Finite sample breakdown point of multivariate regression depth median

Depth induced multivariate medians (multi-dimensional maximum depth estimators) in regression serve as robust alternatives to the traditional least squares and least absolute deviations estimators. The induced median (\bs{\beta}^*_{RD}) from regression depth (RD) of Rousseeuw and Hubert (1999) (RH99) is one of the most prevailing estimators in regression. The maximum regression depth median possesses outstanding robustness similar to the univariate location counterpart. Indeed, the %maximum depth estimator induced from \mbox{RD}, \bs{\beta}^*_{RD} can, asymptotically, resist up to $33\%$ contamination without breakdown, in contrast to the $0\%$ for the traditional estimators %(i.e. they could break down by a single bad point) (see Van Aelst and Rousseeuw, 2000) (VAR00). The results from VAR00 are pioneering and innovative, yet they are limited to regression symmetric populations and the $\epsilon$-contamination and maximum bias model. With finite fixed sample size practice, the most prevailing measure of robustness for estimators is the finite sample breakdown point (FSBP) (Donoho (1982), Donoho and Huber (1983)). A lower bound (LB) of the FSBP for the \bs{\beta}^*_{RD}, which is not sharp, was given in RH99 (in a corollary of a conjecture). An exact FSBP (or even a sharper LB) for the \bs{\beta}^*_{RD} remained open in the last two decades. This article establishes a sharper lower and upper bounds of (and an exact) FSBP for the \bs{\beta}^*_{RD}, revealing an intrinsic connection between the regression depth of \bs{\beta}^*_{RD} and its FSBP. This justifies the employment of the \bs{\beta}^*_{RD} as a robust alternative to the traditional estimators and demonstrating the necessity and the merit of using the FSBP in finite sample real practice instead of an asymptotic breakdown value.Comment: 20 pages, 4 figures, 3 table

Discussion of "Multivariate quantiles and multiple-output regression quantiles: From L1L_1 optimization to halfspace depth"

Discussion of "Multivariate quantiles and multiple-output regression quantiles: From $L_1$ optimization to halfspace depth" by M. Hallin, D. Paindaveine and M. Siman [arXiv:1002.4486]Comment: Published in at http://dx.doi.org/10.1214/09-AOS723B the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric regression using deep neural networks with ReLU activation function

Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to $\log n$-factors) under a general composition assumption on the regression function. The framework includes many well-studied structural constraints such as (generalized) additive models. While there is a lot of flexibility in the network architecture, the tuning parameter is the sparsity of the network. Specifically, we consider large networks with number of potential network parameters exceeding the sample size. The analysis gives some insights into why multilayer feedforward neural networks perform well in practice. Interestingly, for ReLU activation function the depth (number of layers) of the neural network architectures plays an important role and our theory suggests that for nonparametric regression, scaling the network depth with the sample size is natural. It is also shown that under the composition assumption wavelet estimators can only achieve suboptimal rates.Comment: article, rejoinder and supplementary materia

The study of factors that affect cell depth during etching in direct transfer gravure

This thesis is to study the four main factors which affect cell depth in copper cylinder etching in direct transfer gravure, and using ferric chloride as an etchant. The multivariate regression method was used to analyze the sample responses (cell depth) of nine different treatments of an etched-screened-tone scale . The regression equations for predicting cell depths from the four variable factors were calculated and the optimum condition which would give the best tonal gradation was predicted. Also the factors which cause uneven cell depth were detected

Quantile tomography: using quantiles with multivariate data

The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distributions (in particular multivariate normal) with density contours. Relevant questions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approximation/asymptotic properties are studied. It is argued that the analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic interpretation and thus conveys a concrete quantitative meaning; the directional definition can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts---the question that motivated all the development