monin lundgren hierarchy versus the hopf equation in the statistical theory of turbulence a functional differential equation equivalent to the infinitely hierarchic system of the coupled equations for many point velocity distribution functions in turbulence established by monin and lundgren is presented to be compared with the hopf characteristic functional equation which governs the time evolution of a turbulent velocity field it turns out that this functional entirely implies the hopf functional and vice versa this functional allows us to view turbulence in a lagrangian picture simulated by many particle dynamics

consistent distribution free k sample and independence tests for univariate random variables a popular approach for testing if two univariate random variables are statistically independent consists of partitioning the sample space into bins and evaluating a test statistic on the binned data the partition size matters and the optimal partition size is data dependent while for detecting simple relationships coarse partitions may be best for detecting complex relationships a great gain in power can be achieved by considering finer partitions we suggest novel consistent distribution free tests that are based on summation or maximization aggregation of scores over all partitions of a fixed size we show that our test statistics based on summation can serve as good estimators of the mutual information moreover we suggest regularized tests that aggregate over all partition sizes and prove those are consistent too we provide polynomial time algorithms which are critical for computing the suggested test statistics efficiently we show that the power of the regularized tests is excellent compared to existing tests and almost as powerful as the tests based on the optimal yet unknown in practice partition size in simulations as well as on a real data example

measuring and testing dependence by correlation of distances 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

k sample anderson darling tests abstract two k sample versions of an anderson darling rank statistic are proposed for testing the homogeneity of samples their asymptotic null distributions are derived for the continuous as well as the discrete case in the continuous case the asymptotic distributions coincide with the k 1 fold convolution of the asymptotic distribution for the anderson darling one sample statistic the quality of this large sample approximation is investigated for small samples through monte carlo simulation this is done for both versions of the statistic under various degrees of data rounding and sample size imbalances tables for carrying out these tests are provided and their usage in combining independent one or k sample anderson darling tests is pointed out the test statistics are essentially based on a doubly weighted sum of integrated squared differences between the empirical distribution functions of the individual samples and that of the pooled sample one weighting adjusts for the possibly different s

exact pdf equations and closure approximations for advective reactive transport mathematical models of advection reaction phenomena rely on advective flow velocity and bio chemical reaction rates that are notoriously random by using functional integral methods we derive exact evolution equations for the probability density function pdf of the state variables of the advection reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions these pdf equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly correlated random field our analytical solution is then used to investigate the accuracy and robustness of the recently proposed large eddy diffusivity led closure approximation 1 we find that the solution to the led based pdf equation which is exact for uncorrelated reaction rates is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

a kernel two sample test we propose a framework for analyzing and comparing distributions which we use to construct statistical tests to determine if two samples are drawn from different distributions our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel hilbert space rkhs and is called the maximum mean discrepancy mmd we present two distribution free tests based on large deviation bounds for the mmd and a third test based on the asymptotic distribution of this statistic the mmd can be computed in quadratic time although efficient linear time approximations are available our statistic is an instance of an integral probability metric and various classical metrics on distributions are obtained when alternative function classes are used in place of an rkhs we apply our two sample tests to a variety of problems including attribute matching for databases using the hungarian marriage method where they perform strongly excellent performance is also obtained when comparing distributions over graphs for which these are the first such tests

coverage adjusted entropy estimation summary data on neural coding have frequently been analyzed using information theoretic measures these formulations involve the fundamental and generally difficult statistical problem of estimating entropy we review briefly several methods that have been advanced to estimate entropy and highlight a method the coverage adjusted entropy estimator cae due to chao and shen that appeared recently in the environmental statistics literature this method begins with the elementary horvitz thompson estimator developed for sampling from a finite population and adjusts for the potential new species that have not yet been observed in the sample these become the new patterns or words in a spike train that have not yet been observed the adjustment is due to i j good and is called the good turing coverage estimate we provide a new empirical regularization derivation of the coverage adjusted probability estimator which shrinks the maximum likelihood estimate we prove that the cae is consistent and first order optimal with rate op 1 logn in the class of distributions with finite entropy variance and that within the class of distributions with finite qth moment of the log likelihood the good turing coverage estimate and the total probability of unobserved words converge at rate op 1 logn q we then provide a simulation study of the estimator with standard distributions and examples from neuronal data where observations are dependent the results show that with a minor modification the cae performs much better than the mle and is better than the best upper bound estimator due to paninski when the number of possible words m is unknown or infinite copyright q 2007 john wiley sons ltd

nonparametric k sample tests via dynamic slicing k sample testing problems arise in many scientific applications and have attracted statisticians attention for many years we propose an omnibus nonparametric method based on an optimal discretization aka slicing of continuous random variables in the test the novelty of our approach lies in the inclusion of a term penalizing the number of slices i e the resolution of the discretization so as to regularize the corresponding likelihood ratio test statistic an efficient dynamic programming algorithm is developed to determine the optimal slicing scheme asymptotic and finite sample properties such as power and null distribution of the resulting test statistic are studied we compare the proposed testing method with some existing well known methods and demonstrate its statistical power through extensive simulation studies as well as a real data example a dynamic slicing method for the one sample testing problem is further developed and studied under the same framework supplementary materials inclu

symmetric spaces and strongly isotropy irreducible spaces a strongly isotropy irreducible homogeneous space is a connected effective homogeneous space l h where h is a compact subgroup of a lie group l such that the identity component h 0 of h acts irreducibly on t h l h if we assume instead that h but not necessarily h 0 acts irreducibly on tetc l h then l h is called an isotropy irreducible homogeneous space these spaces are interesting because by schur s lemma they all admit an l invariant einstein metric and because they include the irreducible riemannian symmetric spaces as a sub family the nonsymmetric strongly isotropy irreducible homogeneous spaces were classified independently by manturov m1 2 3 wolf wol and later by kr mer kr with some minor omissions in the first two references in addition in wol a detailed study of the geometric properties of strongly isotropy irreducible spaces was carried out the isotropy irreducible spaces which are not strongly isotropy irreducible have recently been classified in wz3 in a different direction we present in this paper a direct proof of a beautiful observation of wall concerning a correspondence between the compact simply connected irreducible symmetric spaces on the one hand and the compact strongly isotropy irreducible quotients of the classical groups on the other hand cf the remarks added in proof in wol pp 147 148 this correspondence has a number of exceptions certain grassmannians and the isotropy irreducible space so 7 g 2 which is diffeomorphic to pt these exceptions appear at first sight to be even more mysterious than the correspondence but our main result is that there is an explanation using general principles of this correspondence as well as all its exceptions in particular this means that the classification of the non symmetric strongly isotropy irreducible quotients of the classical groups may be read off from that of the irreducible symmetric spaces consider a compact simply connected irreducible symmetric space g k where g is the identity component of the isometry group and k is the isotropy group

new evolution equations for the joint response excitation probability density function of stochastic solutions to first order nonlinear pdes by using functional integral methods we determine new evolution equations satisfied by the joint response excitation probability density function pdf associated with the stochastic solution to first order nonlinear partial differential equations pdes the theory is presented for both fully nonlinear and for quasilinear scalar pdes subject to random boundary conditions random initial conditions or random forcing terms particular applications are discussed for the classical linear and nonlinear advection equations and for the advection reaction equation by using a fourier galerkin spectral method we obtain numerical solutions of the proposed response excitation pdf equations these numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi element probabilistic collocation methods it is found that the response excitation approach yields accurate predictions of the statistical properties of the system in addition it allows to directly ascertain the tails of probabilistic distributions thus facilitating the assessment of rare events and associated risks the computational cost of the response excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the pde is subject to high dimensional random boundary or initial conditions the question of high dimensionality for evolution equations involving multidimensional joint response excitation pdfs is also addressed

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

lie symmetry analysis of the lundgren monin novikov equations for multi point probability density functions of turbulent flow the problem of turbulence statistics described by the lundgren monin novikov lmn hierarchy of integro differential equations is studied in terms of its group properties for this we perform a lie group analysis of a truncated lmn chain which presents the first two equations in an infinite set of integro differential equations for the multi point probability density functions pdf s of velocity a complete set of point transformations is derived for the one point pdf s and the independent variables sample space of velocity space and time for this purpose we use a direct method based on the canonical lie backlund operator due to the one way coupling of correlation equations the present results are complete in the sense that no additional symmetries exist for the first leading equation even if the full infinite hierarchy is considered

a consistent multivariate test of association based on ranks of distances summary we consider the problem of detecting associations between random vectors of any dimension few tests of independence exist that are consistent against all dependent alternatives we propose a powerful test that is applicable in all dimensions and consistent against all alternatives the test has a simple form is easy to implement and has good power

equitability mutual information and the maximal information coefficient how should one quantify the strength of association between two random variables without bias for relationships of a specific form despite its conceptual simplicity this notion of statistical equitability has yet to receive a definitive mathematical formalization here we argue that equitability is properly formalized by a self consistency condition closely related to data processing inequality mutual information a fundamental quantity in information theory is shown to satisfy this equitability criterion these findings are at odds with the recent work of reshef et al reshef dn et al 2011 science 334 6062 1518 1524 which proposed an alternative definition of equitability and introduced a new statistic the maximal information coefficient mic said to satisfy equitability in contradistinction to mutual information these conclusions however were supported only with limited simulation evidence not with mathematical arguments upon revisiting these claims we prove that the mathematical definition of equitability proposed by reshef et al cannot be satisfied by any nontrivial dependence measure we also identify artifacts in the reported simulation evidence when these artifacts are removed estimates of mutual information are found to be more equitable than estimates of mic mutual information is also observed to have consistently higher statistical power than mic we conclude that estimating mutual information provides a natural and often practical way to equitably quantify statistical associations in large datasets

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

estimation of entropy and mutual information we present some new results on the nonparametric estimation of entropy and mutual information first we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators the setup is related to grenander s method of sieves and places no assumptions on the underlying probability measure generating the data second we prove a converse to these consistency theorems demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information even given unlimited data this inconsistency theorem leads to an analytical approximation of the bias valid in surprisingly small sample regimes and more accurate than the usual 1 n formula of miller and madow over a large region of parameter space the two most practical implications of these results are negative 1 information estimates in a certain data regime are likely contaminated by bias even if bias corrected estimators are used and 2 confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods finally we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem by casting bias calculation problems in this approximation theory framework we obtain the best possible generalization of known asymptotic bias results more interesting this framework leads to an estimator with some nice properties the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions and this maximum error turns out to be surprisingly small we demonstrate the application of this new estimator on both real and simulated data

equivalence of distance based and rkhs based statistics in hypothesis testing we provide a unifying framework linking two classes of statistics used in two sample and independence testing on the one hand the energy distances and distance covariances from the statistics literature on the other maximum mean discrepancies mmd that is distances between embeddings of distributions to reproducing kernel hilbert spaces rkhs as established in machine learning in the case where the energy distance is computed with a semimetric of negative type a positive definite kernel termed distance kernel may be defined such that the mmd corresponds exactly to the energy distance conversely for any positive definite kernel we can interpret the mmd as energy distance with respect to some negative type semimetric this equivalence readily extends to distance covariance using kernels on the product space we determine the class of probability distributions for which the test statistics are consistent against all alternatives finally we investigate the performance of the family of distance kernels in two sample and independence tests we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels and that other choices from this family can yield more powerful tests

consistent nonparametric tests of independence three simple and explicit procedures for testing the independence of two multi dimensional random variables are described two of the associated test statistics l1 log likelihood are defined when the empirical distribution of the variables is restricted to finite partitions a third test statistic is defined as a kernel based independence measure two kinds of tests are provided distribution free strong consistent tests are derived on the basis of large deviation bounds on the test statistics these tests make almost surely no type i or type ii error after a random sample size asymptotically level tests are obtained from the limiting distribution of the test statistics for the latter tests the type i error converges to a fixed non zero value and the type ii error drops to zero for increasing sample size all tests reject the null hypothesis of independence if the test statistics become large the performance of the tests is evaluated experimentally on benchmark data