arXiv subjects

Daniel Vogel

Publications and source records attributed to Daniel Vogel.

30 records · Page 2Linked to original sources

arXiv2016

Studentized U-quantile processes under dependence with applications to change-point analysis

Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantiles, which do not rely on bootstrapping procedures. We demonstrate this approach in detail at the example of the Hodges-Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good robustness and efficiency properties of the test. Two real-life data sets are analyzed.

math.ST↗

arXiv2015

A Dataset of Naturally Occurring, Whole-Body Background Activity to Reduce Gesture Conflicts

In real settings, natural body movements can be erroneously recognized by whole-body input systems as explicit input actions. We call body activity not intended as input actions "background activity." We argue that understanding background activity is crucial to the success of always-available whole-body input in the real world. To operationalize this argument, we contribute a reusable study methodology and software tools to generate standardized background activity datasets composed of data from multiple Kinect cameras, a Vicon tracker, and two high-definition video cameras. Using our methodology, we create an example background activity dataset for a television-oriented living room setting. We use this dataset to demonstrate how it can be used to redesign a gestural interaction vocabulary to minimize conflicts with the real world. The software tools and initial living room dataset are publicly available (http://www.dgp.toronto.edu/~dustin/backgroundactivity/).

cs.HC↗

arXiv2015

Elliptical graphical modelling

We propose elliptical graphical models based on conditional uncorrelatedness as a general- ization of Gaussian graphical models by letting the population distribution be elliptical instead of normal, allowing the fitting of data with arbitrarily heavy tails. We study the class of propor- tionally affine equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling, leading to a new class of partial correlation estimators and analogues of the classical deviance test. General expressions for the asymptotic variance of partial correla- tion estimators, unconstrained and under decomposable models, are given, and the asymptotic chi square approximation of the pseudo-deviance test statistic is proved. The feasibility of our approach is demonstrated by a simulation study, using, among others, Tyler's scatter estimator, which is distribution-free within the elliptical model. Our approach provides a robustification of Gaussian graphical modelling. The latter is likelihood-based and known to be very sensitive to model misspecification and outlying observations.

stat.ME↗

arXiv2015

Weak convergence of the empirical process and the rescaled empirical distribution function in the Skorokhod product space

We prove the asymptotic independence of the empirical process $α_n = \sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $β_n = n (F_n(τ+\frac{\cdot}{n})-F_n(τ))$, where $F$ is an arbitrary cdf, differentiable at some point $τ$, and $F_n$ the corresponding empricial cdf. This seems rather counterintuitive, since, for every $n \in N$, there is a deterministic correspondence between $α_n$ and $β_n$. Precisely, we show that the pair $(α_n,β_n)$ converges in law to a limit having independent components, namely a time-transformed Brownian bridge and a two-sided Poisson process. Since these processes have jumps, in particular if $F$ itself has jumps, the Skorokhod product space $D(R) \times D(R)$ is the adequate choice for modeling this convergence in. We develop a short convergence theory for $D(R) \times D(R)$ by establishing the classical principle, devised by Yu. V. Prokhorov, that finite-dimensional convergence and tightness imply weak convergence. Several tightness criteria are given. Finally, the convergence of the pair $(α_n,β_n)$ implies convergence of each of its components, thus, in passing, we provide a thorough proof of these known convergence results in a very general setting. In fact, the condition on $F$ to be differentiable in at least one point is only required for $β_n$ to converge and can be further weakened.

math.PR↗

arXiv2015

Asymptotics of the two-stage spatial sign correlation

The spatial sign correlation (Dürre, Vogel and Fried, 2015) is a highly robust and easy-to-compute, bivariate correlation estimator based on the spatial sign covariance matrix. Since the estimator is inefficient when the marginal scales strongly differ, a two-stage version was proposed. In the first step, the observations are marginally standardized by means of a robust scale estimator, and in the second step, the spatial sign correlation of the thus transformed data set is computed. Dürre et al. (2015) give some evidence that the asymptotic distribution of the two-stage estimator equals that of the spatial sign correlation at equal marginal scales by comparing their influence functions and presenting simulation results, but give no formal proof. In the present paper, we close this gap and establish the asymptotic normality of the two-stage spatial sign correlation and compute its asymptotic variance for elliptical population distributions. We further derive a variance-stabilizing transformation, similar to Fisher's z-transform, and numerically compare the small-sample coverage probabilities of several confidence intervals.

stat.ME↗

arXiv2015

Estimation of the variance of partial sums of dependent processes

We study subsampling estimators for the limit variance \[ σ^2=Var(X_1)+2 \sum_{k=2}^\infty Cov(X_1,X_k) \] of partial sums of a stationary stochastic process $(X_k)_{k\geq 1}$. We establish $L_2$-consistency of a non-overlapping block resampling method. Our results apply to processes that can be represented as functionals of strongly mixing processes. Motivated by recent applications to rank tests, we also study estimators for the series $Var(F(X_1))+2 \sum_{k=2}^\infty Cov(F(X_1),F(X_k))$, where $F$ is the distribution function of $X_1$. Simulations illustrate the usefulness of the proposed estimators and of a mean squared error optimal rule for the choice of the block length.

math.ST↗

arXiv2014

On the efficiency of Gini's mean difference

We examine the efficiency of the mean deviation and Gini's mean difference (the mean of all pairwise distances). Our findings support the viewpoint that Gini's mean difference combines the advantages of the mean deviation and the standard deviation.

math.ST↗

arXiv2014

The spatial sign covariance matrix with unknown location

The consistency and asymptotic normality of the spatial sign covariance matrix with unknown location are shown. Simulations illustrate the different asymptotic behavior when using the mean and the spatial median as location estimator.

math.ST↗

arXiv2014

Spatial Sign Correlation

A new robust correlation estimator based on the spatial sign covariance matrix (SSCM) is proposed. We derive its asymptotic distribution and influence function at elliptical distributions. Finite sample and robustness properties are studied and compared to other robust correlation estimators by means of numerical simulations.

stat.ME↗

arXiv2014

A fluctuation test for constant Spearman's rho with nuisance-free limit distribution

A CUSUM type test for constant correlation that goes beyond a previously suggested correlation constancy test by considering Spearman's rho in arbitrary dimensions is proposed. Since the new test does not require the existence of any moments, the applicability on usually heavy-tailed financial data is greatly improved. The asymptotic null distribution is calculated using an invariance principle for the sequential empirical copula process. The limit distribution is free of nuisance parameters and critical values can be obtained without bootstrap techniques. A local power result and an analysis of the behavior of the test in small samples is provided.

stat.ME↗

arXiv2013

Robust estimators for non-decomposable elliptical graphical models

Asymptotic properties of scatter estimators for elliptical graphical models are studied. Such models impose a given pattern of zeros on the inverse of the shape matrix of an elliptically distributed random vector. In particular, we introduce the class of graphical M-estimators and compare them to plug-in M-estimators. It turns out that, under suitable conditions, both approaches yield the same asymptotic efficiency. Furthermore, the results of this paper apply to both decomposable and non-decomposable graphical models and so generalize the results for decomposable models given by Vogel & Fried (2011) for the plug-in M-estimators.

math.ST↗

arXiv2008

RubberEdge: Reducing Clutching by Combining Position and Rate Control with Elastic Feedback

Position control devices enable precise selection, but significant clutching degrades performance. Clutching can be reduced with high control-display gain or pointer acceleration, but there are human and device limits. Elastic rate control eliminates clutching completely, but can make precise selection difficult. We show that hybrid position-rate control can outperform position control by 20% when there is significant clutching, even when using pointer acceleration. Unlike previous work, our RubberEdge technique eliminates trajectory and velocity discontinuities. We derive predictive models for position control with clutching and hybrid control, and present a prototype RubberEdge position-rate control device including initial user feedback.

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