158,689 research outputs found
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
contamination without breakdown, in contrast to the 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 -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 optimization to halfspace depth"
Discussion of "Multivariate quantiles and multiple-output regression
quantiles: From 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 -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
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