaction. Somewhat remarkably, it turns out that the previously known asymptotic null distribution for the type 1 case still applies, but under somewhat stronger conditions than previously recognized. We present Monte Carlo experiments corroborating our theoretical results and showing that standard methods can yield misleading inference when our new, stronger regularity conditions are violated.