on the gaussian approximation of convolutions under multidimensional analogues of s n bernstein s inequality conditions on etudie le taux d approximation de distribution de sommes de vecteurs aleatoires independants par des distributions gaussiennes correspondantes

infinite period bifurcation and global bifurcation branches branches of periodic solutions which exhibit the alternatives of the global hopf bifurcation theorem are calculated for two general systems of differential equations in the first a branch of solu

gaussian approximation of local empirical processes indexed by functions an extended notion of a local empirical process indexed by functions is introduced which includes kernel density and regression function estimators and the conditional empirical process as special cases under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of gaussian processes are established for such processes a compact law of the iterated logarithm lil is then inferred from the corresponding lil for the approximating sequence of gaussian processes a number of statistical applications of our results are indicated

some results on generalized residual entropy shannon s entropy plays an important role in the context of information theory since this entropy is not applicable to a system which has survived for some units of time the concept of residual entropy has been developed in the literature here we generalize the residual entropy by choosing a convex function f with f 1 0 in this paper some orderings and aging properties have been defined in terms of the generalized residual entropy function and their properties have been studied quite a few results available in the literature have been generalized and some distributions viz uniform exponential pareto power series finite range have been characterized through the generalized residual entropy

cumulative tsallis entropy for maximum ranked set sampling with unequal samples abstract in this paper we consider the information content of maximum ranked set sampling procedure with unequal samples mrssu in terms of tsallis entropy which is a non additive generalization of shannon entropy we obtain several results of tsallis entropy including bounds monotonic properties stochastic orders and sharp bounds under some assumptions we also compare the uncertainty and information content of mrssu with its counterpart in the simple random sampling srs data finally we develop some characterization results in terms of cumulative tsallis entropy and residual tsallis entropy of mrssu and srs data

characterization results based on non additive entropy of order statistics abstract we consider a non additive generalization of the shannon entropy measure using order statistics and show that this entropy measure characterizes the distribution function uniquely further we propose a residual non additive entropy measure for order statistics and prove a characterization result for that also

on the lower tail probabilities of some random series the behavior of tail probabilities p s r r 0 is investigated where s is a series s j z j generated by some sequence of positive numbers j and by a sequence z j of independent copies of a positive random variable z we present the exact asymptotic expression for p s r by means of laplace transform eexp s under weak assumptions on the behavior of the tail probabilities of z in the vicinity of zero the bounds of accuracy are also given and under weak supplementary smoothness conditions the asymptotic properties of the density of s are investigated

a quantile approach of tsallis entropy for order statistics abstract the quantile based entropy measures possess some unique properties than its distribution function approach in this article the concept of quantile based tsallis entropy is extended for order statistics we prove that the generalized quantile information between i th order statistics and parent random variable is distribution free also the quantile version of tsallis entropy for residual lifetime has been derived and its monotonicity property is studied further some characterization results based on the tsallis quantile entropy for residual and inactivity time for the series and parallel system have been studied

testing differential networks with applications to the detection of gene gene interactions model organisms and human studies have led to increasing empirical evidence that interactions among genes contribute broadly to genetic variation of complex traits in the presence of gene by gene interactions the dimensionality of the feature space becomes extremely high relative to the sample size this imposes a significant methodological challenge in identifying gene by gene interactions in the present paper through a gaussian graphical model framework we translate the problem of identifying gene by gene interactions associated with a binary trait d into an inference problem on the difference of two high dimensional precision matrices which summarize the conditional dependence network structures of the genes we propose a procedure for testing the differential network globally that is particularly powerful against sparse alternatives in addition a multiple testing procedure with false discovery rate control is developed to infer the specific structure of the differential network theoretical justification is provided to ensure the validity of the proposed tests and optimality results are derived under sparsity assumptions a simulation study demonstrates that the proposed tests maintain the desired error rates under the null and have good power under the alternative the methods are applied to a breast cancer gene expression study

joint testing and false discovery rate control in high dimensional multivariate regression multivariate regression with high dimensional covariates has many applications in genomic and genetic research in which some covariates are expected to be associated with multiple responses this paper considers joint testing for regression coefficients over multiple responses and develops simultaneous testing methods with false discovery rate control the test statistic is based on inverse regression and bias corrected group lasso estimates of the regression coefficients and is shown to have an asymptotic chi squared null distribution a row wise multiple testing procedure is developed to identify the covariates associated with the responses the procedure is shown to control the false discovery proportion and false discovery rate at a prespecified level asymptotically simulations demonstrate the gain in power relative to entrywise testing in detecting the covariates associated with the responses the test is applied to an ovarian cancer dataset to identify the microrna regulators that regulate protein expression

bifurcation in classical bipolar transistor oscillator circuits the results of a systematic asymptotic analysis of two classical oscillator circuits are presented in this paper both circuits are composed of passive components and a single bipolar transistor they are each described by a third order system of nonlinear ordinary differential equations that is singularly perturbed in the limit varepsilon to 0 here varepsilon is the inverse of the circuit s q the two timing method is used to construct the asymptotic behavior of the solutions as varepsilon to 0 these solutions are shown to depend upon amplitudes that evolve on a slow time scale according to amplitude equations these amplitude equations are analyzed and the results are physically interpreted

cluster sets for a generalized law of the iterated logarithm in banach spaces we identify the possible cluster sets for a general law of the iterated logarithm in the banach space setting and show that all the possible limit sets arise as cluster sets for some random vector in an arbitrary separable banach space this extends previous results obtained in finite dimensional euclidean spaces

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

gaussian approximation of suprema of empirical processes we develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of gaussian processes without taking the route of approximating empirical processes themselves in the sup norm we prove an abstract approximation theorem that is applicable to a wide variety of problems primarily in statistics especially the bound in the main approximation theorem is non asymptotic and the theorem does not require uniform boundedness of the class of functions the proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors the proof of which depends on an effective use of stein s method for normal approximation and some new empirical processes techniques we study applications of this approximation theorem to local empirical processes and series estimation in nonparametric regression where the classes of functions change with the sample size and are not donsker type importantly our new technique is able to prove the gaussian approximation for the supremum type statistics under considerably weak regularity conditions especially concerning the bandwidth and the number of series functions in those examples

moderate deviation probabilities for open convex sets nonlogarithmic behavior precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite and infinite dimensional settings our results are based on the existence of dominating points for these sets a related representation formula and asymptotics for the integral term in this formula

a uniform functional law of the logarithm for the local empirical process we prove a uniform functional law of the logarithm for the local empirical process to accomplish this we combine techniques from classical and abstract empirical process theory gaussian distributional approximation and probability on banach spaces the body of techniques we develop should prove useful to the study of the strong consistency of d variate kernel type nonparametric function estimators

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

rotating thermal convection neosteady and neoperiodic solutions this paper considers the secondary and cascading bifurcation of two dimensional steady and period thermal convection states in a rotating box previously developed asymptotic and perturbation methods that rely on the coalescence of two steady convection primary bifurcation points of the conduction state as the taylor number approaches a critical value are employed a multitime analysis is employed to construct asymptotic expansions of the solutions of the initial boundary value problem for the boussinesq theory the small parameter in the expansion is proportional to the deviation of the taylor number from its critical value to leading order the asymptotic expansion of the solution involves the mode amplitudes of the two interacting steady convection states the asymptotic analysis yields a first order system of two coupled ordinary differential equations for the slow time evolution of these amplitudes we investigate the steady states of these amplitude equations and their linearized stability these

two sample tests of high dimensional means for compositional data summary compositional data are ubiquitous in many scientific endeavours motivated by microbiome and metagenomic research we consider a two sample testing problem for high dimensional compositional data and formulate a testable hypothesis of compositional equivalence for the means of two latent log basis vectors we propose a test through the centred log ratio transformation of the compositions the asymptotic null distribution of the test statistic is derived and its power against sparse alternatives is investigated a modified test for paired samples is also considered simulations show that the proposed tests can be significantly more powerful than tests that are applied to the raw and log transformed compositions the usefulness of our tests is illustrated by applications to gut microbiome composition in obesity and crohn s disease

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

two sample test of high dimensional means under dependence summary the paper considers in the high dimensional setting a canonical testing problem in multivariate analysis namely testing the equality of two mean vectors we introduce a new test statistic that is based on a linear transformation of the data by the precision matrix which incorporates the correlations between the variables the limiting null distribution of the test statistic and the power of the test are analysed it is shown that the test is particularly powerful against sparse alternatives and enjoys certain optimality a simulation study is carried out to examine the numerical performance of the test and to compare it with other tests given in the literature the results show that the test proposed significantly outperforms those tests in a range of settings

two sample covariance matrix testing and support recovery in high dimensional and sparse settings in the high dimensional setting this article considers three interrelated problems a testing the equality of two covariance matrices and b recovering the support of and c testing the equality of and row by row we propose a new test for testing the hypothesis h 0 and investigate its theoretical and numerical properties the limiting null distribution of the test statistic is derived and the power of the test is studied the test is shown to enjoy certain optimality and to be especially powerful against sparse alternatives the simulation results show that the test significantly outperforms the existing methods both in terms of size and power analysis of a prostate cancer dataset is carried out to demonstrate the application of the testing procedures when the null hypothesis of equal covariance matrices is rejected it is often of significant interest to further investigate how they differ from each other motivated by applications in genomics we also consider recovering the support of and

gaussian graphical model estimation with false discovery rate control this paper studies the estimation of a high dimensional gaussian graphical model ggm typically the existing methods depend on regularization techniques as a result it is necessary to choose the regularized parameter however the precise relationship between the regularized parameter and the number of false edges in ggm estimation is unclear in this paper we propose an alternative method by a multiple testing procedure based on our new test statistics for conditional dependence we propose a simultaneous testing procedure for conditional dependence in ggm our method can control the false discovery rate fdr asymptotically the numerical performance of the proposed method shows that our method works quite well

some properties of cumulative tsallis entropy abstract the cumulative entropy is an information measure which is alternative to the differential entropy indeed the cumulative entropy of a random lifetime x can be expressed as the expectation of its mean inactivity time evaluated at x in this paper we propose a new generalized cumulative entropy based on tsallis entropy cte and its dynamic version dcte we study some properties and characterization results for this measure

on uncertainty and information properties of ranked set samples ranked set sampling is a sampling design which has a wide range of applications in industrial statistics economics and environmental and ecological studies etc it is well known that ranked set samples provide more fisher information than simple random samples of the same size about the unknown parameters of the underlying distribution in parametric inferences in this paper we consider the uncertainty and information content of ranked set samples in both perfect and imperfect ranking scenarios in terms of shannon entropy renyi and kullback leibler kl information measures it is proved that under these information measures ranked set sampling design performs better than its simple random sampling counterpart of the same size the information content is also a monotone function of the set size in ranked set sampling moreover the effect of ranking error on the information content of the data is investigated

estimates for the closeness of successive convolutions of multidimensional symmetric distributions let 1 2 be i i d random vectors in k with a common distribution i f i 1 2 let s n 1 n we investigate how small is the difference between s n and s n m in the case when i have symmetric distributions