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The distance covariance of Sz\'ekely, et al. [23] and Sz\'ekely and Rizzo [21], a powerful measure of dependence between sets of multivariate random variables, has the crucial feature that it equals zero if and only if the sets are mutually independent. Hence the distance covariance can be applied to multivariate data to detect arbitrary types of non-linear associations between sets of variables. We provide in this article a basic, albeit rigorous, introductory treatment of the distance covariance. Our investigations yield an approach that can be used as the foundation for presentation of this important and timely topic even in advanced undergraduate- or junior graduate-level courses on mathematical statistics., Comment: 12 pages, 2 figures

We initiate the study of goodness-of-fit testing for data consisting of positive definite matrices. Motivated by the appearance of positive definite matrices in numerous applications, including factor analysis, diffusion tensor imaging, volatility models for financial time series, wireless communication systems, and polarimetric radar imaging, we apply the method of Hankel transforms of matrix argument to develop goodness-of-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of the test statistic and a corresponding covariance operator, show that the eigenvalues of the operator satisfy an interlacing property, and apply our test to some financial data. We establish the consistency of the test against a large class of alternative distributions and derive the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We obtain the Bahadur and Pitman efficiency properties of the test statistic and establish a modified version of Wieand’s condition.

We apply the method of Hankel transforms to develop goodness-of-fit tests for gamma distributions with given shape parameters and unknown rate parameters. We derive the limiting null distribution of the test statistic as an integrated squared Gaussian process, obtain the corresponding covariance operator and oscillation properties of its eigenfunctions, show that the eigenvalues of the operator satisfy an interlacing property, and make applications to two data sets. We prove consistency of the test, provide numerical power comparisons with alternative tests, study the test statistic under several contiguous alternatives, and obtain the asymptotic distribution of the test statistic for gamma alternatives with varying rate or shape parameters and for certain contaminated gamma models. We investigate the approximate Bahadur slope of the test statistic under local alternatives, and we establish the validity of the Wieand condition under which approaches through the approximate Bahadur and the Pitman efficiencies are in accord.

The Affordable Care Act introduced a revenue transfer formula that requires insurance plans with generally healthier enrollees to pay funds into a revenue transfer pool to reimburse plans with gene...

© 2019 Elsevier Inc. We consider the construction of insurance premiums that are monotonically increasing with respect to a loading parameter. By introducing weight functions that are totally positive of higher order, we derive higher monotonicity properties of generalized weighted premiums; in particular, we deduce for weight functions that are totally positive of order three a monotonicity property of the variance-to-mean ratio, or index of dispersion, of the loss variable. We derive the higher order total positivity properties of some ratios that arise in actuarial and insurance analysis of combined risks. Further, we examine seven classes of weight functions that have appeared in the literature and we ascertain the higher order total positivity properties of those functions.

We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(\cdot)$, on $\mathcal{L}$. We derive from the supermodularity of $n(\cdot)$ some generalized Fr\'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(\cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fr\'echet inequalities for multidimensional contingency tables.

The Affordable Care Act (ACA) includes a permanent revenue transfer methodology that provides financial incentives to health insurance plans that have higher than average actuarial risk. In this ar...

We consider the problem of calculating distance correlation coefficients between random vectors whose joint distributions belong to the class of Lancaster distributions. We derive under mild convergence conditions a general series representation for the distance covariance for these distributions. To illustrate the general theory, we apply the series representation to derive explicit expressions for the distance covariance and distance correlation coefficients for the bivariate normal distribution and its generalizations of Lancaster type, the multivariate normal distributions, and the bivariate gamma, Poisson, and negative binomial distributions which are of Lancaster type., 32 pages, 1 figure

Multiple root estimation problems in statistical inference arise in many contexts in the literature. In the context of maximum likelihood estimation, the existence of multiple roots causes uncertainty in the computation of maximum likelihood estimators using hill-climbing algorithms, and consequent difficulties in the resulting statistical inference. In this paper, we study the multiple roots phenomenon in maximum likelihood estimation for factor analysis. We prove that the corresponding likelihood equations have uncountably many feasible solutions even in the simplest cases. For the case in which the observed data are two-dimensional and the unobserved factor scores are one-dimensional, we prove that the solutions to the likelihood equations form a one-dimensional real curve.

We construct by using B-spline functions a class of copulas that includes the Bernstein copulas arising in Baker's distributions. The range of correlation of the B-spline copulas is examined, and the Frechet--Hoeffding upper bound is proved to be attained when the number of B-spline functions goes to infinity. As the B-spline functions are well-known to be an order-complete weak Tchebycheff system from which the property of total positivity of any order follows for the maximum correlation case, the results given here extend classical results for the Bernstein copulas. In addition, we derive in terms of the Stirling numbers of the second kind an explicit formula for the moments of the related B-spline functions for nonnegative real numbers.

Summary We study parameter estimation in linear Gaussian covariance models, which are p-dimensional Gaussian models with linear constraints on the covariance matrix. Maximum likelihood estimation for this class of models leads to a non-convex optimization problem which typically has many local maxima. Using recent results on the asymptotic distribution of extreme eigenvalues of the Wishart distribution, we provide sufficient conditions for any hill climbing method to converge to the global maximum. Although we are primarily interested in the case in which n ≫ p, the proofs of our results utilize large sample asymptotic theory under the scheme n/p → γ > 1. Remarkably, our numerical simulations indicate that our results remain valid for p as small as 2. An important consequence of this analysis is that, for sample sizes n≃14p, maximum likelihood estimation for linear Gaussian covariance models behaves as if it were a convex optimization problem.

Truncated multivariate distributions arise extensively in econometric modelling when non-negative random variables are intrinsic to the data-generation process. More broadly, truncated multivariate distributions have appeared in censored and truncated regression models, simultaneous equations modelling, multivariate regression, and applications going back to the now-classic papers of Amemiya (1974) and Heckman (1976). In some applications of truncated multivariate distributions, there arises the problem of characterizing the distribution through correlation and independence properties of sub-vectors. In this paper, we characterize the truncated multivariate normal random vectors for which two complementary sub-vectors are mutually independent. Further, we characterize the multivariate truncated elliptical distributions, proving that if two complementary sub-vectors are mutually independent then the distribution of the joint vector is truncated multivariate normal, as is the distribution of each sub-vector. As an application, we apply the independence criterion to test the hypothesis of independence of the entrance examination scores and subsequent course averages achieved by a sample of university students; to do so, we verify the regularity conditions underpinning a classical theorem of Wilks on the asymptotic null distribution of the likelihood ratio test statistic., 15 pages; updated abstract, additional references and an additional Section #5

© Institute of Mathematical Statistics, 2018. Gaussian graphical models have received considerable attention during the past four decades from the statistical and machine learning communities. In Bayesian treatments of this model, the G-Wishart distribution serves as the conjugate prior for inverse covariance matrices satisfying graphical constraints. While it is straightforward to posit the unnormalized densities, the normalizing constants of these distributions have been known only for graphs that are chordal, or decomposable. Up until now, it was unknown whether the normalizing constant for a general graph could be represented explicitly, and a considerable body of computational literature emerged that attempted to avoid this apparent intractability. We close this question by providing an explicit representation of the G-Wishart normalizing constant for general graphs.

Gaussian particles provide a flexible framework for modelling and simulating three-dimensional star-shaped random sets. In our framework, the radial function of the particle arises from a kernel smoothing, and is associated with an isotropic random field on the sphere. If the kernel is a von Mises--Fisher density, or uniform on a spherical cap, the correlation function of the associated random field admits a closed form expression. The Hausdorff dimension of the surface of the Gaussian particle reflects the decay of the correlation function at the origin, as quantified by the fractal index. Under power kernels we obtain particles with boundaries of any Hausdorff dimension between 2 and 3., Comment: 22 pages, 5 figures, 3 tables; to appear in Advances in Applied Probability

For multiple-input/multiple-output (MIMO) spatial multiplexing with zero-forcing detection (ZF), signal-to-noise ratio (SNR) analysis for Rician fading involves the cumbersome noncentral-Wishart distribution (NCWD) of the transmit sample-correlation (Gramian) matrix. An \textsl{approximation} with a \textsl{virtual} CWD previously yielded for the ZF SNR an approximate (virtual) Gamma distribution. However, analytical conditions qualifying the accuracy of the SNR-distribution approximation were unknown. Therefore, we have been attempting to exactly characterize ZF SNR for Rician fading. Our previous attempts succeeded only for the sole Rician-fading stream under Rician--Rayleigh fading, by writing it as scalar Schur complement (SC) in the Gramian. Herein, we pursue a more general, matrix-SC-based analysis to characterize SNRs when several streams may undergo Rician fading. On one hand, for full-Rician fading, the SC distribution is found to be exactly a CWD if and only if a channel-mean--correlation \textsl{condition} holds. Interestingly, this CWD then coincides with the \textsl{virtual} CWD ensuing from the \textsl{approximation}. Thus, under the \textsl{condition}, the actual and virtual SNR-distributions coincide. On the other hand, for Rician--Rayleigh fading, the matrix-SC distribution is characterized in terms of determinant of matrix with elementary-function entries, which also yields a new characterization of the ZF SNR. Average error probability results validate our analysis vs.~simulation., Comment: 32 pages, 4 figures, 1 table

The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. The asymptotic distribution of the empirical distance standard deviation is derived under the assumption of finite second moments. Applications are provided to hypothesis testing on a data set from materials science and to multivariate statistical quality control. The distance standard deviation is compared to classical scale measures for inference on the spread of heavy-tailed distributions. Inequalities for the distance variance are derived, proving that the distance standard deviation is bounded above by the classical standard deviation and by Gini's mean difference. New expressions for the distance standard deviation are obtained in terms of Gini's mean difference and the moments of spacings of order statistics. It is also shown that the distance standard deviation satisfies the axiomatic properties of a measure of spread.

In the setting of inference with two-step monotone incomplete data drawn from Nd( µ, ∑), a multivariate normal population with mean µ and covariance matrix ∑, we derive a stochastic representation for the exact distribution of a generalization of Hotelling’s T2-statistic, thereby enabling the construction of exact level ellipsoidal confidence regions for µ. By applying the equivariance of [Formula: see text] and [Formula: see text] the maximum likelihood estimators of µ and ∑, respectively, we show that the T2-statistic is invariant under affine transformations. Further, as a consequence of the exact stochastic representation, we derive upper and lower bounds for the cumulative distribution function of the T2-statistic. We apply these results to construct simultaneous confidence regions for linear combinations of µ, and we apply these results to analyze a dataset consisting of cholesterol measurements on a group of Pennsylvania heart disease patients.

A new class of generalized correlation coefficients that contains the Pearson and Kendall statistics as special cases was defined by Chinchilli et al. (2005) and applied to the estimation of correlations coefficients within the context of 2×2 cross-over designs for clinical trials. In this paper, we determine the infinitesimal robustness and local stability properties of these generalized correlation coefficients by deriving their corresponding influence functions. For cases in which the population distribution is a bivariate normal or a mixture of bivariate normal distributions we obtain explicit formulas, and establish monotonicity and sign-reverse rule properties of the generalized correlation coefficients.

We establish a positivity property for the difference of products of certain Schur functions, sλ(x), where λ varies over a fundamental Weyl chamber in ℝn and x belongs to the positive orthant in ℝn. Further, we generalize that result to the difference of certain products of arbitrary numbers of Schur functions. We also derive a log-convexity property of the generalized hypergeometric functions of two Hermitian matrix arguments, and we show how that result may be extended to derive higher-order log-convexity properties.

Letbe a positive Borel measure on R n and pFq(a1,... ,ap;b1,... ,bq;s) be a generalized hypergeometric series. We define a generalized hypergeomet- ric measure, µp,q := pFq(a1,... ,ap;b1,... ,bq;�), as a series of convolution powers of the measure �, and we investigate classes of probability distri- butions which are expressible as such a measure. We show that the Kemp family of distributions (SankhySer. A, 30, (1968), 401-410) is an example of µp,q in whichis a Dirac measure on R. For the case in whichis a Dirac measure on R n , we relate µp,q to the diagonal natural exponential families classified by Bar-Lev, et al. (J. Theoret. Probab. 7 (1994), 883-929). For p < q we show that certain measures µp,q can be expressed as the convo- lution of a sequence of independent multi-dimensional Bernoulli trials. For p = q,q + 1, we show that the measures µp,q are mixture measures with the Dufresne and Poisson-stopped-sum probability distributions as their mixing measures.

This novel display format enables all data in files of any size to be represented on a single screen. The report at the top of the queue or stack in the category of interest is displayed on the right. The numbers in the circles represent the category report count. Reports with critical data are represented in red circles around the inner circle. To improve patient safety and care delivery this structure permits timely critical report acknowledgement and triage of non-critical reports.Display Omitted Invariant single screen EHR visual display.All data within data sets of all sizes represented on a single screen.Data physically partitioned-critical or non-critical for prompt response or work triage.One screen representation of all reports from all EHR systems with interface.Data mining enabled. ObjectiveThe timely acknowledgement of critical patient clinical reports is vital for the delivery of safe patient care. With current EHR systems, critical reports reside on different screens. This leads to treatment delays and inefficient work flows. As a remedy, the R.A.P.I.D. (Root Aggregated Prioritized Information Display) system represents all data on a single screen, and its simple and intuitive "button" array structure allows triaged sign-off/sign-out of critical and non-critical reports. Materials and methodsWith 100 hematology and chemistry reports from each of two EHR systems Meditech (Westwood, MA) and Orchard Labs, Inc. (Carmel, IN), we generated files of the reports in their individual standard display formats (enhanced Meditech-EM and enhanced Orchard-EO). We also displayed the same 200 reports in the R.A.P.I.D. format. We then conducted a randomized trial to compare the time and accuracy of acknowledgement of critical and non-critical results. ResultsThe sign-off times for reviewing the results for physician and non-physician providers, respectively, in seconds (with 95% confidence intervals) were for EM 1.78 (1.40-2.26) and 1.99 (1.72-2.30), for EO 2.69 (2.12-3.42) and 2.78 (2.40-3.21), and for R.A.P.I.D. 0.83 (0.70-0.98) and 1.58 (1.43-1.76). Non-physician providers reassigned system-defined non-critical results as critical with a frequency of 15.2% for EM, 18.4% for EO, and 7.83% for R.A.P.I.D., and critical results as non-critical with a frequency of 14.7%, 5.6%, and 5.8% respectively. DiscussionThe new display system was superior to two standard EHR systems that were significantly enhanced by first collecting the reports from their usual distributed locations and then by creating for each of the two standard EHRs a single file of reports for acknowledgement. ConclusionsFrom a single screen display of all reports, the new display system enables timely acknowledgement of critical reports for patient safety and non-critical report triage for improved provider work flows.

We study zero-forcing detection (ZF) for multiple-input/multiple-output (MIMO) spatial multiplexing under transmit-correlated Rician fading for an N_R X N_T channel matrix with rank-1 line-of-sight (LoS) component. By using matrix transformations and multivariate statistics, our exact analysis yields the signal-to-noise ratio moment generating function (m.g.f.) as an infinite series of gamma distribution m.g.f.'s and analogous series for ZF performance measures, e.g., outage probability and ergodic capacity. However, their numerical convergence is inherently problematic with increasing Rician K-factor, N_R , and N_T. We circumvent this limitation as follows. First, we derive differential equations satisfied by the performance measures with a novel automated approach employing a computer-algebra tool which implements Groebner basis computation and creative telescoping. These differential equations are then solved with the holonomic gradient method (HGM) from initial conditions computed with the infinite series. We demonstrate that HGM yields more reliable performance evaluation than by infinite series alone and more expeditious than by simulation, for realistic values of K , and even for N_R and N_T relevant to large MIMO systems. We envision extending the proposed approaches for exact analysis and reliable evaluation to more general Rician fading and other transceiver methods., Accepted for publication by the IEEE Transactions on Wireless Communications, on April 7th, 2016; this is the final revision before publication

We show that the distribution of the scalar Schur complement in a noncentral Wishart matrix is a mixture of central chi-square distributions with different degrees of freedom. For the case of a rank-1 noncentrality matrix, the weights of the mixture representation arise from a noncentral beta mixture of Poisson distributions.

In reading Professor Belden’s comments we find that the author has indirectly aimed those comments toward the article by Durand (1957) which is, in part, the basis for our article. In framing our response we therefore begin with biographical details about Durand’s professional achievements. Subsequently, we turn to a response to the comments in question. We also received from Professor H. C. Tijms (Department of Econometrics and Operations Research, Free University, Amsterdam, The Netherlands) some remarks on the comments we made in our article regarding results of Whitworth (1901). We will provide further details of Whitworth’s results, including their historical context.

Recent articles have described the advantages of teaching elementary statistics and probability classes using approaches that encourage greater student engagement, including experimentation, with the subject matter. We describe our experiences in introducing the subject of random walks to small groups of highschool and first-year college students. As we show in this article, the topic of random walks provides a superb way for instructors to introduce a class to elementary simulation problems, calculation of expectations and measures of variability for geometric distributions, real-world interpretation and consequences for the divergence of infinite series, and the behavior of random walks on restricted sets in the plane. Most enchantingly, all facets of this journey are entirely accessible to an involved class of students equipped with minimal knowledge of calculus. Based on our experiences, we strongly recommend student involvement in the teaching of introductory concepts to small classes.

During the late 1990s high technology growth stock prices were raised to unprecedented levels by avid stock purchasers around the world. In early 2000, share prices subsequently underwent prolonged declines, leaving many purchasers with devastating losses. This article reviews some aspects of the history of the St. Petersburg paradox and some related games. We recount a remarkable article by Durand in which the valuation of growth stocks is related to the St. Petersburg paradox. Our conclusion is that the run-up in stock prices in the late 1990s and the subsequent declines in 2000 could have been avoided by an analysis and application of the St. Petersburg paradox.

In multivariate statistical analysis, several authors have studied the total positivity properties of the generalized (0F1) hypergeometric function of two real symmetric matrix arguments. In this paper, we make use of zonal polynomial expansions to obtain a new proof of a result that these 0F1 functions fail to satisfy certain pairwise total positivity properties; this proof extends both to arbitrary generalized ( rFs) functions of two matrix arguments and to the generalized hypergeometric functions of Hermitian matrix arguments. In the case of the generalized hypergeometric functions of two Hermitian matrix arguments, we prove that these functions satisfy certain modified pairwise TP2 properties; the proofs of these results are based on Sylvester’s formula for compound determinants and the condensation formula of C. L. Dodgson [Lewis Carroll] (1866).

A classical theorem of Boas, Kac, and Krein states that a characteristic function φ with φ(x) = 0 for |x| > T admits a representation of the form φ(x) = ∫u(y)u(y + x) dy, x ∈ R, where the convolution root u ∈ L 2 (R) is complex-valued with u(x) = 0 for |x| ≥ τ/2. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If φ is real-valued and even, can the convolution root u be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of φ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on R d is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turan's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if f is a probability density on R d whose characteristic function φ vanishes outside the unit ball, then ∫|x| 2 f(x) dx = -Δφ(0) ≥ 4j 2 (d-2)/2 where j v denotes the first positive zero of the Bessel function J v , and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in R 2 does not exist.

Periodicities of radio flaring activity from the Algol systems β Per and δ Lib and the RS CVn systems V711 Tau and UX Ari were determined from a continuous 5-year survey. The radio continuum fluxes at 2.3 GHz and 8.3 GHz were monitored with the NRAO—Green Bank Interferometer for 2096 days from 1995 January to 2000 October. The maximum detected flare strengths at 8.3 GHz were 1.17 Jy for β Per, 0.034 Jy for δ Lib, 1.44 Jy for V711 Tau, and 0.82 Jy for UX Ari. Power Spectrum Analysis and Phase Dispersion Minimization were used to determine the periodicity of flaring activity in each binary. The strongest periodicities found were 48.9 days for β Per, 120.7 days for V711 Tau, and 141.4 days for UX Ari. No significant periodicities were found for δ Lib. The continuous survey has demonstrated that there are active and quiescent flaring cycles in V711 Tau and β Per.

We report on the longest-running continuous radio flare survey of two Algol-type systems (β Per and δ Lib) and two RS CVn systems (V711 Tau and UX Ari). All four systems have late-type components, and all were known to display radio flaring activity. The primary aim of the campaign was to determine the timescales for flaring activity in these systems. The radio continuum flux at 2.3 and 8.3 GHz was monitored with the NRAO-Green Bank Interferometer from 1995 January to 2000 October. The survey spanned 2096 days with interruptions during maintenance runs and temporary closings of the interferometer. Many strong flares were detected with continuum fluxes at 8.3 GHz as high as 1.17 Jy in β Per, 1.44 Jy in V711 Tau, and 0.82 Jy in UX Ari. Only two flares were detected from δ Lib during 1123 days of monitoring, and the continuum flux reached a maximum of only 0.034 Jy at 8.3 GHz. The independent techniques of Power Spectrum Analysis and Phase Dispersion Minimization were used to determine the periodicity of flaring activity in each binary. The strongest periodicities found were 48.9 ± 1.7 days for β Per, 120.7 ± 3.4 days for V711 Tau, and 141.4 ± 4.5 days for UX Ari, with other significant periodicities of 80.8 ± 2.5 days for V711 Tau and 52.6 ± 0.7 days for UX Ari. In the case of δ Lib, the strongest periodicities were related to the duration of the two monitoring cycles within the data set and are not real. The continuous survey has demonstrated that there are active and quiescent flaring cycles in V711 Tau and β Per. During both of these cycles, β Per had more flares than V711 Tau, but its strongest flares were typically weaker than those of V711 Tau.

We generalize an integral which arises in several areas in probability and statistics and which is at the core of the field of distance correlation, a concept developed by Sz\'ekely, Rizzo and Bakirov (2007) to measure dependence between random variables. Let $m$ be a positive integer and let ${\cos_m}(u)$, $u \in \mathbb{R}$, be the truncated Maclaurin expansion of ${\cos}(u)$, where the expansion is truncated at the $m$th summand. For $t, x \in \mathbb{R}^d$, let $\langle t,x\rangle$ and $\|x\|$ denote the standard Euclidean inner product and norm, respectively. We establish the integral formula: For $\alpha \in \mathbb{C}$ and $x \in \mathbb{R}^d$, $\int_{{\mathbb{R}}^d} [\cos_m(\langle t,x\rangle) - \cos(\langle t,x\rangle)] \,{\rm d}t/{\|t\|^{d+\alpha}} = C(d,\alpha) \, \|x\|^{\alpha}$, with absolute convergence if and only if $2(m-1) < \Re(\alpha) < 2m$. Moreover, the constant $C(d,\alpha)$ does not depend on $m$., Comment: 7 pages; to appear in Statistics and Probability Letters, 2015

The accurate classification of galaxies in large-sample astrophysical databases of galaxy clusters depends sensitively on the ability to distinguish between morphological types, especially at higher redshifts. This capability can be enhanced through a new statistical measure of association and correlation, called the {\it distance correlation coefficient}, which has more statistical power to detect associations than does the classical Pearson measure of linear relationships between two variables. The distance correlation measure offers a more precise alternative to the classical measure since it is capable of detecting nonlinear relationships that may appear in astrophysical applications. We showed recently that the comparison between the distance and Pearson correlation coefficients can be used effectively to isolate potential outliers in various galaxy datasets, and this comparison has the ability to confirm the level of accuracy associated with the data. In this work, we elucidate the advantages of distance correlation when applied to large databases. We illustrate how the distance correlation measure can be used effectively as a tool to confirm nonlinear relationships between various variables in the COMBO-17 database, including the lengths of the major and minor axes, and the alternative redshift distribution. For these outlier pairs, the distance correlation coefficient is routinely higher than the Pearson coefficient since it is easier to detect nonlinear relationships with distance correlation. The V-shaped scatterplots of Pearson versus distance correlation coefficients also reveal the patterns with increasing redshift and the contributions of different galaxy types within each redshift range., 5 pages, 2 tables, 3 figures; published in Astrophysical Journal Letters, 784, L34 (2014)

We consider the problem of testing multivariate normality when the data consists of a random sample of two-step monotone incomplete observations. We define for such data a generalization of Mardia's statistic for measuring kurtosis, derive the asymptotic non-null distribution of the statistic under certain regularity conditions and against a broad class of alternatives, and give an application to a well-known data set on cholesterol measurements., Comment: 20 pages

Denote by the class of probability density functions on ℝ with a nonnegative and integrable characteristic function. For each p ∈ , there exists an adjoint density , which is proportional to the characteristic function of p. The products λ(p) = Var(p) Var() have a greatest lower bound Λ known as Laue's constant. In this paper we improve the previous estimates of Λ, proving that 0.543 < 0.85024. Further results and related estimates are also obtained for a multivariate version of the uncertainty relation which is based on univariate projections.

Suppose that the random vector (X1, …, Xq) follows a Dirichlet distribution on Rq+ with parameter (p1, …, pq)∈Rq+. For f1, …, fq>0, it is well-known that E(f1X1+…+fqXq)−(p1+…+pq)=f−p11…f−pqq. In this paper, we generalize this expectation formula to the singular and non-singular multivariate Dirichlet distributions as follows. Let Ωr denote the cone of all r×r positive-definite real symmetric matrices. For x∈Ωr and 1⩽j⩽r, let detjx denote the jth principal minor of x. For s=(s1, …, sr)∈Rr, the generalized power function of x∈Ωr is the function Δs(x)=(det1x)s1−s2(det2x)s2−s3…(detr−1x)sr−1−sr(detrx)sr; further, for any t∈R, we denote by s+t the vector (s1+t, …, sr+t). Suppose X1, …, Xq∈Ωr are random matrices such that (X1, …, Xq) follows a multivariate Dirichlet distribution with parameters p1, …, pq. Then we evaluate the expectation E[Δs1(X1)…Δsq(Xq)Δs1+…+sq+p((a+f1X1+…+fqXq)−1)], where a∈Ωr, p=p1+…+pq, f1, …, fq>0, and s1, …, sq each belong to an appropriate subset of Rr+. The result obtained is parallel to that given above for the univariate case, and remains valid even if some of the Xj's are singular. Our derivation utilizes the framework of symmetric cones, so that our results are valid for multivariate Dirichlet distributions on all symmetric cones.