Robust M-estimation of multivariate GARCH models

The Gaussian quasi-maximum likelihood estimator of Multivariate GARCH models is shown to be very sensitive to outliers in the data. A class of robust M-estimators for MGARCH models is developed. To increase the robustness of the estimators, the use of volatility models with the property of bounded innovation propagation is recommended. The Monte Carlo study and an empirical application to stock returns document the good robustness properties of the M-estimator with a fat-tailed Student t loss function.     

Smoothed additive estimators for non-error rates in multiple discriminant analysis

A generalization of smoothed additive estimators for non-error rates to the case of more than two groups is discussed. Several properties the smoothing should have are shown to be satisfied. The problem of choosing a smoothing parameter is considered and a parameter choice depending on the sample is proposed. In simulation experiments with normal, uniform and discrete distributions the smoothed additive estimators with fixed and variable smoothing parameter are compared to the leaving-one out method and the resubstitution method with respect to bias and variance.     

Robust linear and support vector regression

The robust Huber M-estimator, a differentiable cost function that is quadratic for small errors and linear otherwise, is modeled exactly, in the original primal space of the problem, by an easily solvable simple convex quadratic program for both linear and nonlinear support vector estimators. Previous models were significantly more complex or formulated in the dual space and most involved specialized numerical algorithms for solving the robust Huber linear estimator. Numerical test comparisons with these algorithms indicate the computational effectiveness of the new quadratic programming model for both linear and nonlinear support vector problems. Results are shown on problems with as many as 20000 data points, with considerably faster running times on larger problems     

A combined scheme of edge-based and node-based smoothed finite element methods for Reissner–Mindlin flat shells

In this paper, a combined scheme of edge-based smoothed finite element method (ES-FEM) and node-based smoothed finite element method (NS-FEM) for triangular Reissner–Mindlin flat shells is developed to improve the accuracy of numerical results. The present method, named edge/node-based S-FEM (ENS-FEM), uses a gradient smoothing technique over smoothing domains based on a combination of ES-FEM and NS-FEM. A discrete shear gap technique is incorporated into ENS-FEM to avoid shear-locking phenomenon in Reissner–Mindlin flat shell elements. For all practical purpose, we propose an average combination (aENS-FEM) of ES-FEM and NS-FEM for shell structural problems. We compare numerical results obtained using aENS-FEM with other existing methods in the literature to show the effectiveness of the present method.     

A generalization of Tyler’s M-estimators to the case of incomplete data

Many different robust estimation approaches for the covariance or shape matrix of multivariate data have been established. Tyler’s M-estimator has been recognized as the ‘most robust’ M-estimator for the shape matrix of elliptically symmetric distributed data. Tyler’s M-estimators for location and shape are generalized by taking account of incomplete data. It is shown that the shape matrix estimator remains distribution-free under the class of generalized elliptical distributions. Its asymptotic distribution is also derived and a fast algorithm, which works well even for high-dimensional data, is presented. A simulation study with clean and contaminated data covers the complete-data as well as the incomplete-data case, where the missing data are assumed to be MCAR, MAR, and NMAR.     

Relations for moments of progressively Type-II censored order statistics from half-logistic distribution with applications to inference

In this paper, we establish several recurrence relations for the single and product moments of progressively Type-II right censored order statistics from a half-logistic distribution. The use of these relations in a systematic recursive manner would enable one to compute all the means, variances and covariances of progressively Type-II right censored order statistics from the half-logistic distribution for all sample sizes n, effective sample sizes m, and all progressive censoring schemes (R₁,…,R_m). The results established here generalize the corresponding results for the usual order statistics due to Balakrishnan (1985). These moments are then utilized to derive best linear unbiased estimators of the scale and location-scale parameters of the half-logistic distribution. A comparison of these estimators with the maximum likelihood estimates is then made. The best linear unbiased predictors of censored failure times is then discussed briefly. Finally, two numerical examples are presented to illustrate all the inferential methods developed here.     

An analytical C3 continuous tool path corner smoothing algorithm for 6R robot manipulator

Tool path smoothness is important to guarantee good dynamic and tracking performance of robot manipulators. An analytical C³ continuous tool path corner smoothing algorithm is proposed for robot manipulators with 6 rotational (6R) joints. The tool tip position is smoothed directly in the workpiece coordinate system (WCS). The tool orientation is smoothed after transferring the tool orientation matrix as three rotary angles. Micro-splines of the tool tip position and tool orientation are constructed under the constraints of the maximum deviation error tolerances in the WCS. Then the tool orientation and tool tip position are synchronized to the tool tip displacement with C³ continuity by replacing the remaining linear segments using specially constructed B-splines. Control points of the locally inserted micro-splines are all evaluated analytically without any iterative calculations. Simulation and experimental results show that the proposed algorithm satisfies constraints of the preset tool tip position and the tool orientation tolerances. The proposed corner smoothing algorithm achieves smoother and lower jerks than C² continuous corner smoothing algorithm. Experimental results show that the tracking errors associated to the execution of the C³ continuous tool path are up to 10% smaller than C² continuous path errors.     

Effect of different surface processing protocols in three-dimensional images for rapid prototyping

The aim of the study was to evaluate the accuracy of virtual three-dimensional reconstructions of human dry mandibles, produced with different surface processing protocols. Three-dimensional images were built from computed tomography scans of 10 dry mandibles, and the surface was smoothed, refined, or both, generating 30 different images. Linear measurements from anatomical landmarks were calculated and compared with the corresponding measurements of the original dry mandible (gold standard). The results showed no differences between the models that were just refined or just smoothed (p > 0.05), when compared to the gold standard. When these two tools were applied together, there was a statistically significant difference (p < 0.01). In conclusion, the application of a single processing tool (smoothing or refinement) in the virtual models does not affect the anatomical measures. However, the simultaneous application of both tools increases the differences between the reconstructions and the original anatomical parts.     

M-estimator and D-optimality model construction using orthogonal forward regression.

This correspondence introduces a new orthogonal forward regression (OFR) model identification algorithm using D-optimality for model structure selection and is based on an M-estimators of parameter estimates. M-estimator is a classical robust parameter estimation technique to tackle bad data conditions such as outliers. Computationally, The M-estimator can be derived using an iterative reweighted least squares (IRLS) algorithm. D-optimality is a model structure robustness criterion in experimental design to tackle ill-conditioning in model structure. The orthogonal forward regression (OFR), often based on the modified Gram-Schmidt procedure, is an efficient method incorporating structure selection and parameter estimation simultaneously. The basic idea of the proposed approach is to incorporate an IRLS inner loop into the modified Gram-Schmidt procedure. In this manner, the OFR algorithm for parsimonious model structure determination is extended to bad data conditions with improved performance via the derivation of parameter M-estimators with inherent robustness to outliers. Numerical examples are included to demonstrate the effectiveness of the proposed algorithm.     

Comparing penalized splines and fractional polynomials for flexible modelling of the effects of continuous predictor variables

P(enalized)-splines and fractional polynomials (FPs) have emerged as powerful smoothing techniques with increasing popularity in applied research. Both approaches provide considerable flexibility, but only limited comparative evaluations of the performance and properties of the two methods have been conducted to date. Extensive simulations are performed to compare FPs of degree 2 (FP2) and degree 4 (FP4) and two variants of P-splines that used generalized cross validation (GCV) and restricted maximum likelihood (REML) for smoothing parameter selection. The ability of P-splines and FPs to recover the “true” functional form of the association between continuous, binary and survival outcomes and exposure for linear, quadratic and more complex, non-linear functions, using different sample sizes and signal to noise ratios is evaluated. For more curved functions FP2, the current default setting in implementations for fitting FPs in R, STATA and SAS, showed considerable bias and consistently higher mean squared error (MSE) compared to spline-based estimators and FP4, that performed equally well in most simulation settings. FPs however, are prone to artefacts due to the specific choice of the origin, while P-splines based on GCV reveal sometimes wiggly estimates in particular for small sample sizes. Application to a real dataset illustrates the different features of the two approaches.