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In this paper three unrelated problems will be discussed. What connects them is the rich methodology of classical probability theory. In the first two problems we have a complete answer to the problems raised; in the third case, what we call the Hungarian roulette problem, we only have a conjecture with heuristic justification., Comment: 13 pages, 1 figure. To appear in Annales Univ. Sci. Budapest., Sec. Comp., vol. 56

Since Pearson's correlation was introduced at the end of the 19th century many dependence measures have appeared in the literature. Recently we have suggested four simple axioms for dependence measures of random variables that take values in Hilbert spaces. We showed that distance correlation satisfies all these axioms. We still need a new measure of dependence because existing measures either do not work in general metric spaces (that are not Hilbert spaces) or they do not satisfy our four simple axioms. The earth mover's correlation introduced in this paper applies in general metric spaces and satisfies our four axioms (two of them in a weaker form).

Attempts to quantify dependence between random elements X and Y via maximal correlation go back to Gebelein (1941) and R\'{e}nyi (1959). After summarizing properties (including some new) of the R\'{e}nyi measure of dependence, a calibrated scale of dependence is introduced. It is based on the ``complexity`` of approximating functions of X by functions of Y.

Deblurring is a fundamental inverse problem in bioimaging. It requires modelling the point spread function (PSF), which captures the optical distortions entailed by the image formation process. The PSF limits the spatial resolution attainable for a given microscope. However, recent applications require a higher resolution, and have prompted the development of super-resolution techniques to achieve sub-pixel accuracy. This requirement restricts the class of suitable PSF models to analog ones. In addition, deblurring is computationally intensive, hence further requiring computationally efficient models. A custom candidate fitting both requirements is the Gaussian model. However, this model cannot capture the rich tail structures found in both theoretical and empirical PSFs. In this paper, we aim at improving the reconstruction accuracy beyond the Gaussian model, while preserving its computational efficiency. We introduce a new class of analog PSF models based on Gaussian mixtures. The number of Gaussian kernels controls both the modelling accuracy and the computational efficiency of the model: the lower the number of kernels, the lower accuracy and the higher efficiency. To explore the accuracy--efficiency trade-off, we propose a variational formulation of the PSF calibration problem, where a convex sparsity-inducing penalty on the number of Gaussian kernels allows trading accuracy for efficiency. We derive an efficient algorithm based on a fully-split formulation of alternating split Bregman. We assess our framework on synthetic and real data and demonstrate a better reconstruction accuracy in both geometry and photometry in point source localisation---a fundamental inverse problem in fluorescence microscopy., Comment: This paper has been accepted in the IEEE Transactions on Image Processing

Kymographs are widely used to represent and anal- yse spatio-temporal dynamics of fluorescence markers along curvilinear biological compartments. These objects have a sin- gular geometry, thus kymograph reconstruction is inherently an analog image processing task. However, the existing approaches are essentially digital: the kymograph photometry is sampled directly from the time-lapse images. As a result, such kymographs rely on raw image data that suffer from the degradations entailed by the image formation process and the spatio-temporal resolution of the imaging setup. In this work, we address these limitations and introduce a well-grounded Bayesian framework for the analog reconstruction of kymographs. To handle the movement of the object, we introduce an intrinsic description of kymographs using differential geometry: a kymograph is a photometry defined on a parameter space that is embedded in physical space by a time-varying map that follows the object geometry. We model the kymograph photometry as a L\'evy innovation process, a flexible class of non-parametric signal priors. We account for the image formation process using the virtual microscope framework. We formulate a computationally tractable representation of the associated maximum a posteriori problem and solve it using a class of efficient and modular algorithms based on the alternating split Bregman. We assess the performance of our Bayesian framework on synthetic data and apply it to reconstruct the fluorescence dynamics along microtubules in vivo in the budding yeast S. cerevisiae. We demonstrate that our framework allows revealing patterns from single time-lapse data that are invisible on standard digital kymographs., Comment: This paper has been accepted in the IEEE Transactions on Image Processing

Given a distribution in the unite square and having iid sample from it the first question what a statistician might do to test the hypothesis that the sample is iid. For this purpose an extension of the Plancherel measure is introduced. Recent literature on asymptotic behavior of Plancherel measure is discussed with extension to the new set up. Models for random permutations are described and the power of different tests is compared., Comment: 10 pages + 13pages of figures, figures are contained in 3 pdf files, the first one contains the 11 pages of Figure 1

Sufficient conditions for comparing the convolutions of heterogeneous gamma random variables in terms of the usual stochastic order are established. Such comparisons are characterized by the Schur convexity properties of the cumulative distribution function of the convolutions. Some examples of the practical applications of our results are given.

Distance covariance and distance correlation have been widely adopted in measuring dependence of a pair of random variables or random vectors. If the computation of distance covariance and distance correlation is implemented directly accordingly to its definition then its computational complexity is O($n^2$) which is a disadvantage compared to other faster methods. In this paper we show that the computation of distance covariance and distance correlation of real valued random variables can be implemented by an O(n log n) algorithm and this is comparable to other computationally efficient algorithms. The new formula we derive for an unbiased estimator for squared distance covariance turns out to be a U-statistic. This fact implies some nice asymptotic properties that were derived before via more complex methods. We apply the fast computing algorithm to some synthetic data. Our work will make distance correlation applicable to a much wider class of applications., Comment: 38 pages, 6 tables, 5 figures. arXiv admin note: text overlap with arXiv:1205.4701 by other authors

Let $(X, \mathbf{Z})$ be a continuous random vector in $\mathbb{R} \times \mathbb{R}^d$, $d \ge 1$. In this paper, we define the notion of a nonparametric residual of $X$ on $\mathbf{Z}$ that is always independent of the predictor $\mathbf{Z}$. We study its properties and show that the proposed notion of residual matches with the usual residual (error) in a multivariate normal regression model. Given a random vector $(X, Y, \mathbf{Z})$ in $\mathbb{R} \times \mathbb{R} \times \mathbb{R}^d$, we use this notion of residual to show that the conditional independence between $X$ and $Y$, given $\mathbf{Z}$, is equivalent to the mutual independence of the residuals (of $X$ on $\mathbf{Z}$ and $Y$ on $\mathbf{Z}$) and $\mathbf{Z}$. This result is used to develop a test for conditional independence. We propose a bootstrap scheme to approximate the critical value of this test. We compare the proposed test, which is easily implementable, with some of the existing procedures through a simulation study., Comment: 19 pages, 2 figures

This article considers stochastic algorithms for efficiently solving a class of large scale non-linear least squares (NLS) problems which frequently arise in applications. We propose eight variants of a practical randomized algorithm where the uncertainties in the major stochastic steps are quantified. Such stochastic steps involve approximating the NLS objective function using Monte-Carlo methods, and this is equivalent to the estimation of the trace of corresponding symmetric positive semi-definite (SPSD) matrices. For the latter, we prove tight necessary and sufficient conditions on the sample size (which translates to cost) to satisfy the prescribed probabilistic accuracy. We show that these conditions are practically computable and yield small sample sizes. They are then incorporated in our stochastic algorithm to quantify the uncertainty in each randomized step. The bounds we use are applications of more general results regarding extremal tail probabilities of linear combinations of gamma distributed random variables. We derive and prove new results concerning the maximal and minimal tail probabilities of such linear combinations, which can be considered independently of the rest of this paper.

Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions, and applications of distance correlation have been discussed in the recent literature, but the problem of defining the partial distance correlation has remained an open question of considerable interest. The problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space. For the definition of partial distance correlation we introduce a new Hilbert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hilbert space, and develop and implement a test for zero partial distance correlation. Our intermediate results provide an unbiased estimator of squared distance covariance, and a neat solution to the problem of distance correlation for dissimilarities rather than distances.

In classical analysis of variance, dispersion is measured by considering squared distances of sample elements from the sample mean. We consider a measure of dispersion for univariate or multivariate response based on all pairwise distances between-sample elements, and derive an analogous distance components (DISCO) decomposition for powers of distance in $(0,2]$. The ANOVA F statistic is obtained when the index (exponent) is 2. For each index in $(0,2)$, this decomposition determines a nonparametric test for the multi-sample hypothesis of equal distributions that is statistically consistent against general alternatives., Comment: Published in at http://dx.doi.org/10.1214/09-AOAS245 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Rejoinder to "Brownian distance covariance" by G\'abor J. Sz\'ekely and Maria L. Rizzo [arXiv:1010.0297], Comment: Published in at http://dx.doi.org/10.1214/09-AOAS312REJ the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Distance correlation is a new class of multivariate dependence coefficients applicable to random vectors of arbitrary and not necessarily equal dimension. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but generalize and extend these classical bivariate measures of dependence. Distance correlation characterizes independence: it is zero if and only if the random vectors are independent. The notion of covariance with respect to a stochastic process is introduced, and it is shown that population distance covariance coincides with the covariance with respect to Brownian motion; thus, both can be called Brownian distance covariance. In the bivariate case, Brownian covariance is the natural extension of product-moment covariance, as we obtain Pearson product-moment covariance by replacing the Brownian motion in the definition with identity. The corresponding statistic has an elegantly simple computing formula. Advantages of applying Brownian covariance and correlation vs the classical Pearson covariance and correlation are discussed and illustrated., Comment: This paper discussed in: [arXiv:0912.3295], [arXiv:1010.0822], [arXiv:1010.0825], [arXiv:1010.0828], [arXiv:1010.0836], [arXiv:1010.0838], [arXiv:1010.0839]. Rejoinder at [arXiv:1010.0844]. Published in at http://dx.doi.org/10.1214/09-AOAS312 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Distance correlation is a new measure of dependence between random vectors. Distance covariance and distance correlation are analogous to product-moment covariance and correlation, but unlike the classical definition of correlation, distance correlation is zero only if the random vectors are independent. The empirical distance dependence measures are based on certain Euclidean distances between sample elements rather than sample moments, yet have a compact representation analogous to the classical covariance and correlation. Asymptotic properties and applications in testing independence are discussed. Implementation of the test and Monte Carlo results are also presented., Comment: Published in at http://dx.doi.org/10.1214/009053607000000505 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Generalized t-tests are constructed under weaker than normal conditions. In the first part of this paper we assume only the symmetry (around zero) of the error distribution (i). In the second part we assume that the error distribution is a Gaussian scale mixture (ii). The optimal (smallest) critical values can be computed from generalizations of Student's cumulative distribution function (cdf), $t_n(x)$. The cdf's of the generalized $t$-test statistics are denoted by (i) $t_n^S(x)$ and (ii) $t_n^G(x)$, resp. As the sample size $n\to \infty$ we get the counterparts of the standard normal cdf $\Phi(x)$: (i) $\Phi^S(x):=\operatorname {lim}_{n\to \infty}t_n^S(x)$, and (ii) $\Phi^G(x):=\operatorname {lim}_{n\to \infty}t_n^G(x)$. Explicit formulae are given for the underlying new cdf's. For example $\Phi^G(x)=\Phi(x)$ iff $|x|\ge \sqrt{3}$. Thus the classical 95% confidence interval for the unknown expected value of Gaussian distributions covers the center of symmetry with at least 95% probability for Gaussian scale mixture distributions. On the other hand, the 90% quantile of $\Phi^G$ is $4\sqrt{3}/5=1.385... >\Phi^{-1}(0.9)=1.282...$., Comment: Published at http://dx.doi.org/10.1214/074921706000000365 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)