Minimum Asymptotic Variance Research Articles

Reading the Candlesticks: An OK Estimator for Volatility

Abstract We propose an Optimal candlesticK (OK) estimator for the spot volatility using high-frequency candlestick observations. Under a standard infill asymptotic setting, we show that the OK estimator is asymptotically unbiased and has minimal asymptotic variance within a class of linear estimators. Its estimation error can be coupled by a Brownian functional, which permits valid inference. Our theoretical and numerical results suggest that the proposed candlestick-based estimator is much more accurate than the conventional spot volatility estimator based on high-frequency returns. An empirical illustration documents the intraday volatility dynamics of various assets during the Fed chairman’s recent congressional testimony.

Covariate adjustment in experiments with matched pairs

This paper studies inference for the average treatment effect (ATE) in experiments in which treatment status is determined according to “matched pairs” and it is additionally desired to adjust for observed, baseline covariates to gain further precision. By a “matched pairs” design, we mean that units are sampled i.i.d. from the population of interest, paired according to observed, baseline covariates, and finally, within each pair, one unit is selected at random for treatment. Importantly, we presume that not all observed, baseline covariates are used in determining treatment assignment. We study a broad class of estimators based on a “doubly robust” moment condition that permits us to study estimators with both finite-dimensional and high-dimensional forms of covariate adjustment. We find that estimators with finite-dimensional, linear adjustments need not lead to improvements in precision relative to the unadjusted difference-in-means estimator. This phenomenon persists even if the adjustments interact with treatment; in fact, doing so leads to no changes in precision. However, gains in precision can be ensured by including fixed effects for each of the pairs. Indeed, we show that this adjustment leads to the minimum asymptotic variance of the corresponding ATE estimator among all finite-dimensional, linear adjustments. We additionally study an estimator with a regularized adjustment, which can accommodate high-dimensional covariates. We show that this estimator leads to improvements in precision relative to the unadjusted difference-in-means estimator and also provides conditions under which it leads to the “optimal” nonparametric, covariate adjustment. A simulation study confirms the practical relevance of our theoretical analysis, and the methods are employed to reanalyze data from an experiment using a “matched pairs” design to study the effect of macroinsurance on microenterprise.

Regression-adjusted estimation of quantile treatment effects under covariate-adaptive randomizations

Datasets from field experiments with covariate-adaptive randomizations (CARs) usually contain extra covariates in addition to the strata indicators. We propose to incorporate these additional covariates via auxiliary regressions in the estimation and inference of unconditional quantile treatment effects (QTEs) under CARs. We establish the consistency and limit distribution of the regression-adjusted QTE estimator and prove that the use of multiplier bootstrap inference is non-conservative under CARs. The auxiliary regression may be estimated parametrically, nonparametrically, or via regularization when the data are high-dimensional. Even when the auxiliary regression is misspecified, the proposed bootstrap inferential procedure still achieves the nominal rejection probability in the limit under the null. When the auxiliary regression is correctly specified, the regression-adjusted estimator achieves the minimum asymptotic variance. We also discuss forms of adjustments that can improve the efficiency of the QTE estimators. The finite sample performance of the new estimation and inferential methods is studied in simulations, and an empirical application to a well-known dataset concerned with expanding access to basic bank accounts on savings is reported.

Central limit theorems for discretized occupation time functionals

The approximation of integral type functionals is studied for discrete observations of a continuous Itô semimartingale. Based on novel approximations in the Fourier domain, central limit theorems are proved for L2-Sobolev functions with fractional smoothness. An explicit L2(P)-lower bound shows that already lower order quadrature rules, such as the trapezoidal rule and the classical Riemann estimator, are rate optimal, but only the trapezoidal rule is efficient, achieving the minimal asymptotic variance.

Angular Superresolution of Real Aperture Radar With High-Dimensional Data: Normalized Projection Array Model and Adaptive Reconstruction

Angular resolution of real aperture radar (RAR) can be improved using deconvolution methods to achieve enhanced target information based on the convolution relationship between target scatterings and an antenna pattern. However, depending on the wide scanning scope and dense sampling angular interval, the computational complexity of the deconvolution methods will drastically increase as the dimension of azimuthal data increases. In this paper, to efficiently improve the angular resolution of RAR, a generalized adaptive asymptotic minimum variance (GAAMV) estimator that relies on a normalized projection array (NPA) model is proposed. On the one hand, the traditional convolution model of RAR is transformed into an NPA model to compress the data dimension. The proposed NPA model can normalize the signal model to make it independent of the sampling parameters. On the other hand, based on the NPA model, a GAAMV estimator is proposed to efficiently reconstruct the targets by adaptively updating each grid. Moreover, the penalty parameter is extended as a generalized case to improve its adaptability to different scenes. Based on the proposed model and method, the computational complexity can be decreased, especially for high-dimensional azimuthal data. Simulations and experimental data verify the proposed model and method.

GMM estimation of a spatial autoregressive model with autoregressive disturbances and endogenous regressors

This paper considers the generalized method of moments (GMM) estimation of a spatial autoregressive (SAR) model with SAR disturbances, where we allow for endogenous regressors in addition to a spatial lag of the dependent variable. We do not assume any reduced form of the endogenous regressors, thus we allow for spatial dependence and heterogeneity in endogenous regressors, and allow for nonlinear relations between endogenous regressors and their instruments. Innovations in the model can be homoscedastic or heteroskedastic with unknown forms. We prove that GMM estimators with linear and quadratic moments are consistent and asymptotically normal. In the homoscedastic case, we derive the best linear and quadratic moments that can generate an optimal GMM estimator with the minimum asymptotic variance.

Designing efficient randomized trials: power and sample size calculation when using semiparametric efficient estimators.

Trials enroll a large number of subjects in order to attain power, making them expensive and time-consuming. Sample size calculations are often performed with the assumption of an unadjusted analysis, even if the trial analysis plan specifies a more efficient estimator (e.g. ANCOVA). This leads to conservative estimates of required sample sizes and an opportunity for savings. Here we show that a relatively simple formula can be used to estimate the power of any two-arm, single-timepoint trial analyzed with a semiparametric efficient estimator, regardless of the domain of the outcome or kind of treatment effect (e.g. odds ratio, mean difference). Since an efficient estimator attains the minimum possible asymptotic variance, this allows for the design of trials that are as small as possible while still attaining design power and control of type I error. The required sample size calculation is parsimonious and requires the analyst to provide only a small number of population parameters. We verify in simulation that the large-sample properties of trials designed this way attain their nominal values. Lastly, we demonstrate how to use this formula in the "design" (and subsequent reanalysis) of a real randomized trial and show that fewer subjects are required to attain the same design power when a semiparametric efficient estimator is accounted for at the design stage.

Regression-Adjusted Estimation of Quantile Treatment Effects under Covariate-Adaptive Randomizations

Datasets from field experiments with covariate-adaptive randomizations (CARs) usually contain extra covariates in addition to the strata indicators. We propose to incorporate these additional covariates via auxiliary regressions in the estimation and inference of unconditional quantile treatment effects (QTEs) under CARs. We establish the consistency and limit distribution of the regression-adjusted QTE estimator and prove that the use of multiplier bootstrap inference is non-conservative under CARs. The auxiliary regression may be estimated parametrically, nonparametrically, or via regularization when the data are high-dimensional. Even when the auxiliary regression is misspecified, the proposed bootstrap inferential procedure still achieves the nominal rejection probability in the limit under the null. When the auxiliary regression is correctly specified, the regression-adjusted estimator achieves the minimum asymptotic variance. We also discuss forms of adjustments that can improve the efficiency of the QTE estimators. The finite sample performance of the new estimation and inferential methods is studied in simulations and an empirical application to a well-known dataset concerned with expanding access to basic bank accounts on savings is reported.

Efficient sparse parameter estimation based methods for two‐dimensional DOA estimation of coherent signals

This study addresses the problem of direction-of-arrival (DOA) estimation of coherent signals via sparse parameter estimation. Since many sparse methods provide good performances regardless of signal correlations and array geometry, they can be considered as candidates for DOA estimation of coherent signals impinging on a sensor array with arbitrary geometry. However, their straightforward applications require high computational loads especially for two-dimensional (2D) DOA estimation. Two efficient methods based on sparse parameter estimation are herein presented; one is a combined approach of sparse estimation and the RELAX algorithm extended for 2D DOA estimation and the other relies on the adaptive 2D grid refinement and power update control. Numerical simulations are performed to demonstrate the efficiency of the proposed methods using a uniform circular array for both 1D and 2D DOA estimation cases. It is shown that sparse asymptotic minimum variance (SAMV)-RELAX, a combined approach of SAMV and RELAX, outperforms SAMV and multi-stage SAMV in 2D scenarios without suffering from plateau effects for off-grid signals and that its computational load is significantly lower than those of SAMV and multi-stage SAMV. In addition, SAMV-RELAX does not require the difficult selection of grid parameters for fine DOA estimation unlike the multi-stage approach.

Off-grid DOA estimation of correlated sources for nonuniform linear array through hierarchical sparse recovery in a Bayesian framework and asymptotic minimum variance criterion

This paper provides a method to solve off-grid direction-of-arrival (DOA) estimation for nonuniform linear array (NLA) correlated source condition through hierarchical sparse recovery and asymptotic minimum variance (AMV) criterion. In this method, space is firstly divided into a discretized grid. Most rows and columns of the signal covariance matrix modeled on this grid are zero vectors because the number of sources is considerably smaller than that of grid points. Hence, the vectorized signal covariance matrix is regarded as a block-sparse vector, and active blocks are sparse vectors. Based on this, a hierarchical sparse prior is then assigned on the vectorized signal covariance matrix to encourage the sparsity between and within blocks. Finally, the variational Bayesian inference is applied to estimate the vectorized signal covariance matrix. Furthermore, first-order Taylor series expansion is applied to approximate the steering vector as a function of the grid error between the true DOA and the closest grid point. Grid error is estimated under the AMV criterion and applied to modify the grid iteratively, thus alleviating the basis mismatch. Simulation results show that the proposed method achieves high estimation accuracy for the NLA correlated source condition.

Multi-objective optimization design of accelerated degradation test based on Wiener process

A multi-objective optimization method for the accelerated degradation test based on Wiener process is proposed in this article in order to solve the problem that a single objective optimization cannot solve the difficulty or even conflicting test configurations caused by different optimization objective functions. An accelerated degradation model is established based on Wiener process, and the unknown parameters are solved by a two-step maximum likelihood estimation method. Considering both the accuracies of life estimation and the model parameter estimation, a multi-objective optimization model is established with the optimization goals of the minimum asymptotic variance of P-quantile of lifetime and the maximum determinant of Fisher information matrix. The Pareto solutions are set by the multi-objective genetic algorithm, and the test configurations for multi-objectives are obtained. Under the step stress accelerated degradation test and the constant stress accelerated degradation test, the effectiveness of the proposed method is verified by an optimization example of LEDs accelerated degradation test.

Efficiency of subspace-based estimators for elliptical symmetric distributions

Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameter-dependent subspace have proved very useful in many applications in signal processing. The purpose of this paper is to complement theoretical results already available on the asymptotic (in the number of measurements) performance of subspace-based estimators derived in the Gaussian context to real elliptical symmetric (RES), circular complex elliptical symmetric (C-CES) and non-circular CES (NC-CES) distributed observations in the same framework. First, the asymptotic distribution of M-estimates of the orthogonal projection matrix is derived from those of the M-estimates of the covariance matrix. This allows us to characterize the asymptotically minimum variance (AMV) estimator based on estimates of orthogonal projectors associated with different M-estimates of the covariance matrix. A closed-form expression is then given for the AMV bound on the parameter of interest characterized by the column subspace of the mixing matrix of general linear mixture models. We also specify the conditions under which the AMV bound based on Tyler’s M-estimate attains the stochastic Cramér-Rao bound (CRB) for the complex Student t and complex generalized Gaussian distributions. Finally, we prove that the AMV bound attains the stochastic CRB in the case of maximum likelihood (ML) M-estimate of the covariance matrix for RES, C-CES and NC-CES distributed observations, which is equal to the semiparametric CRB (SCRB) recently introduced.

High‐resolution time delay estimation via sparse parameter estimation methods

This study addresses the high-resolution time delay estimation (TDE) via sparse parameter estimation methods. Two representative algorithms, Sparse Asymptotic Minimum Variance (SAMV) and SParse Iterative Covariance-based Estimation are devised in both the time and frequency domains for application to the TDE of spread-spectrum signals and their performances are analysed in various multipath environments. The authors also proposes the combined approach of SAMV and weighted RELAX, referred to as SAMV-WRELAX, to reduce the computational load. Numerical examples demonstrate that the frequency-domain approaches with a proper type of snapshots not only outperform the corresponding time-domain approaches but also mitigate the problem of the noise correlation encountered in time-domain processing. They also show that the computational load of SAMV-WRELAX with a grid size of 0.1T c decreases up to a few tenths of that of SAMV with a fine grid, e.g. a size of 0.01T c , without any performance degradations.

Extended State Filter Based Disturbance and Uncertainty Mitigation for Nonlinear Uncertain Systems With Application to Fuel Cell Temperature Control

The filter design for nonlinear uncertain systems is quite challenging since efficient estimation is required against stochastic noises, nonlinear uncertain dynamics as well as their concurrent effects. To this end, this article develops a novel filter algorithm by augmenting the disturbance as well as unknown nonlinear dynamics as an extended state and constructing consistent Kalman–Bucy algorithm. The proposed extended state based Kalman–Bucy filter (KBF) is shown to be of bounded estimation error, and the estimation accuracy can be online evaluated. More importantly, the estimation of asymptotic minimum variance is realized in condition that the changing rate of uncertainty approaches to zero. Therefore, the proposed extended state filter enables effective mitigation of disturbance and unknown nonlinear dynamics in real time by feedback control. The proposed algorithm is experimentally verified via a temperature control application in proton exchange membrane fuel cell, in which the thermocouple noise and the electrochemical uncertainty are seriously presented. The temperature variation of the extended state based KBF-based control is greatly reduced, in comparison with the conventional control. The results in this article depict a promising prospect of the proposed method for industrial control applications to handle both noises and nonlinear uncertain dynamics.

Robust direction‐of‐arrival estimation based on sparse asymptotic minimum variance

This study proposes a direction-of-arrival (DOA) estimation algorithm named robust sparse asymptotic minimum variance (RSAMV) to solve the current DOA algorithms' problems, such as the difficulty in weak target estimation, low resolution and the incapacity of separating coherent signal estimation. Through utilising a virtual weak target, the algorithm carries out dynamic diagonal loading to the sampling covariance matrix of SAMV in the iterative process, which effectively reduces weak target loss. Meanwhile, showing the feature of ultra-low side lobe and high sparseness, the spatial spectrum of RSAMV can easily achieve the high-resolution estimation of space target in the circumstances of coherent interference. Simulation results show that, compared with other algorithms, the RSAMV algorithm has higher spatial resolution ability and weak target detection ability. Its spatial spectrum has higher sparseness than other sparse algorithms and its performance is more robust than other SAMV algorithms. The Bering-Time Recording map processed by results that experiment on sea demonstrate the superiority of RSAMV algorithm.

Root sparse asymptotic minimum variance for off-grid direction-of-arrival estimation

Sparsity-based direction-of-arrival (DOA) estimation algorithms have received much attention due to their good performances. However, most of these algorithms suffer from the so-called basis mismatch if the true DOAs deviate from the discrete grid points. This paper provides an off-grid DOA estimation method which iteratively updates the grid until some of the grid points coincide with the DOAs. Based on the first-order Taylor series expansion of the true steering vectors, the deviations of these grid points from the true DOAs, i.e., the so-called grid errors, are linked to the steering vectors defined on the grid by a linear relation approximately. Using this linear relation, the grid errors can be estimated under the asymptotic minimum variance criterion. The grid is modified by adding the grid errors to the grid points, leading to them closer to the true DOAs, and consequently mitigating the basis mismatch. It is shown by simulations that the proposed method achieves high performance both in terms of estimation accuracy and computational efficiency.

An Efficient Nonparametric Estimate for Spatially Correlated Functional Data

Functional data are often generated by modern biomedical technologies where features related to the pathophysiology and pathogenesis of a disease are interrogated repeatedly over time and at multiple spatially interdependent regions. To reduce model complexity and simplify the resulting inference, possible spatial correlation among neighboring regions is often neglected. In this article, we propose a weighted kernel smoothing estimate of the mean function that leverages the spatial and temporal correlation. We also address the companion problem of developing a simultaneous prediction method for individual curves using discrete samples. We establish the asymptotic properties of the proposed estimate, including its unique maximum efficiency achieving minimum asymptotic variance. The proposed method improves estimation and prediction in the presence of sparse observations, and therefore, is advantageous to biomedical applications that utilize markers to identify features intrinsic to a particular disease at multiple interdependent sites within an organ. Our simulation and case studies show that the proposed method outperforms conventional approaches for characterizing the dynamic functional imaging data, with the maximum benefit achieved in the presence of a small number of repeated scans.

Multi-Objective Optimization Design of Constant Stress Accelerated Degradation Test Using Inverse Gaussian Process

A multi-objective optimization method for constant stress accelerated degradation test is proposed in order to solve the problem of different or even conflicted test configuration under different optimization objectives. The inverse Gaussian process is used as a degradation model, and the unknown parameters are solved by maximum likelihood estimation. The two optimization criteria of the maximum determinant of the information matrix and the minimum asymptotic variance of P-quantile are considered. The improved multi-objective particle swarm optimization algorithm is proposed to search test optimal configuration, and the Pareto solution set for dual-objectives is obtained. Finally, the effectiveness of the method is illustrated by a group of examples of electrical connectors. Compared with the single-objective optimization design, the proposed method is more reasonable and convenient for test configuration. The performance index of the test function indicates that the optimal algorithm we proposed has some obvious advantages over the NSGA-II in diversity and convergence of the Pareto solutions and it is significant in guiding engineering practice.

Source localization based on matrix filter and sparse asymptotic minimum variance

Wideband direction of arrival (DOA) estimation plays an important role in passive sonar signal processing. Recently, sparsity-based DOA estimation method has attracted considerable attention because of its high resolution in the condition of few snapshots and low signal-to-noise ratio. However, the localization accuracy is seriously affected by the interferences. Matrix filter (MF) has been widely used in passive sonar systems as a useful tool to passband the targets-of-interest while attenuating the interferences, but the output of the MF seriously affected subsequent DOA estimation when the power of the interferences after filtering is still stronger than the weak targets. In this paper, a method based on MF and sparse asymptotic minimum variance (SAMV) is given to localize the weak targets in a strong interference environment. The given method improves the ability of SAMV on weak targets localization and achieves high localization accuracy even in the condition that the power of the interferences after filtering remains stronger than the weak targets, which is verified by simulation and experimental results.

On extended state based Kalman filter design for a class of nonlinear time-varying uncertain systems

This paper considers the filtering problem for a class of multi-input multi-output systems with nonlinear time-varying uncertain dynamics, random process andmeasurement noise. An extended state based Kalman filter, with the idea of timely estimating the unknown dynamics, is proposed for better robustness and higher estimation precision. The stability of the proposed filter is rigorously proved for nonlinear time-varying uncertain system with weaker stability condition than the extended Kalman filter, i.e., the initial estimation error, the uncertain dynamics and the noises are only required to be bounded rather than small enough. Moreover, quantitative precision of the proposed filter is theoretically evaluated. The proposed algorithm is proved to be the asymptotic unbiased minimum variance filter for constant uncertainty. The simulation results of some benchmark examples demonstrate the feasibility and effectiveness of the method.