Average Estimator Research Articles

Frequentist model averaging under inequality constraints

Motivated by the inability of the Akaike information criterion to distinguish between statistical models with inequality constraints on parameters, a series of papers have considered alternative model selection criteria for models under inequality constraints. Among the criteria proposed, the order-restricted information criterion and its generalised version have received the most attention. This paper considers the alternative approach of model averaging, where estimates are obtained based not on one single linear model, but a weighted ensemble of linear models each subject to a distinct set of linear inequality constraints. We develop a J-fold cross validation criterion for choosing model weights that reflect the support of the models by the data, and prove that the resultant model average estimator has an optimal asymptotic property. We further prove that for the special case of a finite number of regressors, the sum of the weights obtained by the proposed method for models with the correct inequality constraints converges to one in probability. In our limited simulation study and real data analysis, it is observed that the proposed model average estimator frequently outperforms the estimator that arises from the generalised order-restricted information-based model selection criterion in terms of average MSE and median MSE.

Distributed fixed-time average-tracking for multi-agent systems with mismatched and matched disturbances

In this paper, distributed fixed-time average-tracking problem is discussed for second-order multi-agent systems (MASs) with mismatched and matched disturbances. To address this issue, we first propose a novel distributed fixed-time average estimator for each agent to estimate the average value of multiple time-varying reference signals (TVRSs). Then, to tackle with the case in the presence of external mismatched and matched disturbances, a disturbance observer is constructed to estimate them in a fixed time. Finally, based on the disturbance observer, a fixed-time average-tracking protocol is designed to drive each agent to track the average value of multiple TVRSs within a fixed time. Simulation experiments are conducted to verify the effectiveness of the proposed fixed-time anti-disturbance average-tracking algorithm.

Contour Maps for Simultaneous Increase in Yield Strength and Elongation of Hot Extruded Aluminum Alloy 6082

In this paper, the Conditional Average Estimator artificial neural network (CAE ANN) was used to analyze the influence of chemical composition in conjunction with selected process parameters on the yield strength and elongation of an extruded 6082 aluminum alloy (AA6082) profile. Analysis focused on the optimization of mechanical properties as a function of casting temperature, casting speed, addition rate of alloy wire, ram speed, extrusion ratio, and number of extrusion strands on one side, and different contents of chemical elements, i.e., Si, Mn, Mg, and Fe, on the other side. The obtained results revealed very complex non-linear relationships between all of these parameters. Using the proposed approach, it was possible to identify the combinations of chemical composition and process parameters as well as their values for a simultaneous increase of yield strength and elongation of extruded profiles. These results are a contribution of the presented study in comparison with published research results of similar studies in this field. Application of the proposed approach, either in the research and/or in industrial aluminum production, suggests a further increase in the relevant mechanical properties.

Model selection and model averaging for matrix exponential spatial models

In this paper, we focus on a model specification problem in spatial econometric models when an empiricist needs to choose from a pool of candidates for the spatial weights matrix. We propose a model selection (MS) procedure for the matrix exponential spatial specification (MESS), when the true spatial weights matrix may not be in the set of candidate spatial weights matrices. We show that the selection estimator is asymptotically optimal in the sense that asymptotically it is as efficient as the infeasible estimator that uses the best candidate spatial weights matrix. The proposed selection procedure is also consistent in the sense that when the data generating process involves spatial effects, it chooses the true spatial weights matrix with probability approaching one in large samples. We also propose a model averaging (MA) estimator that compromises across a set of candidate models. We show that it is asymptotically optimal. We further flesh out how to extend the proposed selection and averaging schemes to higher order specifications and to the MESS with heteroscedasticity. Our Monte Carlo simulation results indicate that the MS and MA estimators perform well in finite samples. We also illustrate the usefulness of the proposed MS and MA schemes in a spatially augmented economic growth model.

Reactive power sharing and voltage restoration in islanded AC microgrids

Microgrids (MG) are a new and innovative concept in modern distribution networks. Several challenges are associated with the operation and control of MG networks. Active and reactive power sharing among energy resources interfaced through power electronic conversion stages is a major challenge. Although active power sharing can be achieved under varying scenarios, sharing of reactive power between distributed generation units is difficult to achieve. This paper presents a novel and innovative control scheme to ensure sharing of reactive power between Distributed generation units within an autonomous, islanded AC microgrid. A framework composed of novel multiagent moving average estimators is proposed to make the participating nodes share active and reactive power among themselves. Detailed case studies are carried out in MATLAB and Simulink environment to verify the efficacy of the proposed control scheme. Furthermore, a lab scale MG set up is implemented to verify the operation of the control scheme in an MG environment.

Optimal model averaging estimator for multinomial logit models

In this paper, we study optimal model averaging estimators of regression coefficients in a multinomial logit model, which is commonly used in many scientific fields. A Kullback–Leibler (KL) loss-based weight choice criterion is developed to determine averaging weights. Under some regularity conditions, we prove that the resulting model averaging estimators are asymptotically optimal. When the true model is one of the candidate models, the averaged estimators are consistent. Simulation studies suggest the superiority of the proposed method over commonly used model selection criterions, model averaging methods, as well as some other related methods in terms of the KL loss and mean squared forecast error. Finally, the website phishing data is used to illustrate the proposed method.

Local Sound Speed Estimation for Pulse-Echo Ultrasound in Layered Media.

Our previous methodology in local sound speed estimation utilized time delays measured by the cross correlation of delayed full-synthetic aperture channel data to estimate the average speed of sound. However, focal distortions in this methodology lead to biased estimates of the average speed of sound, which, in turn, leads to biased estimates of the local speed of sound. Here, we demonstrate the bias in the previous methodology and introduce a coherence-based average sound speed estimator that eliminates this bias and is computationally much cheaper in practice. Because this coherence-based approach estimates the average sound speed in the medium over an equally spaced grid in depth rather than time, we derive a refined model that relates the local and average speeds of sound as a function of depth in layered media. A fast, closed-form inversion of this model yields highly accurate local sound speed estimates. The root-mean-square (rms) error of local sound speed reconstruction in simulations of two-layer media is 4.6 and 2.5 m/s at 4 and 8 MHz, respectively. This work examines the impact of frequency, f -number, aberration, and reverberation on sound speed estimation. Phantom and in vivo experiments in rats further validate the coherence-based sound speed estimator.

Model Averaging Estimation Method by Kullback–Leibler Divergence for Multiplicative Error Model

In this paper, we propose the model averaging estimation method for multiplicative error model and construct the corresponding weight choosing criterion based on the Kullback–Leibler divergence with a hyperparameter to avoid the problem of overfitting. The resulting model average estimator is proved to be asymptotically optimal. It is shown that the Kullback–Leibler model averaging (KLMA) estimator asymptotically minimizes the in‐sample Kullback–Leibler divergence and improves the forecast accuracy of out‐of‐sample even under different loss functions. In simulations, we show that the KLMA estimator compares favorably with smooth‐AIC estimator (SAIC), smooth‐BIC estimator (SBIC), and Mallows model averaging estimator (MMA), especially when some nonlinear noise is added to the data generation process. The empirical applications in the daily range of S&P500 and price duration of IBM show that the out‐of‐sample forecasting capacity of the KLMA estimator is better than that of other methods.

Semiparametric Single-Index Estimation for Average Treatment Effects

We propose a semiparametric method to estimate the average treatment effect under the assumption of unconfoundedness given observational data. Our estimation method alleviates misspecification issues of the propensity score function by estimating the single-index link function involved through Hermite polynomials. Our approach is computationally tractable and allows for moderately large dimension covariates. We provide the large sample properties of the estimator and show its validity. Also, the average treatment effect estimator achieves the parametric rate and asymptotic normality. Our extensive Monte Carlo study shows that the proposed estimator is valid in finite samples. We also provide an empirical analysis on the effect of maternal smoking on babies' birth weight and the effect of job training program on future earnings.

Average Treatment Effect Estimation in Observational Studies with Functional Covariates

Functional data analysis is an important area in modern statistics and has been successfully applied in many fields. Although many scientific studies aim to find causations, a predominant majority of functional data analysis approaches can only reveal correlations. In this paper, average treatment effect estimation is studied for observational data with functional covariates. This paper generalizes various state-of-art propensity score estimation methods for multivariate data to functional data. The resulting average treatment effect estimators via propensity score weighting are numerically evaluated by a simulation study and applied to a real-world dataset to study the causal effect of duloxitine on the pain relief of chronic knee osteoarthritis patients.

Robust and efficient estimation for nonlinear model based on composite quantile regression with missing covariates

<abstract><p>In this article, two types of weighted quantile estimators were proposed for nonlinear models with missing covariates. The asymptotic normality of the proposed weighted quantile average estimators was established. We further calculated the optimal weights and derived the asymptotic distributions of the correspondingly resulted optimal weighted quantile estimators. Numerical simulations and a real data analysis were conducted to examine the finite sample performance of the proposed estimators compared with other competitors.</p></abstract>

Bayesian learning with Wasserstein barycenters

We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasserstein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection framework, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provide explicit conditions granting the existence and statistical consistency of the BWB, and discuss some of its general and specific properties, providing insights into its advantages compared to usual choices, such as the model average estimator. Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper, and provide a numerical example for experimental validation of the proposed method.

Efficient Estimation for the Derivative of Nonparametric Function by Optimally Combining Quantile Information

In this article, we focus on the efficient estimators of the derivative of the nonparametric function in the nonparametric quantile regression model. We develop two ways of combining quantile regression information to derive the estimators. One is the weighted composite quantile regression estimator based on the quantile weighted loss function; the other is the weighted quantile average estimator based on the weighted average of quantile regression estimators at a single quantile. Furthermore, by minimizing the asymptotic variance, the optimal weight vector is computed, and consequently, the optimal estimator is obtained. Furthermore, we conduct some simulations to evaluate the performance of our proposed estimators under different symmetric error distributions. Simulation studies further illustrate that both estimators work better than the local linear least square estimator for all the symmetric errors considered except the normal error, and the weighted quantile average estimator performs better than the weighted composite quantile regression estimator in most situations.

Model averaging for interval-valued data

In recent years, model averaging, by which estimates are obtained based on not one single model but a weighted ensemble of models, has received growing attention as an alternative to model selection. To-date, methods for model averaging have been developed almost exclusively for point-valued data, despite the fact that interval-valued data are commonplace in many applications and the substantial body of literature on estimation and inference methods for interval-valued data. This paper focuses on the special case of interval time series data, and assumes that the mid-point and log-range of the interval values are modelled by a two-equation vector autoregressive with exogenous covariates (VARX) model. We develop a methodology for combining models of varying lag orders based on a weight choice criterion that minimises an unbiased estimator of the squared error risk of the model average estimator. We prove that this method yields predictors of mid-points and ranges with an optimal asymptotic property. In addition, we develop a method for correcting the range forecasts, taking into account the forecast error variance. An extensive simulation experiment examines the performance of the proposed model averaging method in finite samples. We apply the method to an interval-valued data series on crude oil future prices.

Assessing the performance of sustainable entrepreneurship and environmental corporate social responsibility: revisited environmental nexus from business firms.

Society is currently contemplating sustainable growth strategies, which have become somewhat apprehensive by associating entrepreneurship, innovation, and sustainable development. In this perspective, this article's objective is to connect sustainable development to environmentally sensitive entrepreneurship via scientific proof of developing nations. Therefore, this research objective is to confirm the hypothesis of corporate social responsibility (CSR) to confirm that Pakistan achieves its objectives for sustainable development. The combined average estimator and pooled mean group (PMG) model of the self-release lag model determines a lengthy-term combination of factors and environmental analysis in the Kuznets. We can see the U-shaped ecological arcs in Pakistan. Further results show that the pooled mean group autoregressive distributed lag (PMG-ARDL) estimator has a long-term relationship. This indicates that a 1% rise in per-capita income at some stage of industrial growth will reduce environmental emissions by 2.88%, 4.54%, and 2.48%. The results show that (1) CSR has a substantial and robust link to the two factors of organizational success (employee engagement and credibility); (ii) respectively socio-cultural and ecological CSR make a positive contribution to the success of Pakistani companies; (iii) the ecological dimensions of CSR being the essential relevance to Pakistani companies' credibility and engagement of employees. This research attempts to include additional analytical information on the contribution of CSR to profitable growth. This also has academic and empirical ramifications, showing how domestic and international companies in developing countries can do CSR. This research is expected to give guidance to policymakers.

A note on point estimation and interval estimation of the relative treatment effect under a simple crossover design.

To increase power or reduce the number of patients needed for a parallel groups design, the crossover design has been often used to study treatments for noncurable chronic diseases. However, in the presence of carry-over effect caused by treatments, the commonly-used estimator which ignores the carry-over effect leads to a biased estimator for estimating the treatment effect difference. A two-stage test approach aimed to address carry-over effect proposed was found to be potentially misleading. In this paper, we propose a weighted average of the commonly-used estimator and an unbiased estimator that uses only the first period of the data. We derive an optimal weight that minimizes the mean squared error (MSE) and its modified estimator. We apply Monte Carlo simulation to evaluate the performance of the proposed estimators in a variety of situations. In the simulations, we examine the estimated MSE (EMSE), percentile interval length, and coverage probability calculated from the percentile intervals among considered estimators. Simulation results show that our proposed weighted average estimator and its modified estimator lead to smaller EMSEs on average comparing to the two commonly used estimators. The coverage probabilities using our proposed estimators are reasonably close to the nominal confidence level and the interval lengths are shorter comparing to the use of the unbiased estimator that uses only the first period of the data. We apply an example that was to evaluate the efficacy of two type of bronchodilators for asthma treatment to demonstrate the use of the proposed estimators.

Optimal model averaging for multivariate regression models

In this paper, frequentist model averaging is considered in the context of a multivariate multiple regression model. We propose a weight choice criterion based on a plug-in counterpart of the quadratic risk of the model average estimator that involves an approximation of the distribution of a ratio of quadratic forms by an F distribution. We establish an asymptotic theory for the resultant model average estimator for both the general and restricted weight sets, and derive the convergence rate of the model weights to the quadratic risk-based optimal weights. The merits of our approach are illustrated by a simulation study and an application based on from the Sixth National Population Census of China.

Asymptotic Analysis for a Stochastic Second-Order Cone Programming and Applications

Stochastic second-order cone programming (SSOCP) is an extension of deterministic second-order cone programming, which demonstrates underlying uncertainties in practical problems arising in economics engineering and operations management. In this paper, asymptotic analysis of sample average approximation estimator for SSOCP is established. Conditions ensuring the asymptotic normality of sample average approximation estimators for SSOCP are obtained and the corresponding covariance matrix is described in a closed form. Based on the analysis, the method to estimate the confidence region of a stationary point of SSOCP is provided and three examples are illustrated to show the applications of the method.

Modified Regularized Wavelength Average Velocity Estimator for normal excitation setup.

Crawling Waves Sonoelastography (CWS) is an ultrasound elastography approach for the Shear Waves Speed (SWS) estimation. Several studies show promising results for tissue characterization. The algorithms used to calculate the SWS have been commonly implemented considering an opposing vibration sources to the side of the tissue of interest. However, implementing this mechanical setup has important limitations considering the geometry of the body. For that reason, a propagation from the top to the surface can be useful. Previous estimators such as Phase Derivative have been modified and tested in phantom studies, however, the presences of artifacts limited the performed of the SWS map. In this study, the Regularized Wavelength Average Velocity Estimator (R-WAVE) technique is modified and evaluated (RWm) to be used for normal propagation. The results of heterogeneous simulations and phantoms experiments showed consistent results with the literature (ie: Simulations Max Bias PDm 11.64 % • RWm 10.21 %, Max CNR PDm 37.82 dB • RWm 44.42 dB, Phantom Experiments Max Bias PDm 15.42 % • RWm 13.99 %, Max CNR PDm 24.14 dB • RWm 26.40 dB). The result of this study shows the potential of RWm to characterize the stiffness of the tissue as well as to differentiate tumors on in vivo applications.Clinical relevance This study presents a modification of the regularized shear wave speed estimator based on crawling waves sonoelastography approach for medical tissue analysis. This technique can be used to discriminate benignant from malignant tumors.

Shear Wave Speed estimator using Continuous Wavelet Transform for Crawling Wave Sonoelastography.

Crawling Wave Sonoelastography (CWS) is an elastography ultrasound-based imaging approach that provides tissue stiffness information through the calculation of Shear Wave Speed (SWS). Many SWS estimators have been developed; however, they report important limitations such as the presence of artifacts, border effects or high computational cost. In addition, these techniques require a moving interference pattern which could be challenging for in vivo applications. In this study, a new estimator based on the Continuous Wavelet Transform (CWT) is proposed. This allows the generation of a SWS image for every sonoelasticity video frame. Testing was made with data acquired from experiments conducted on a gelatin phantom with a circular inclusion. It was excited with two vibration sources placed at both sides with frequencies ranging from 200 Hz to 360 Hz in steps of 20 Hz. Results show small variation of the SWS image across time. Additionally, images were compared with the Phase Derivative method (PD) and the Regularized Wavelength Average Velocity Estimator (R-WAVE). Similar SWS values were obtained for the three estimators within a certain region of interest in the inclusion (At 360 Hz, CWT: 5.01±0.2m/s, PD: 5.11±0.28m/s, R-WAVE: 4.51±0.62m/s) and in the background (At 360 Hz, CWT: 3.67±0.15m/s, PD: 3.69±0.23m/s, R-WAVE: 3.58±0.24m/s). CWT also presented the lowest coefficient of variation and the highest contrast-to-noise ratio for most frequencies, which allows better discrimination between regions.Clinical relevance-This study presents a new Shear Wave Speed estimator for Crawling Wave Sonoelastography, which can be useful to characterize soft tissue and detect lesions.