to how many simultaneous hypothesis tests can normal student s t or bootstrap calibration be applied in the analysis of microarray data and in some other contemporary statistical problems it is not uncommon to apply hypothesis tests in a highly simultaneous way the number n say of tests used can be much larger than the sample sizes n to which the tests are applied yet we wish to calibrate the tests so that the overall level of the simultaneous test is accurate often the sampling distribution is quite different for each test so there may not be an opportunity to combine data across samples in this setting how large can n be as a function of n before level accuracy becomes poor here we answer this question in cases where the statistic under test is of student s t type we show that if either the normal or student t distribution is used for calibration then the level of the simultaneous test is accurate provided that log n increases at a strictly slower rate than n 1 3 as n diverges on the other hand if bootstrap methods are used for calibration then we may choose log n almost as large as

applications of linking to hammerstein integral equations in this paper we establish some results for asymptotic linear hammerstein integral equations using the notion of linking both homotopical and co homological

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

functional coefficient autoregressive models in this article we propose a new class of models for nonlinear time series analysis investigate properties of the proposed model and suggest a modeling procedure for building such a model the proposed modeling procedure makes use of ideas from both parametric and nonparametric statistics a consistency result is given to support the procedure for illustration we apply the proposed model and procedure to several data sets and show that the resulting models substantially improve postsample multi step ahead forecasts over other models

efficient estimation of copula based semiparametric markov models this paper considers the efficient estimation of copula based semiparametric strictly stationary markov models these models are characterized by nonparametric invariant one dimensional marginal distributions and parametric bivariate copula functions where the copulas capture temporal dependence and tail dependence of the processes the markov processes generated via tail dependent copulas may look highly persistent and are useful for financial and economic applications we first show that markov processes generated via clayton gumbel and student s t copulas and their survival copulas are all geometrically ergodic we then propose a sieve maximum likelihood estimation mle for the copula parameter the invariant distribution and the conditional quantiles we show that the sieve mles of any smooth functional is root n consistent asymptotically normal and efficient and that their sieve likelihood ratio statistics are asymptotically chi square distributed monte carlo studies indicate that even for markov models generated via tail dependent copulas and fat tailed marginals our sieve mles perform very well

weak convergence for weighted empirical processes of dependent sequences in this paper we establish weak convergence theorems for weighted empirical processes of strong mixing p mixing and associated sequences we apply these results to obtain weak convergence of integral functionals of empirical processes and of mean residual life processes in reliability theory to carry out the proofs we develop two rosenthal type inequalities for strong mixing and associated sequences

compactness of the set of trajectories of the controllable system described by an affine integral equation in this paper the control system with integral constraint on the controls is studied it is assumed that the behavior of the system is described by an affine volterra integral equation which is nonlinear with respect to the state vector and is linear with respect to the control vector the closed ball of the space l p p 1 with radius m 0 and centered at the origin is chosen as the set of admissible control functions it is proved that the set of trajectories is a bounded closed precompact and hence compact subset of the space of continuous functions

em algorithms for ml factor analysis the details of em algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models the algorithm is essentially the same for both cases and involves only simple least squares regression operations the largest matrix inversion required is for aq q symmetric matrix whereq is the matrix of factors the example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation

robustness and accuracy of methods for high dimensional data analysis based on student s t statistic summary student s t statistic is finding applications today that were never envisaged when it was introduced more than a century ago many of these applications rely on properties e g robustness against heavy tailed sampling distributions that were not explicitly considered until relatively recently we explore these features of the t statistic in the context of its application to very high dimensional problems including feature selection and ranking the simultaneous testing of many different hypotheses and sparse high dimensional signal detection robustness properties of the t ratio are highlighted and it is established that those properties are preserved under applications of the bootstrap in particular bootstrap methods correct for skewness and therefore lead to second order accuracy even in the extreme tails indeed it is shown that the bootstrap and also the more popular but less accurate t distribution and normal approximations are more effective in the tails than towards the middle of the distribution these properties motivate new methods e g bootstrap based techniques for signal detection that confine attention to the significant tail of a statistic

self normalized cramer type large deviations for independent random variables let x 1 x 2 be independent random variables with zero means and finite variances it is well known that a finite exponential moment assumption is necessary for a cramer type large deviation result for the standardized partial sums in this paper we show that a cramer type large deviation theorem holds for self normalized sums only under a finite 2 th moment 0 1 in particular we show p s n v n x 1 x 1 o 1 1 x 2 d 2 n for 0 x d n where d n n i 1 ex 2 i 1 2 n i 1 e x i 2 1 2 and v n n i 1 x 2 i 1 2 applications to the studentized bootstrap and to the self normalized law of the iterated logarithm are discussed

nonlinear additive arx models abstract we consider in this article a class of nonlinear additive autoregressive models with exogenous variables for nonlinear time series analysis and propose two modeling procedures for building such models the procedures proposed use two backfitting techniques the ace and bruto algorithms to identify the nonlinear functions involved and use the methods of best subset regression and variable selection in regression analysis to determine the final model simulated and real examples are used to illustrate the analysis

approximation of the integral funnel of the urysohn type integral operator abstract the integral funnel of the closed ball of the space lp p 1 with radius r and centered at the origin under urysohn type integral operator is defined as the set of graphs of the images of all functions from given ball approximation of the integral funnel is considered the closed ball of the space lp p 1 with radius r is replaced by the set consisting of a finite number of functions the hausdorff distance between integral funnel and the set consisting of the sections of the graphs of images of a finite number of functions is evaluated it is proved that in the case of appropriate choosing of the discretization parameters the approximating sets converges to the integral funnel

a note on blow up solutions to some nonlinear volterra integral equations abstract in this paper we investigate certain nonlinear volterra integral equations with the power nonlinearity the basic results provide a criterion involving the kernel and the nonlinearity for the existence of the nontrivial and blow up solution

consistency of the maximum likelihood estimator for general hidden markov models consider a parametrized family of general hidden markov models where both the observed and unobserved components take values in a complete separable metric space we prove that the maximum likelihood estimator mle of the parameter is strongly consistent under a rather minimal set of assumptions as special cases of our main result we obtain consistency in a large class of nonlinear state space models as well as general results on linear gaussian state space models and finite state models a novel aspect of our approach is an information theoretic technique for proving identifiability which does not require an explicit representation for the relative entropy rate our method of proof could therefore form a foundation for the investigation of mle consistency in more general dependent and non markovian time series also of independent interest is a general concentration inequality for v uniformly ergodic markov chains

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

self normalized cramer type moderate deviations under dependence we establish a cramer type moderate deviation result for self normalized sums of weakly dependent random variables where the moment requirement is much weaker than the non self normalized counterpart the range of the moderate deviation is shown to depend on the moment condition and the degree of dependence of the underlying processes we consider three types of self normalization the equal block scheme the big block small block scheme and the interlacing scheme simulation study shows that the latter can have a better finite sample performance our result is applied to multiple testing and construction of simultaneous confidence intervals for ultra high dimensional time series mean vectors

minimum time with bounded energy minimum energy with bounded time necessary and sufficient conditions for the regularity of the minimum time function and minimum energy function for a control system with controls in l p 0 infty bbb r m and p ge 1 are given in terms of topological properties of the reachable sets in particular standard local controllability assumptions are sufficient to yield the continuity of both value functions for linear systems and p ge1 and the holder continuity of the minimum time function for nonlinear systems and p 1

on the continuity properties of the attainable sets of control systems with integral constraints on control in the paper the attainable sets of the control systems with integral constraints on control are studied it is assumed that the system is nonlinear with respect to the phase state vector and linear with respect to the control vector the admissible control functions are chosen in lp from the ball centered at the origin with given radius it is proved that the attainable sets of the system are continuous with respect to p

convergence rates in the strong law for bounded mixing sequences speed of convergence is studied for a marcinkiewicz zygmund strong law for partial sums of bounded dependent random variables under conditions on their mixing rate though mixing is also considered the most interesting result concerns absolutely regular sequences the results are applied to renewal theory to show that some of the estimates obtained by other authors on coupling are best possible another application sharpens a result for averaging a function along a random walk

a two sample test for high dimensional data with applications to gene set testing we proposed a two sample test for means of high dimensional data when the data dimension is much larger than the sample size the classical hotelling s t 2 test does not work for this large p small n situation the proposed test does not require explicit conditions on the relationship between the data dimension and sample size this offers much flexibility in analyzing high dimensional data an application of the proposed test is in testing significance for sets of genes which we demonstrate in an empirical study on a leukemia data set

nonlinearity and temporal dependence nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models we study this link using three measures of temporal dependence mixing mixing and mixing stationary diffusions that are mixing have mixing coefficients that decay exponentially to zero when they fail to be mixing they are still mixing and mixing but coefficient decay is slower than exponential for such processes we find transformations of the markov states that have finite variances but infinite spectral densities at frequency zero the resulting spectral densities behave like those of stochastic processes with long memory finally we show how state dependent poisson sampling alters the temporal dependence