Weak Assumptions Research Articles

Simple Inference on Functionals of Set-Identified Parameters Defined by Linear Moments

This article proposes a new approach to obtain uniformly valid inference for linear functionals or scalar subvectors of a partially identified parameter defined by linear moment inequalities. The procedure amounts to bootstrapping the value functions of randomly perturbed linear programming problems, and does not require the researcher to grid over the parameter space. The low-level conditions for uniform validity rely on genericity results for linear programs. The unconventional perturbation approach produces a confidence set with a coverage probability of 1 over the identified set, but obtains exact coverage on an outer set, is valid under weak assumptions, and is computationally simple to implement.

Optimized laser models with Heisenberg-limited coherence and sub-Poissonian beam photon statistics

Recently it has been shown that it is possible for a laser to produce a stationary beam with a coherence (quantified as the mean photon number at spectral peak) which scales as the fourth power of the mean number of excitations stored within the laser, this being quadratically larger than the standard or Schawlow-Townes limit [1]. Moreover, this was analytically proven to be the ultimate quantum limit (Heisenberg limit) scaling under defining conditions for CW lasers, plus a strong assumption about the properties of the output beam. In Ref. [2], we show that the latter can be replaced by a weaker assumption, which allows for highly sub-Poissonian output beams, without changing the upper bound scaling or its achievability. In this Paper, we provide details of the calculations in Ref. [2], and introduce three new families of laser models which may be considered as generalizations of those presented in that work. Each of these families of laser models is parameterized by a real number, $p$, with $p=4$ corresponding to the original models. The parameter space of these laser families is numerically investigated in detail, where we explore the influence of these parameters on both the coherence and photon statistics of the laser beams. Two distinct regimes for the coherence may be identified based on the choice of $p$, where for $p>3$, each family of models exhibits Heisenberg-limited beam coherence, while for $p<3$, the Heisenberg limit is no longer attained. Moreover, in the former regime, we derive formulae for the beam coherence of each of these three laser families which agree with the numerics. We find that the optimal parameter is in fact $p\approx4.15$, not $p=4$.

Birth Death Swap population in random environment and aggregation with two timescales

This paper deals with the stochastic modeling of a class of heterogeneous population in random environment, structured by discrete subgroups, and called birth–death–swap. In addition to demographic events modifying the population size, swap events, i.e. moves between subgroups, occur in the population. Event intensities are random functionals of the population. In the first part, we show that the complexity of the problem is significantly reduced by modeling the jump measure of the population as a multivariate counting process. This process is defined as the solution of a stochastic differential system with random coefficients, driven by a multivariate Poisson random measure. The solution is obtained under weak assumptions, by the thinning of a strongly dominating point process generated by the same Poisson measure. This key construction relies on a general strong comparison result, of independent interest.The second part is dedicated to averaging results when swap events are significantly more frequent than demographic events. An important ingredient is the stable convergence, which is well-adapted to the general random environment. The pathwise construction by domination yields tightness results straightforwardly. At the limit, the demographic intensity functionals are averaged against random kernels resulting from swap events. Finally, under a natural assumption, we show the convergence of the aggregated population to a birth–death process in random environment, with non-linear intensity functionals.

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.

Estimation of multivariate tail quantities

For d≥2 risk variables, three methods have been proposed to estimate the multivariate tail quantities, including multivariate tail probabilities, tail dependence functions and tail quantile sets. The methods are based on weak assumptions on the joint tails of the copulas of the d variables. The first method is developed based on the tail expansion of copula along different directions to the joint upper or lower orthant. The latter two methods are based on the asymptotic expansion of a family of tail-weighted functions defined from the copula. Extensive simulation experiments are conducted to evaluate and compare the three methods under different scenarios. The simulation results show that the methods yield accurate estimates of the tail quantities and effectively distinguish the tail properties, such as reflection asymmetry, permutation asymmetry, and heterogeneous tail dependence. One data example is presented to illustrate the applicability of the proposed methods as inference and diagnostic tools.

Superpolynomial quantum-classical separation for density modeling

Density modelling is the task of learning an unknown probability density function from samples, and is one of the central problems of unsupervised machine learning. In this work, we show that there exists a density modelling problem for which fault-tolerant quantum computers can offer a super-polynomial advantage over classical learning algorithms, given standard cryptographic assumptions. Along the way, we provide a variety of additional results and insights, of potential interest for proving future distribution learning separations between quantum and classical learning algorithms. Specifically, we (a) provide an overview of the relationships between hardness results in supervised learning and distribution learning, and (b) show that any weak pseudo-random function can be used to construct a classically hard density modelling problem. The latter result opens up the possibility of proving quantum-classical separations for density modelling based on weaker assumptions than those necessary for pseudo-random functions.

On the reliability of published findings using the regression discontinuity design in political science

The regression discontinuity (RD) design offers identification of causal effects under weak assumptions, earning it a position as a standard method in modern political science research. But identification does not necessarily imply that causal effects can be estimated accurately with limited data. In this paper, we highlight that estimation under the RD design involves serious statistical challenges and investigate how these challenges manifest themselves in the empirical literature in political science. We collect all RD-based findings published in top political science journals in the period 2009–2018. The distribution of published results exhibits pathological features; estimates tend to bunch just above the conventional level of statistical significance. A reanalysis of all studies with available data suggests that researcher discretion is not a major driver of these features. However, researchers tend to use inappropriate methods for inference, rendering standard errors artificially small. A retrospective power analysis reveals that most of these studies were underpowered to detect all but large effects. The issues we uncover, combined with well-documented selection pressures in academic publishing, cause concern that many published findings using the RD design may be exaggerated.

Deep nonparametric regression on approximate manifolds: Nonasymptotic error bounds with polynomial prefactors

We study the properties of nonparametric least squares regression using deep neural networks. We derive nonasymptotic upper bounds for the excess risk of the empirical risk minimizer of feedforward deep neural regression. Our error bounds achieve minimax optimal rate and improve over the existing ones in the sense that they depend polynomially on the dimension of the predictor, instead of exponentially on dimension. We show that the neural regression estimator can circumvent the curse of dimensionality under the assumption that the predictor is supported on an approximate low-dimensional manifold or a set with low Minkowski dimension. We also establish the optimal convergence rate under the exact manifold support assumption. We investigate how the prediction error of the neural regression estimator depends on the structure of neural networks and propose a notion of network relative efficiency between two types of neural networks, which provides a quantitative measure for evaluating the relative merits of different network structures. To establish these results, we derive a novel approximation error bound for the Hölder smooth functions using ReLU activated neural networks, which may be of independent interest. Our results are derived under weaker assumptions on the data distribution and the neural network structure than those in the existing literature.

Saturation of the magnetorotational instability and the origin of magnetically elevated accretion discs

ABSTRACT We propose that the strength of angular momentum transport in accretion discs threaded by net vertical magnetic field is determined by a self-regulation mechanism: the magnetorotational instability (MRI) grows until its own turbulent resistivity damps the fastest growing mode on the scale of the disc thickness. Given weak assumptions as to the structure of MRI-derived turbulence, supported by prior simulation evidence, the proposed mechanism reproduces the known scaling of the viscous α-parameter, $\alpha \propto \beta _z^{-1/2}$. Here, $\beta _z = 8\pi p_{\rm g}/B_{z0}^2$ is the initial plasma β-parameter on the disc mid-plane, Bz0 is the net field, and pg is the mid-plane gas pressure. We generalize the argument to discs with strong suprathermal toroidal magnetic fields, where the MRI growth rate is modified from the weak-field limit. Additional sources of turbulence are required if such discs are to become magnetically elevated, with the increased scale heights near the mid-plane that are seen in simulations. We speculate that tearing modes, associated with current sheets broadened by the effective resistivity, are a possible source of enhanced turbulence in elevated discs.

Optimal estimating function for weak location‐scale dynamic models

Estimating functions provide a very general framework for the statistical inference of dynamic models under weak assumptions. We consider a class of time series models consisting in the parametrization of the first two conditional moments which – by contrast with classical location‐scale dynamic models – do not impose further constraints on the conditional distribution/moments. Quasi‐likelihood estimators (QLE) are obtained by solving estimating equations deduced from those two conditional moments. Conditions ensuring the existence and asymptotic properties (consistency and asymptotic normality) of such estimators are provided. We propose optimal QLEs in Godambe's sense, deduced from a condition obtained by Chandra and Taniguchi (2001, Annals of the Institute of Statistical Mathematics 53, 125–141). The particular case of the quasi‐maximum likelihood estimators is considered. For pure location models, a data‐driven procedure for optimally choosing the QLE is proposed. Our results are illustrated via Monte Carlo experiments and real financial data.

From transformation to normalisation: An exploratory study of a large-scale agile transformation

Following the highly pervasive and effective use of agile methods at the team level, many software organisations now wish to replicate this success at the organisational level, adopting large-scale agile methods. However, an analysis of extant literature reveals (i) a lack of theoretically grounded studies of large-scale agile transformations; (ii) weak assumptions underpinning those that do exist; and (iii) an overly simplistic and often problematic over-reliance on simple adherence to agile methods. In addition, there is a lack of insights on why large-scale agile transformations fail. To explore this, we apply normalisation process theory (NPT) to examine key challenges in normalising (i.e. the process of embedding and sustaining) large-scale agile methods. We present an in-depth case study of large-scale agile transformation and demonstrate how failing to normalise a large-scale agile method can unravel an organisation’s transformation efforts. There are four key contributions from this research. First, this study shows that the standard focus on initial adoption is one of the main reasons for the widespread failure of digital transformations and in this case a large-scale agile transformation. Second, this is the first study to introduce and empirically apply NPT to information system development (ISD) as an illustrative guide for those across information systems field to follow. Third, we present key recommendations based on NPT to guide researchers and practitioners on the normalisation of large-scale transformations. Fourth, we present an agenda for future research on large-scale agile transformations.

Cullen Numbers in Sums of Terms of Recurrence Sequence

Let $$(U_n)_{n\ge 0}$$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $$\ell $$ , let $$\ell \cdot 2^{\ell } + 1$$ be a Cullen number. Recently in Bilu et al. (J Number Theory 202:412–425, 2019), generalized Cullen numbers in terms of linear recurrence sequence $$(U_n)_{n\ge 0}$$ under certain weak assumptions has been studied. In this paper, we consider the more general Diophantine equation $$U_{n_1} + \cdots + U_{n_k} = \ell \cdot x^{\ell } + Q(x)$$ , for a given polynomial $$Q(x) \in \mathbb {Z}[x]$$ and prove an effective finiteness result. Furthermore, we demonstrate our method by an example.

Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation

AbstractGradient‐type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD‐id) is one of such methods. The convergence behavior of the PSD‐id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non‐asymptotic estimates indicate a superlinear convergence of the PSD‐id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD‐id using a restricted formulation of the PSD‐id. More importantly, we extend the new convergence analysis of the PSD‐id to a practically preferred block version of the PSD‐id, or BPSD‐id, and show the cluster robustness of the BPSD‐id. Numerical examples are provided to validate the theoretical estimates.

Backward Importance Sampling for Online Estimation of State Space Models

This article proposes a new Sequential Monte Carlo algorithm to perform online estimation in the context of state space models when either the transition density of the latent state or the conditional likelihood of an observation given a state is intractable. In this setting, obtaining low variance estimators of expectations under the posterior distributions of the unobserved states given the observations is a challenging task. Following recent theoretical results for pseudo-marginal sequential Monte Carlo smoothers, a pseudo-marginal backward importance sampling step is introduced to estimate such expectations. This new step allows to reduce very significantly the computational time of the existing numerical solutions based on an acceptance–rejection procedure for similar performance, and to broaden the class of eligible models for such methods. For instance, in the context of multivariate stochastic differential equations, the proposed algorithm makes use of unbiased estimates of the unknown transition densities under much weaker assumptions than most standard alternatives. The performance of this estimator is assessed for high-dimensional discrete-time latent data models, for recursive maximum likelihood estimation in the context of Partially Observed Diffusion process (POD), and in the case of a bidimensional partially observed stochastic Lotka-Volterra model. Supplementary materials for this article are available online.

Mediation and instrumental variable analyses for vaccine-induced antibody titer against influenza B

ObjectiveImmune correlate analyses for vaccine trials have been applied to investigate associations of vaccine efficacy and surrogate markers such as vaccine-induced antibodies. However, the role of antibody as a surrogate marker in predicting the outcome can vary by time, and surrogate-outcome confounding may have resulted in bias even in randomized trials. We provide a framework for surrogate marker assessment to address the aforementioned issues. Study Design and SettingWe reanalyzed the vaccine randomized trial for influenza B. We conducted a mediation analysis that enables estimation of vaccine efficacy, mediation effects and proportion of mediation on disease probabilities at various follow-up times. We proposed instrumental variable (IV) analyses with randomized vaccination as an IV accounting for potential unmeasured confounding. ResultsThe mediation effect of vaccine efficacy by hemagglutination inhibition (HAI) titer was significantly protective at 181 days after vaccination: 63.2% [95% confidence interval, (CI) = (39.9%, 82.0%)], and HAI titer explained 61.1% [95% CI = (36.7%, 96.2%)] of the protective effect of vaccination. ConclusionsMost of vaccine efficacy is mediated by HAI titer, particularly in children 10 years and older. Our contribution is to provide causal analytics for the role of surrogate marker with weaker assumptions regarding surrogate–disease causation.

Comparing random effects models, ordinary least squares, or fixed effects with cluster robust standard errors for cross-classified data.

Cross-classified random effects modeling (CCREM) is a common approach for analyzing cross-classified data in psychology, education research, and other fields. However, when the focus of a study is on the regression coefficients at Level 1 rather than on the random effects, ordinary least squares regression with cluster robust variance estimators (OLS-CRVE) or fixed effects regression with CRVE (FE-CRVE) could be appropriate approaches. These alternative methods are potentially advantageous because they rely on weaker assumptions than those required by CCREM. We conducted a Monte Carlo Simulation study to compare the performance of CCREM, OLS-CRVE, and FE-CRVE in models, including conditions where homoscedasticity assumptions and exogeneity assumptions held and conditions where they were violated, as well as conditions with unmodeled random slopes. We found that CCREM out-performed the alternative approaches when its assumptions are all met. However, when homoscedasticity assumptions are violated, OLS-CRVE and FE-CRVE provided similar or better performance than CCREM. When the exogeneity assumption is violated, only FE-CRVE provided adequate performance. Further, OLS-CRVE and FE-CRVE provided more accurate inferences than CCREM in the presence of unmodeled random slopes. Thus, we recommend two-way FE-CRVE as a good alternative to CCREM, particularly if the homoscedasticity or exogeneity assumptions of the CCREM might be in doubt. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

The total factor carbon emission productivity in China’s industrial Sectors: An analysis based on the global Malmquist-Luenberger index

In this paper, we construct a new Global Malmquist-Luenberger Index to measure the total factor carbon emission productivity (TFCEP). Specifically, by invoking the weak disposability assumptions on both energy inputs and carbon dioxide emissions, the index constructed in this study fully considers the relationship between energy inputs and carbon dioxide emissions. This method avoids the distortion of estimates present in the existing techniques due to their neglect of the inherent relationship between energy inputs and carbon dioxide emissions. Employing this index, the TFCEP of 41 industrial sectors in China from 2013 to 2016 was measured and analyzed. Our main results are as follows. First, for most industrial sectors, the TFCEP increased yearly, but the growth rate was low. Second, the growth and decline of TFCEP were mainly driven by technological progress or deterioration. Third, the change of technical efficiency in the industrial sectors was not synchronized with the technological change. However, higher efficiency changes were often accompanied by a lower best practice gap change (BPC) and vice versa.

Gradient blowup behavior for a viscous Hamilton-Jacobi equation with degenerate gradient nonlinearity

The paper is concerned with gradient blowup behavior for a semilinear parabolic equationut−Δu=δm(x)|∇u|p+ΛinΩ×(0,T) with the zero Dirichlet condition. Two results with p>m+2 and m⩾0 are established. One of which is that any gradient blowup solution follows a global ODE type behavior, with domination of normal derivatives over the tangential derivatives. The other is about time-increasing solutions. Zhang and Hu (2010) [50] obtained the precise gradient blowup rate in one-dimensional case, but the higher dimensional case was left as an open problem. Here we solve this problem by establishing the gradient blowup rate, for any small γ>0,C(T−t)−m+1p−m−2⩽‖∇u‖∞⩽Cγ(T−t)−m+1p−m−2−γ for suitable ranges of p and m, which extends the result of (Attouchi and Souplet (2020) [3]) to the case m≠0. As an important by-product which is of independent interests itself, the gradient estimate near boundary for the corresponding elliptic equation is derived under weaker assumptions on the inhomogeneous term.

Manifolds with nonnegative curvature operator of the second kind

We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is diffeomorphic to a spherical space form, improving a recent result of Cao–Gursky–Tran assuming two-positivity. The second one states that a closed Riemannian manifold with three-nonnegative curvature operator of the second kind is either diffeomorphic to a spherical space form, or flat, or isometric to a quotient of a compact irreducible symmetric space. This settles the nonnegativity part of Nishikawa’s conjecture under a weaker assumption.

Solution of the Modified Time Fractional Coupled Burgers Equations Using Laplace Adomian Decompostion Method

Abstract In this work, a coupled system of time-fractional modified Burgers’ equations is considered. Three different fractional operators: Caputo, Caputo-Fabrizio and Atangana-Baleanu operators are implemented for the equations. Also, two different scenarios are examined for each fractional operator: when the initial conditions are u(x, y, 0) = sin(xy), v(x, y, 0) = sin(xy), and when they are u(x, y, 0) = e{−kxy}, v(x, y, 0) = e{−kxy}, where k, α are some positive constants. With the aid of computable Adomian polynomials, the solutions are obtained using Laplace Adomian decomposition method (LADM). The method does not need linearization, weak nonlinearity assumptions or perturbation theory. Simulations are also presented to support theoretical results, and the behaviour of the solutions under the three different fractional operators compared.