Finite Sample Bias of Robust Estimators in Segmentation of Closely Spaced Structures: A Comparative Study

This paper presents the design and implementation of a new comparative analytical framework for studying the usability of modern high breakdown robust estimators. The emphasis is on finding the intrinsic limits, in terms of size and relative spatial accuracy, of such techniques in solving the emerging challenges of the segmentation of fine structures. A minimum threshold for the distance between separable structures is shown to depend mainly on the scale estimation error. A scale invariant performance measure is introduced to quantify the finite sample bias of the scale estimate of a robust estimator and the measure is evaluated for some state-of-the-art high breakdown robust estimators using datasets containing at least two close but distinct structures with varying distances and inlier ratios. The results show that the new generation of density-based robust estimators (such as pbM-estimator and TSSE) have a poorer performance in problems with datasets containing only a small number of samples in each structure compared with ones based on direct processing of the residuals (such as MSSE). An important message of this paper is that an estimator that performs best in some circumstances, may not be competitive in others: particularly performance on data structures that are relatively large and/or well-separated vs closely spaced fine structures.     

On a fast, robust estimator of the mode: Comparisons to other robust estimators with applications

Advances in computing power enable more widespread use of the mode, which is a natural measure of central tendency since it is not influenced by the tails in the distribution. The properties of the half-sample mode, which is a simple and fast estimator of the mode of a continuous distribution, are studied. The half-sample mode is less sensitive to outliers than most other estimators of location, including many other low-bias estimators of the mode. Its breakdown point is one half, equal to that of the median. However, because of its finite rejection point, the half-sample mode is much less sensitive to outliers that are all either greater or less than the other values of the sample. This is confirmed by applying the mode estimator and the median to samples drawn from normal, lognormal, and Pareto distributions contaminated by outliers. It is also shown that the half-sample mode, in combination with a robust scale estimator, is a highly robust starting point for iterative robust location estimators such as Huber's M-estimator. The half-sample mode can easily be generalized to modal intervals containing more or less than half of the sample. An application of such an estimator to the finding of collision points in high-energy proton–proton interactions is presented.     

Robust variance estimation for random effects meta-analysis

In random effects meta-analysis, an overall effect is estimated using a weighted mean, with weights based on estimated marginal variances. The variance of the overall effect is often estimated using the inverse of the sum of the estimated weights, and inference about the overall effect is typically conducted using this ‘usual’ variance estimator, which is not robust to errors in the estimated marginal variances. In this paper, robust estimation for the asymptotic variance of a weighted overall effect estimate is explored by considering a robust variance estimator in comparison with the usual variance estimator and another less frequently used estimator, a weighted version of the sample variance. Three illustrative examples are presented to demonstrate and compare the three estimation methods. Furthermore, a simulation study is conducted to assess the robustness of the three variance estimators using estimated weights. The simulation results show that the robust variance estimator and the weighted sample variance estimator both estimate the variance of an overall effect more accurately than the usual variance estimator when the weights are imprecise due to the use of estimated marginal variances, as is typically the case in practice.Therefore, we argue that inference about an overall effect should be based on the robust variance estimator or the weighted sample variance, which provide protection against the practice of using estimated weights in meta-analytical inference.     

Robust, reduced-order, nonstrictly proper state estimation via the optimal projection equations with Petersen—Hollot bounds

A state-estimation design problem involving parametric plant uncertainties is considered. An error bound suggested by recent work of Petersen and Hollot is utilized for guaranteeing robust estimation. Necessary conditions which generalize the optimal projection equations for reduced-order state estimation are used to characterize the estimator which minimizes the error bound. The design equations thus effectively serve as sufficient conditions for synthesizing robust estimators. An additional feature is the presence of a static estimation gain in conjunction with the dynamic (Kalman) estimator, i. e., a nonstrictly proper estimator.     

Worst-case estimation for econometric models with unobservable components

A worst-case estimator for econometric models containing unobservable components, based on minimax principles for optimal selection of parameters, is proposed. Worst-case estimators are robust against the averse effects of unobservables. Computing worst-case estimators involves solving a minimax continuous problem, which is quite a challenging task. Large sample theory is considered, and a Monte Carlo study of finite-sample properties is conducted. A financial application is considered.     

One-step minimum Hellinger distance estimation

It is well known now that the minimum Hellinger distance estimation approach introduced by Beran (Beran, R., 1977. Minimum Hellinger distance estimators for parametric models. Ann. Statist. 5, 445-463) produces estimators that achieve efficiency at the model density and simultaneously have excellent robustness properties. However, computational difficulties and algorithmic convergence problems associated with this method have hampered its application in practice, particularly when the method is applied to models with high-dimensional parameter spaces. A one-step minimum Hellinger distance (MHD) procedure is investigated in this paper to overcome computational drawbacks of the fully iterative MHD method. The idea is to start with an initial estimator, and then iterate the Newton-Raphson equation once related to the Hellinger distance. The resulting estimator can be considered a one-step MHD estimator. We show that the proposed one-step MHD estimator has the same asymptotic behavior as the MHD estimator, as long as the initial estimators are reasonably good. Furthermore, our theoretical and numerical studies also demonstrate that the proposed one-step MHD estimator also retains excellent robustness properties of the MHD estimators. A real data example is analyzed as well.     

On second order efficient robust inference

General strategies for constructing second order efficient robust distances from suitable properties of the residual adjustment functions (RAF) are discussed. Based on those properties families of estimators are constructed using the truncated polynomial, negative exponential and sigmoidal functions as RAFs and their efficiency and robustness properties are investigated. The estimators have full asymptotic efficiency, and are automatically second order efficient. Many of the proposed estimators are competitive or better than the minimum Hellinger distance estimator (MHDE) and minimum negative exponential disparity estimator (MNEDE) under the combined goals of asymptotic efficiency with strong robustness properties. Hence the proposed families give the user the flexibility to choose from a large class of robust second order efficient estimators based upon specific needs.     

Generalized method of moments estimation for cointegrated vector autoregressive models

In this study, a generalized method of moments (GMM) for the estimation of nonstationary vector autoregressive models with cointegration is considered. Two iterative methods are considered: a simultaneous estimation method and a switching estimation method. The asymptotic properties of the GMM estimators of these methods are found to be the same as those of the Gaussian reduced-rank estimator. Through Monte Carlo simulation, the small-sample properties of the GMM estimators are studied and compared with those of the Gaussian reduced-rank estimator and the maximum likelihood estimator considered by other researchers. In the case of small samples, the GMM estimators are more robust to deviations from normality assumptions, particularly to outliers.     

Robustness of the estimators of transition rates for size-classified matrix models

Matrix models are often used to model the dynamics of age-structured or size-structured populations. The Usher model is an important particular case that relies on the following hypothesis: between time steps t and t+1, individuals either remain in the same class, move up to the following class, or die. There are then two ways of handling data that do not meet this condition: either remove them prior to data analysis or rectify them. These two ways correspond to two estimators of transition parameters. The former, which corresponds to the classical estimator, is obtained from the latter by a data trimming. The two estimators of transition parameters are compared on the basis of their robustness in order to obtain a criterion of choice between the two estimators. The influence curve of both estimators is first computed, then their gross sensitivity and their asymptotic variance. The untrimmed estimator is more robust than the classical one. Its asymptotic variance can be lower or greater than that of the classical estimator depending on the boundaries used for data trimming. The results are applied to a tropical rain forest in French Guiana, with a discussion on the role of the class width.     

Nonparametric conditional hazard rate estimation: A local linear approach

A new nonparametric estimator for the conditional hazard rate is proposed, which is defined as the ratio of local linear estimators for the conditional density and survivor function. The resulting hazard rate estimator is shown to be pointwise consistent and asymptotically normally distributed under appropriate conditions. Furthermore, plug-in bandwidths based on normal and uniform reference distributions and minimizing the asymptotic mean squared error are derived. In terms of the mean squared error the new estimator is highly competitive in comparison to existing estimators for the conditional hazard rate. Moreover, its smoothing parameters are relatively robust to misspecification of the reference distributions, which facilitates bandwidth selection. Additionally, the new hazard rate estimator is conveniently calculated using standard software for local linear regression. The use of the local linear hazard rate is illustrated in an application to kidney transplant data.     

Maximum trimmed likelihood estimator for multivariate mixed continuous and categorical data

In this article, we apply the maximum trimmed likelihood (MTL) approach [Hadi, A.S., Luceño, A., 1997. Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms. Comput. Statist. Data Anal. 25, 251-272] to obtain the robust estimators of multivariate location and shape, especially for data mixed with continuous and categorical variables. The forward search algorithm [Atkinson, A.C., 1994. Fast very robust methods for the detection of multiple outliers. J. Amer. Statist. Assoc. 89, 1329-1339] is adapted to compute the proposed MTL estimates. A simulation study shows that the proposed estimator outperforms the classical maximum likelihood estimator when outliers exist in data. Real data sets are also used to illustrate the method and results of the detection of the outliers.     

Estimating quantiles under sampling on two occasions with arbitrary sample designs

A practical problem related to the estimation of quantiles in double sampling with arbitrary sampling designs in each of the two phases is investigated. In practice, this scheme is commonly used for official surveys, in which quantile estimation is often required when the investigation deals with variables such as income or expenditure. A class of estimators for quantiles is proposed and some important properties, such as asymptotic unbiasedness and asymptotic variance, are established. The optimal estimator, in the sense of minimizing the asymptotic variance, is also presented. The proposed class contains several known types of estimators, such as ratio and regression estimators, which are of practical application and are therefore derived. Assuming several populations, the proposed estimators are compared with the direct estimator via an empirical study. Results show that a gain in efficiency can be obtained.     

Estimating quantiles under sampling on two occasions with arbitrary sample designs

A practical problem related to the estimation of quantiles in double sampling with arbitrary sampling designs in each of the two phases is investigated. In practice, this scheme is commonly used for official surveys, in which quantile estimation is often required when the investigation deals with variables such as income or expenditure. A class of estimators for quantiles is proposed and some important properties, such as asymptotic unbiasedness and asymptotic variance, are established. The optimal estimator, in the sense of minimizing the asymptotic variance, is also presented. The proposed class contains several known types of estimators, such as ratio and regression estimators, which are of practical application and are therefore derived. Assuming several populations, the proposed estimators are compared with the direct estimator via an empirical study. Results show that a gain in efficiency can be obtained.     

Robust adaptive-scale parametric model estimation for computer vision

Robust model fitting essentially requires the application of two estimators. The first is an estimator for the values of the model parameters. The second is an estimator for the scale of the noise in the (inlier) data. Indeed, we propose two novel robust techniques: the two-step scale estimator (TSSE) and the adaptive scale sample consensus (ASSC) estimator. TSSE applies nonparametric density estimation and density gradient estimation techniques, to robustly estimate the scale of the inliers. The ASSC estimator combines random sample consensus (RANSAC) and TSSE, using a modified objective function that depends upon both the number of inliers and the corresponding scale. ASSC is very robust to discontinuous signals and data with multiple structures, being able to tolerate more than 80 percent outliers. The main advantage of ASSC over RANSAC is that prior knowledge about the scale of inliers is not needed. ASSC can simultaneously estimate the parameters of a model and the scale of the inliers belonging to that model. Experiments on synthetic data show that ASSC has better robustness to heavily corrupted data than least median squares (LMedS), residual consensus (RESC), and adaptive least Kth order squares (ALKS). We also apply ASSC to two fundamental computer vision tasks: range image segmentation and robust fundamental matrix estimation. Experiments show very promising results.     

Notes on interval estimation of risk difference in stratified noncompliance randomized trials: A Monte Carlo evaluation

When comparing an experimental treatment with a standard treatment in a randomized clinical trial (RCT), we often use the risk difference (RD) to measure the efficacy of an experimental treatment. In this paper, we have developed four asymptotic interval estimators for the RD in a stratified RCT with noncompliance. These include an asymptotic interval estimator based on the weighted-least-squares (WLS) estimator of the RD, an asymptotic interval estimator using tanh⁻¹(x) transformation with the WLS optimal weight, an asymptotic interval estimator derived from Fieller’s Theorem, and an asymptotic interval estimator using a randomization-based approach. Based on Monte Carlo simulations, we have compared these four asymptotic interval estimators with the asymptotic interval estimator recently proposed elsewhere. We have found that when the probability of compliance is high, the interval estimator using a randomization-based approach is probably more accurate than the others, especially when the stratum size is not large. When the probability of compliance is moderate, the interval estimator using tanh⁻¹(x) transformation is likely to be the best among all interval estimators considered here. We note that the interval estimator proposed elsewhere can be of use when the underlying RD is small, but lose accuracy when the RD is large. We also note that when the number of patients per assigned treatment is large, the four asymptotic interval estimators developed here are essentially equivalent; they are all appropriate for use. Finally, to illustrate the use of these interval estimators, we consider the data taken from a large field trial studying the effect of a multifactor intervention program on reducing the mortality of coronary heart disease in middle-aged men.     

Corner-based estimation of tire forces and vehicle velocities robust to road conditions

Recent developments in vehicle stability control and active safety systems have led to an interest in reliable vehicle state estimation on various road conditions. This paper presents a novel method for tire force and velocity estimation at each corner to monitor tire capacities individually. This is entailed for more demanding advanced vehicle stability systems and especially in full autonomous driving in harsh maneuvers. By integrating the lumped LuGre tire model and the vehicle kinematics, it is shown that the proposed corner-based estimator does not require knowledge of the road friction and is robust to model uncertainties. The stability of the time-varying longitudinal and lateral velocity estimators is explored. The proposed method is experimentally validated in several maneuvers on different road surface frictions. The experimental results confirm the accuracy and robustness of the state estimators.     

A method of robust multi-rate state estimation

The inadequacy of the standard notions of detectability and observability to ascertain robust state estimation is shown. The notion of robust state estimation is defined, and for a class of processes the conditions under which the robust state estimation is possible, are given. A method of robust, nonlinear, multi-rate, state estimator design is presented. It can be used to improve robustness in an existing estimator or design a new robust estimator. Estimator tuning guidelines that ensure the asymptotic stability of the estimator error dynamics are given. To ensure that estimation error does not exceed a desired limit, the sampling period of infrequent measurements should be less than an upper bound that depends on factors such as the size of the process dominant time constant, the magnitude of measurement noise, and the level of process–model mismatch. An expression that can be used to calculate the upper bound on the sampling period of infrequent measurements, is presented. The upper bound is the latest time at which the next infrequent measurements should arrive to ensure that estimation error does not exceed a desired limit. The expression also allows one to calculate the highest quality of estimation achievable in a given process. A binary distillation flash tank and a free-radical polymerization reactor are considered to show the application and performance of the estimator.     

Robust estimation of constrained covariance matrices for confirmatory factor analysis

Confirmatory factor analysis (CFA) is a data analysis procedure that is widely used in social and behavioral sciences in general and other applied sciences that deal with large quantities of data (variables). The classical estimator (and inference) procedures are based either on the maximum likelihood (ML) or generalized least squares (GLS) approaches which are known to be nonrobust to departures from the multivariate normal assumption underlying CFA. A natural robust estimator is obtained by first estimating the (mean and) covariance matrix of the manifest variables and then “plug-in” this statistic into the ML or GLS estimating equations. This two-stage method however does not fully take into account the covariance structure implied by the CFA model. An S-estimator for the parameters of the CFA model that is computed directly from the data is proposed instead and the corresponding estimating equations and an iterative procedure are derived. It is also shown that the two estimators have different asymptotic properties. A simulation study compares the finite sample properties of both estimators showing that the proposed direct estimator is more stable (smaller MSE) than the two-stage estimator.     

A robust estimator for the tail index of Pareto-type distributions

In extreme value statistics, the extreme value index is a well-known parameter to measure the tail heaviness of a distribution. Pareto-type distributions, with strictly positive extreme value index (or tail index) are considered. The most prominent extreme value methods are constructed on efficient maximum likelihood estimators based on specific parametric models which are fitted to excesses over large thresholds. Maximum likelihood estimators however are often not very robust, which makes them sensitive to few particular observations. Even in extreme value statistics, where the most extreme data usually receive most attention, this can constitute a serious problem. The problem is illustrated on a real data set from geopedology, in which a few abnormal soil measurements highly influence the estimates of the tail index. In order to overcome this problem, a robust estimator of the tail index is proposed, by combining a refinement of the Pareto approximation for the conditional distribution of relative excesses over a large threshold with an integrated squared error approach on partial density component estimation. It is shown that the influence function of this newly proposed estimator is bounded and through several simulations it is illustrated that it performs reasonably well at contaminated as well as uncontaminated data.     

Semiparametric multivariate density estimation for positive data using copulas

The estimation of density functions for positive multivariate data is discussed. The proposed approach is semiparametric. The estimator combines gamma kernels or local linear kernels, also called boundary kernels, for the estimation of the marginal densities with parametric copulas to model the dependence. This semiparametric approach is robust both to the well-known boundary bias problem and the curse of dimensionality problem. Mean integrated squared error properties, including the rate of convergence, the uniform strong consistency and the asymptotic normality are derived. A simulation study investigates the finite sample performance of the estimator. The proposed estimator performs very well, also for data without boundary bias problems. For bandwidths choice in practice, the univariate least squares cross validation method for the bandwidth of the marginal density estimators is investigated. Applications in the field of finance are provided.