Asymptotic Efficiency Research Articles

An Efficient Data-Independent Priority Queue and its Application to Dark Pools

We introduce a secure data-independent priority queue which supports polylogarithmic-time insertion operations and constant-time deletions and read-front (aka peek) operations as opposed to the originally introduced queue by Toft (PODC '11). Moreover, we minimize the number of comparisons required to perform different operations on Toft's priority queue. Data-independent data structures—first identified explicitly by Toft, and further elaborated by Mitchell and Zimmerman (STACS '14)—serve the purpose of computing on encrypted data without executing branching code which can be used to avoid prohibitively expensive operations in secure computation applications. Focusing on the costly sorting operations, we show significant asymptotic improvements over prior privacy preserving dark pool applications. Dark pools are securities-trading venues which attain ad-hoc order privacy, by matching orders outside of publicly visible exchanges via the so-called dark pool operators. In this paper, we describe an efficient and secure dark pool (implementing a full continuous double auction) based on our new priority queue. Our construction's security guarantees are cryptographic based on secure multiparty computation (MPC), and do not require that the dark pool operators are trusted. Our construction improves upon the asymptotic efficiency attained by previous efforts. Existing cryptographic dark pools process new orders in time which grows linearly in the size of the standing order book; ours does so in polylogarithmic time. We describe a concrete implementation of our MPC protocol with malicious security in the honest majority setting. We also report benchmarks of our implementation and compare them to prior works. Our protocol reduces the total running time by several orders of magnitude over prior secure dark pool solutions.

Highly Efficient Estimators with High Breakdown Point for Linear Models with Structured Covariance Matrices

A unified approach is provided for a method of estimation of the regression parameter in balanced linear models with a structured covariance matrix that combines a high breakdown point with high asymptotic efficiency at models with multivariate normal errors. Of main interest are linear mixed effects models, but our approach also includes several other standard multivariate models, such as multiple regression, multivariate regression, and multivariate location and scatter. Sufficient conditions are provided for the existence of the estimators and corresponding functionals, strong consistency and asymptotic normality is established, and robustness properties are derived in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities. Some results are new and others are more general than existing ones in the literature. In this way, results on high breakdown estimation with high efficiency in a wide variety of multivariate models are completed and improved.

Algorithm 1034: An Accelerated Algorithm to Compute the Q n Robust Statistic, with Corrections to Constants

The robust scale estimator Q n developed by Croux and Rousseeuw [ 3 ], for the computation of which they provided a deterministic algorithm, has proven to be very useful in several domains including in quality management and time series analysis. It has interesting mathematical (50% breakdown, 82% Asymptotic Relative Efficiency) and computing ( O(nlogn) time, O ( n ) space) properties. While working on a faster algorithm to compute Q n , we have discovered an error in the computation of the d constant, and as a consequence in the d n constants that are used to scale the statistic for consistency with the variance of a normal sample. These errors have been reproduced in several articles including in the International Standard Organisation 13,528 [ 12 ] document. In this article, we fix the errors and present a new approach, which includes a new algorithm, allowing computations to run 1.3 to 4.5 times faster when n grows from 10 to 100,000.

On Fixed-Accuracy Confidence Intervals for the Parameters of Lindley Distribution and Its Extensions

The purpose of the present paper is to deal with sequential estimation of the parameter θ in a Lindley distribution. A fixed-accuracy confidence interval for θ with a preassigned confidence coefficient is developed. It is established that, no fixed sample size procedure can solve the estimation problem and hence a purely sequential methodology is proposed to deal with the situation. The first-order asymptotic efficiency and consistency properties associated with our purely sequential strategy are derived. Similar estimation strategies are also outlined for a few other extensions of the Lindley distribution. Extensive simulation analysis is carried out to validate the theoretical findings. We also provide a real data example, where we estimate the parameter related to the “initial mass function" for a particular cluster of stars.

Asymptotically efficient estimation for diffusion processes with nonsynchronous observations

We study maximum-likelihood-type estimation for diffusion processes when the coefficients are nonrandom and observations occur in nonsynchronous manner. The problem of nonsynchronous observations is important when we consider the analysis of high-frequency data in a financial market. Constructing a quasi-likelihood function to define the estimator, we adaptively estimate the parameter for the diffusion part and the drift part. We consider the asymptotic theory when the terminal time point T_n and the observation frequency goes to infinity, and show the consistency and the asymptotic normality of the estimator. Moreover, we show local asymptotic normality for the statistical model, and asymptotic efficiency of the estimator as a consequence. To show the asymptotic properties of the maximum-likelihood-type estimator, we need to control the asymptotic behaviors of some functionals of the sampling scheme. Though it is difficult to directly control those in general, we study tractable sufficient conditions when the sampling scheme is generated by mixing processes.

Reprint of: Initial conditions and moment restrictions in dynamic panel data models

Estimation of the dynamic error components model is considered using two alternative linear estimators that are designed to improve the properties of the standard first-differenced GMM estimator. Both estimators require restrictions on the initial conditions process. Asymptotic efficiency comparisons and Monte Carlo simulations for the simple AR(1) model demonstrate the dramatic improvement in performance of the proposed estimators compared to the usual first-differenced GMM estimator, and compared to non-linear GMM. The importance of these results is illustrated in an application to the estimation of a labour demand model using company panel data.

On Huber's contaminated model

Huber's contaminated model is a basic model for data with outliers. This paper aims at addressing several fundamental problems about this model. We first study its identifiability properties. Several theorems are presented to determine whether the model is identifiable for various situations. Based on these results, we discuss the problem of estimating the parameters with observations drawn from Huber's contaminated model. A definition of estimation consistency is introduced to handle the general case where the model may be unidentifiable. This consistency is a strong robustness property. After showing that existing estimators cannot be consistent in this sense, we propose a new estimator that possesses the consistency property under mild conditions. Its adaptive version, which can simultaneously possess this consistency property and optimal asymptotic efficiency, is also provided. Numerical examples show that our estimators have better overall performance than existing estimators no matter how many outliers in the data.

Wasserstein information matrix

We study information matrices for statistical models by the $$L^2$$ -Wasserstein metric. We call them Wasserstein information matrices (WIMs), which are analogs of classical Fisher information matrices. We introduce Wasserstein score functions and study covariance operators in statistical models. Using them, we establish Wasserstein–Cramer–Rao bounds for estimations and explore their comparisons with classical results. We next consider the asymptotic behaviors and efficiency of estimators. We derive the online asymptotic efficiency for Wasserstein natural gradient. Besides, we establish a Poincaré efficiency for Wasserstein natural gradient of maximal likelihood estimation. Several analytical examples of WIMs are presented, including location-scale families, independent families, rectified linear unit (ReLU) generative models.

Adaptive normalization for IPW estimation

Abstract Inverse probability weighting (IPW) is a general tool in survey sampling and causal inference, used in both Horvitz–Thompson estimators, which normalize by the sample size, and Hájek/self-normalized estimators, which normalize by the sum of the inverse probability weights. In this work, we study a family of IPW estimators, first proposed by Trotter and Tukey in the context of Monte Carlo problems, that are normalized by an affine combination of the sample size and a sum of inverse weights. We show how selecting an estimator from this family in a data-dependent way to minimize asymptotic variance leads to an iterative procedure that converges to an estimator with connections to regression control methods. We refer to such estimators as adaptively normalized estimators. For mean estimation in survey sampling, the adaptively normalized estimator has asymptotic variance that is never worse than the Horvitz–Thompson and Hájek estimators. Going further, we show that adaptive normalization can be used to propose improvements of the augmented IPW (AIPW) estimator, average treatment effect (ATE) estimators, and policy learning objectives. Appealingly, these proposals preserve both the asymptotic efficiency of AIPW and the regret bounds for policy learning with IPW objectives, and deliver consistent finite sample improvements in simulations for all three of mean estimation, ATE estimation, and policy learning.

Covariate adjustment in multiarmed, possibly factorial experiments

AbstractRandomized experiments are the gold standard for causal inference and enable unbiased estimation of treatment effects. Regression adjustment provides a convenient way to incorporate covariate information for additional efficiency. This article provides a unified account of its utility for improving estimation efficiency in multiarmed experiments. We start with the commonly used additive and fully interacted models for regression adjustment in estimating average treatment effects (ATE), and clarify the trade-offs between the resulting ordinary least squares (OLS) estimators in terms of finite sample performance and asymptotic efficiency. We then move on to regression adjustment based on restricted least squares (RLS), and establish for the first time its properties for inferring ATE from the design-based perspective. The resulting inference has multiple guarantees. First, it is asymptotically efficient when the restriction is correctly specified. Second, it remains consistent as long as the restriction on the coefficients of the treatment indicators, if any, is correctly specified and separate from that on the coefficients of the treatment-covariate interactions. Third, it can have better finite sample performance than the unrestricted counterpart even when the restriction is moderately misspecified. It is thus our recommendation when the OLS fit of the fully interacted regression risks large finite sample variability in case of many covariates, many treatments, yet a moderate sample size. In addition, the newly established theory of RLS also provides a unified way of studying OLS-based inference from general regression specifications. As an illustration, we demonstrate its value for studying OLS-based regression adjustment in factorial experiments. Importantly, although we analyse inferential procedures that are motivated by OLS, we do not invoke any assumptions required by the underlying linear models.

Selection of linear mixed‐effects models for clustered data

AbstractWe consider model selection for linear mixed‐effects models with clustered structure, where conditional Kullback–Leibler (CKL) loss is applied to measure the efficiency of the selection. We estimate the CKL loss by substituting the empirical best linear unbiased predictors (EBLUPs) into random effects with model parameters estimated by maximum likelihood. Although the BLUP approach is commonly used in predicting random effects and future observations, selecting random effects to achieve asymptotic loss efficiency concerning CKL loss is challenging and has not been well studied. In this paper, we propose addressing this difficulty using a conditional generalized information criterion (CGIC) with two tuning parameters. We further consider a challenging but practically relevant situation where the number, , of clusters does not go to infinity with the sample size. Hence the random‐effects variances are not consistently estimable. We show that via a novel decomposition of the CKL risk, the CGIC achieves consistency and asymptotic loss efficiency, whether is fixed or increases to infinity with the sample size. We also conduct numerical experiments to illustrate the theoretical findings.

Robust testing procedures for scale differences in paired data

Several statistical tests have been proposed in the literature to assess the difference between two scale parameters in paired data. If the location parameters of each sample are different, many of these test statistics are adjusted using the estimated location parameters. Then, the performance of the test statistics depends on the selection of the location estimators. In this study, the asymptotically distribution-free test based on the U-statistic, which does not depend on two location parameters, is constructed. The asymptotic normality and the asymptotic power of the proposed test are examined. Furthermore, the proposed test is compared with the existing tests in terms of Pitman's asymptotic relative efficiency. To improve the Type I error rate in the small sample size, the robust test procedure based on the percentile bootstrap is also proposed. Our simulation studies show that the Type I error rate of the proposed test is more controlled and robust than existing tests in many situations. Further, the empirical power of the proposed test is shown to be high for heavy-tailed or asymmetric distributions.

Information Flow Design and Asymptotic Efficiency

Information Flow Design and Asymptotic Efficiency

Sieve estimation of a class of partially linear transformation models with interval-censored competing risks data.

In this paper, we consider a class of partially linear transformation models with interval-censored competing risks data. Under a semiparametric generalized odds rate specification for the cause-specific cumulative incidence function, we obtain optimal estimators of the large number of parametric and nonparametric model components via maximizing the likelihood function over a joint B-spline and Bernstein polynomial spanned sieve space. Our specification considers a relatively simpler finite-dimensional parameter space, approximating the infinite-dimensional parameter space as n → ∞, thereby allowing us to study the almost sure consistency, and rate of convergence for all parameters, and the asymptotic distributions and efficiency of the finite-dimensional components. We study the finite sample performance of our method through simulation studies under a variety of scenarios. Furthermore, we illustrate our methodology via application to a dataset on HIV-infected individuals from sub-Saharan Africa.

On the Robust Estimation of Spatial Autoregressive Models

The spatial autoregressive (SAR) model is commonly used for cross-sectional data with spatial dependence in the response variable. A robust M-estimator for the SAR model with independent normal random errors is proposed, which is resistant to extreme contamination or outliers in the data. The proposed estimator is defined by a set of robust estimating functions derived by modifying the estimating functions for the standard maximum likelihood (ML) estimator using the Huber function. An iterative algorithm based on iteratively reweighted least squares and a fixed point method is provided to compute the robust M-estimator. The proposed estimator is shown to be Fisher-consistent under the core model, has a bounded influence function, and is asymptotically normally distributed. A closed-form expression for the asymptotic covariance matrix estimator under the core model is derived, and used to study the asymptotic relative efficiency of the proposed estimator. Simulation studies demonstrate that the proposed robust M-estimator greatly outperforms the ML estimator as well as several other non-robust estimators of the SAR model, in terms of both estimation and inference under different contamination scenarios. Additionally, the proposed estimator offers comparable performance to some robust estimators of the SAR model under contamination. Applying the robust M-estimator for the SAR model to the U.S. federal grants and U.S. Nielsen Local Television View datasets reveals strong positive spatial dependence in the grant value among counties, insignificant positive effect of news watching on grant distribution, and evidence of outliers in the data. This results in different conclusions from those obtained using the ML estimator.

Asymptotic relative efficiency of the Kendall and Spearman correlation statistics

Необходимое и достаточное условие для того, чтобы асимптотическая относительная эффективность Питмена для корреляционных статистик Кендалла и Спирмена для критерия независимости равнялась $1$, дано в терминах некоторых свойств гладкости и невырожденности модели. Получены соответствующие простые в использовании и широко применимые достаточные условия. Эти условия выполняются для большинства известных моделей зависимости.

Composite smoothed quantile regression

Composite quantile regression (CQR) is an efficient method to estimate parameters of the linear model with non‐Gaussian random noise. The non‐smoothness of CQR loss prevents many efficient algorithms from being used. In this paper, we propose the composite smoothed quantile regression (CSQR) model and investigate the inference problem for a large‐scale dataset, in which the dimensionality is allowed to increase with the sample size while . After applying the convolution smoothing technique to the composite quantile loss, we obtain the convex and twice differentiable CSQR loss function, which can be optimized via the gradient descent algorithm. Theoretically, we establish the non‐asymptotic error bound for the CSQR estimators and further provide the Bahadur representation and the Berry–Esseen bound, from which the asymptotic normality of CSQR estimator can be immediately derived. To make valid inference, we construct the confidence intervals that based on the asymptotic distribution. Besides, we also explore the asymptotic relative efficiency of the CSQR estimator with respect to the standard CQR estimator. At last, we provide extensive numerical experiments on both simulated and real data to demonstrate the good performance of our CSQR estimator compared with some baselines.

On the asymptotic rate of convergence of Stochastic Newton algorithms and their Weighted Averaged versions

Most machine learning methods can be regarded as the minimization of an unavailable risk function. To optimize the latter, with samples provided in a streaming fashion, we define general (weighted averaged) stochastic Newton algorithms, for which a theoretical analysis of their asymptotic efficiency is conducted. The corresponding implementations are shown not to require the inversion of a Hessian estimate at each iteration under a quite flexible framework that covers the case of linear, logistic or softmax regressions to name a few. Numerical experiments on simulated and real data give the empirical evidence of the pertinence of the proposed methods, which outperform popular competitors particularly in case of bad initializations.

Conjuring Power from a Theory of Change: The PWRD Method for Trials with Anticipated Variation in Effects

We present an aggregation scheme that increases power in randomized controlled trials and quasi-experiments when the intervention possesses a robust and well-articulated theory of change. Intervention studies using longitudinal data often include multiple observations on individuals, some of which may be more likely to manifest a treatment effect than others. An intervention’s theory of change provides guidance as to which of those observations are best situated to exhibit that treatment effect. Our power-maximizing weighting for repeated-measurements with delayed-effects scheme, PWRD aggregation, converts the theory of change into a test statistic with improved asymptotic relative efficiency, delivering tests with greater statistical power. We illustrate this method on an IES-funded cluster randomized trial testing the efficacy of a reading intervention designed to assist early elementary students at risk of falling behind their peers. The salient theory of change holds program benefits to be delayed and non-uniform, experienced after a student’s performance stalls. In this instance, the PWRD technique’s effect on power is found to be comparable to that of doubling the number of clusters in the experiment.

Nonparametric Two-Sample Tests of High Dimensional Mean Vectors via Random Integration

Testing the equality of the means in two samples is a fundamental statistical inferential problem. Most of the existing methods are based on the sum-of-squares or supremum statistics. They are possibly powerful in some situations, but not in others, and they do not work in a unified way. Using random integration of the difference, we develop a framework that includes and extends many existing methods, especially in high-dimensional settings, without restricting the same covariance matrices or sparsity. Under a general multivariate model, we can derive the asymptotic properties of the proposed test statistic without specifying a relationship between the data dimension and sample size explicitly. Specifically, the new framework allows us to better understand the test’s properties and select a powerful procedure accordingly. For example, we prove that our proposed test can achieve the power of 1 when nonzero signals in the true mean differences are weakly dense with nearly the same sign. In addition, we delineate the conditions under which the asymptotic relative Pitman efficiency of our proposed test to its competitor is greater than or equal to 1. Extensive numerical studies and a real data example demonstrate the potential of our proposed test. Supplementary materials for this article are available online.