iterative time limited signal restoration the purpose of this paper is to show that time limited restoration of shift invariant blurred signals can be done by means of fixed point solutions of contraction mappings under rather general conditions for the distortion operator all our results are valid for multidimensional signals an application of these results to the iterative extrapolation of band limited discrete images is shown

the berry esseen bound for student s statistic we prove the berry esseen bound for the student t statistic under the assumption of a third moment this bound coincides up to an absolute constant with the classical berry esseen bound for the mean in general the distribution of the student statistic converges to the standard normal distribution function at least as fast as the distribution of the mean and sometimes faster for example rates of convergence can be proved if the underlying distribution is in the domain of attraction of the normal law

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

bootstrapping the student t statistic let x 1 x n n 1 be independent identically distributed random variables and consider the student t statistic t n based upon these random variables gine gotze and mason 1997 proved that t n converges in distribution to a standard normal random variable if and only if x is in the domain of attraction of a normal random variable and ex 0 we shall show that roughly the same holds true for the bootstrapped student t statistic t n in the process we shall disclose all the possible subsequential limiting laws of t n the proofs introduce a number of amusing tricks that may be of independent interest

precise asymptotics in the self normalized law of the iterated logarithm abstract let x x 1 x 2 be i i d nondegenerate random variables with zero means s n j 1 n x j and v n 2 j 1 n x j 2 we investigate the precise asymptotics in the law of the iterated logarithm for self normalized sums s n v n also for the maximum of self normalized sums max 1 k n s k v n when x belongs to the domain of attraction of the normal law

towards a universal self normalized moderate deviation this paper is an attempt to establish a universal moderate deviation for self normalized sums of independent and identically distributed random variables without any moment condition the exponent term in the moderate deviation is specified when the distribution is in the centered feller class an application to the law of the iterated logarithm is given

improving generalised estimating equations using quadratic inference functions summary generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified however under such misspecification the estimator of the regression parameter can be inefficient in this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter and that remains optimal even if the working correlation structure is misspecified the idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices a representation that is valid for the working correlations most commonly used both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations this approach also provides a chi squared inference function for testing nested models and a chi squared regression misspecification test furthermore the test statistic follows a chi squared distribution asymptotically whether or not the working correlation structure is correctly specified

darling erdős theorem for self normalized sums let x x1 x2 be i i d nondegenerate random variables sn nj 1xj and v2n nj 1x2j we investigate the asymptotic vspace 1pt behavior in distribution of the maximum of self normalized sums max1 k nsk vk and the law of the iterated logarithm for self normalized sums sn vn when x belongs to the domain of attraction of the normal law in this context we establish a darling erd s type theorem as well as an erd s feller kolmogorov petrovski type test for self normalized sums

self normalized processes exponential inequalities moment bounds and iterated logarithm laws self normalized processes arise naturally in statistical applications being unit free they are not affected by scale changes moreover self normalization often eliminates or weakens moment assumptions in this paper we present several exponential and moment inequalities particularly those related to laws of the iterated logarithm for self normalized random variables including martingales tail probability bounds are also derived for random variables bt 0 and at let yt exp at 2bt2 2 we develop inequalities for the moments of at bt or supt 0at bt log logbt 1 2 and variants thereof when eyt 1 or when yt is a supermartingale for all belonging to some interval our results are valid for a wide class of random processes including continuous martingales with at mt and b t sqrt langle m rangle t and sums of conditionally symmetric variables di with at i 1tdi and b t sqrt sum i 1 t d i 2 a sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in rm m 1 is also established another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self normalized by the square root of the sum of squares the key ingredient in this development is a new exponential supermartingale involving i 1tdi and i 1tdi2 a compact law of the iterated logarithm for self normalized martingales is also derived in this connection

a combinatorial particle swarm optimization for solving multi mode resource constrained project scheduling problems the particle swarm optimization pso has been widely used to solve continuous problems the discrete problems have just begun to be also solved by the discrete pso however the combinatorial problems remain a prohibitive area to the pso mainly in case of integer values in this paper we propose a combinatorial pso cpso algorithm that we take up challenge to use in order to solve a multi mode resource constrained project scheduling problem mrcpsp the results that have been obtained using a standard set of instances after extensive experiments prove to be very competitive in terms of number of problems solved to optimality by comparing average deviations and percentages of optima found our cpso algorithm outperforms the simulated annealing algorithm and it is close to the pso algorithm

an exponential nonuniform berry esseen bound for self normalized sums in this paper we shall derive exponential nonuniformberry esseen bounds in the central limit theorem for self normalized sums we show that the size of the error can be reduced considerably by replacing the usual standardization by self normalization in particular we establish the exponential bounds for the probability of the self normalized sums under the condition that the third moment is finite whereas an exponential moment assumption is required for the standardized sums applications to t statistics and the probabilities of moderate deviations of self normalized sums are also discussed

when is the student t statistic asymptotically standard normal let x x i i in mathbb n be independent identically distributed random variables it is shown that the student t statistic based upon the sample x i i 1 n is asymptotically n 0 1 if and only if x is in the domain of attraction of the normal law it is also shown that for any x if the self normalized sums u n sum i 1 n x i sum i 1 n x i 2 1 2 n in mathbb n are stochastically bounded then they are uniformly subgaussian that is sup n mathbb e exp lambda u n 2 infty for some lambda 0

a linear time varying system framework for noniterative discrete time band limited signal extrapolation a technique for the analysis and design of noniterative algorithms for discrete time band limited signal extrapolation is described the approach involves modeling the extrapolation process as a linear time varying ltv system or filter together with a previously developed fourier theory for ltv systems this model provides a frequency domain transfer function representation for the extrapolation system this representation serves as a powerful tool for characterizing and comparing the reconstruction properties of several well known least squares optimal algorithms for band limited extrapolation moreover the frequency domain setting provides a conceptually attractive means for understanding the process of extrapolation itself additionally a least squares approximation methodology for designing ltv filters for band limited extrapolation is developed the design technique is shown to unify a broad class of algorithms for extrapolating discrete time data and further to provide a means for designing new and improved extrapolation algorithms

continuation techniques for a certain class of analytic functions this paper deals with the numerical continuation problem of analytic functions g z given by g z int omega k z t f t dt where k defines a bounded operator on l 2 omega it is assumed that g is known over a finite segment a of the real line where g is to be sampled our continuation techniques emerged partly from a generalization of the landweber iteration we show that a certain discrete approximation of the proposed iterative technique yields two step noniterative algorithms for solving the continuation problem we also prove convergence of these approximations to the sought function g special emphasis on the continuation problem for the case k z t e 2 pi izt is given and some related numerical examples are presented the continuation problem when the known part of g z is contaminated with some noise is addressed and some techniques for solving this problem are also provided

necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix let xk i i 1 k 1 be a double array of nondegenerate i i d random variables and let pn n 1 be a sequence of positive integers such that n pn is bounded away from 0 and in this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry l n rm max 1 leq i 0 we show that the following six statements are equivalent bf i quad lim n to infty n 2 int limits n log n 1 4 infty left f n 1 x f n 1 left frac sqrt n log n x right right df x 0 bf ii quad n mathbb p left max 1 leq i j leq n x 1 i x 1 j geq sqrt n log n right to 0 quad rm as n to infty bf iii quad frac w mu n sqrt n log n stackrel mathbb p rightarrow 2 sigma 2 bf iv quad left frac n log n right 1 2 l n stackrel mathbb p rightarrow 2 bf v quad lim n rightarrow infty mathbb p left frac w mu n 2 n sigma 4 a n leq t right exp left frac 1 sqrt 8 pi e t 2 right infty t infty bf vi quad lim n rightarrow infty mathbb p left n l n 2 a n leq t right exp left frac 1 sqrt 8 pi e t 2 right infty t infty where mu mathbb e x 1 1 sigma 2 mathbb e x 1 1 mu 2 and an 4 log pn log log pn the equivalences between i ii iii and v assume that only mathbb e x 1 1 2 1 2 and n 1 alpha l n stackrel mathbb p rightarrow 0 1 2 alpha leq 1 respectively the current work thus provides weak limit analogues of the strong limit theorems of li and rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of ln obtained by jiang some open problems are also posed

non finite fisher information and homogeneity an em approach summary even simple examples of finite mixture models can fail to fulfil the regularity conditions that are routinely assumed in standard parametric inference problems many methods have been investigated for testing for homogeneity in finite mixture models for example but all rely on regularity conditions including the finiteness of the fisher information and the space of the mixing parameter being a compact subset of some euclidean space very simple examples where such assumptions fail include mixtures of two geometric distributions and two exponential distributions and more generally mixture models in scale distribution families to overcome these difficulties we propose and study an em test statistic which has a simple limiting distribution for examples in this paper simulations show that the em test has accurate type i errors and is more efficient than existing methods when they are applicable a real example is also included

precise asymptotics in complete moment convergence for self normalized sums let x x 1 x 2 be a sequence of nondegenerate i i d random variables with zero means set s n x 1 x n and v n 2 x 1 2 x n 2 ex 2i x x is a slowly varying function at we prove that for any b 2 d 2 b 1 email protected email protected b d 1 email protected n 2 logn d 2 bne s n v n 2i s n email protected n logn 1 b email protected n b d 1 b d 1 2 where n is a standard normal random variable

donsker s theorem for self normalized partial sums processes let x x1 x2 be a sequence of nondegenerate i i d random variables with zero means in this paper we show that a self normalized version of donsker s theorem holds only under the assumption that x belongs to the domain of attraction of the normal law a thus resulting extension of the arc sine law is also discussed we also establish that a weak invariance principle holds true for self normalized self randomized partial sums processes of independent random variables that are assumed to be symmetric around mean zero if and only if max1 j n xj vn p0 as n where v2n nj 1x2j

asymptotics for the random coefficient first order autoregressive model with possibly heavy tailed innovations consider a random coefficient ar 1 model x t n n x t 1 u t where n n 1 is a sequence of real numbers n n 1 is a sequence of random variables and the innovations of the model form a sequence of i i d random variables belonging to the domain of attraction of the normal law by imposing some weaker conditions the conditional least squares estimator of the autoregressive coefficient n is achieved and shown to be asymptotically normal by allowing the second moment of the innovation to be possibly infinite

exact convergence rate and leading term in central limit theorem for student s t statistic the leading term in the normal approximation to the distribution of student s t statistic is derived in a general setting with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law the form of the leading term is shown to have its origin in the way in which extreme data influence properties of the studentized sum the leading term approximation is used to give the exact rate of convergence in the central limit theorem up to order n 1 2 where n denotes sample size it is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points moreover the exact rate is identical to that for the non studentized sum when the latter is normalized for scale using a truncated form of variance but when the corresponding truncated centering constant is omitted examples of characterizations of convergence rates are also given it is shown that in some instances their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points

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

interpolation and the discrete papoulis gerchberg algorithm analyze the performance of an iterative algorithm similar to the discrete papoulis gerchberg algorithm and which can be used to recover missing samples in finite length records of band limited data no assumptions are made regarding the distribution of the missing samples in contrast with the often studied extrapolation problem in which the known samples are grouped together indeed it is possible to regard the observed signal as a sampled version of the original one and to interpret the reconstruction result studied as a sampling result the authors show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data they give upper and lower bounds for the error as a function of the number of iterations together with the signals for which the bounds are attained also they analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix from this analysis they infer the optimum value of the relaxation constant they also point out among all sampling sets with the same density those for which the convergence rate of the recovery algorithm is maximum or minimum for low pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer the worst convergence rates generally occur when the missing samples are contiguous

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

asymptotics for self normalized random products of sums of i i d random variables abstract let x x i i 1 be a sequence of independent and identically distributed positive random variables which is in the domain of attraction of the normal law and t n be a positive integer random variable denote s n i 1 n x i v n 2 i 1 n x i x 2 where x denotes the sample mean then we show that the self normalized random product of the partial sums k 1 t n s k k v t n is still asymptotically lognormal under a suitable condition about t n

a kind of complete moment convergence for self normalized sums let x x n n 1 be a sequence of independent and identically distributed i i d random variables with x in the domain of attraction of the normal law and ex 0 in this note we obtain a new kind of complete moment convergence for self normalized sums

discrete and continuous band limited signal extrapolation this paper has two main purposes first some new algorithms for the extrapolation of multidimensional band limited sequences are presented these algorithms extend those given in 1 in two ways 1 we do not impose any restrictions on either the shape of the region containing the set of samples of the two dimensional signal to be extrapolated or on the shape of its passband zone 2 we obtain a class of sequences which are band limited extrapolations of the given data including the minimum norm solution of 1 as a special case the second objective of out paper is to relate the discrete extrapolation problem to the continuous signal extrapolation problem 2 specifically we prove that the solution obtained by using our approach for the discrete extrapolation problem tends to the solution of the continuous problem when the sampling rate used in the known part of the continuous signal to be extrapolated approaches infinity

convergence of iterative nonexpansive signal reconstruction algorithms iterative algorithms for signal reconstruction from partial time and frequency domain knowledge have proven useful in a number of application areas in this paper a general convergence proof applicable to a general class of such iterative reconstruction algorithms is presented the proof relies on the concept of a nonexpansive mapping in both the time and frequency domains two examples studied in detail are time limited extrapolation equivalently band limited extrapolation and phase only signal reconstruction the proof of convergence for the phase only iteration is a new result obtained by this method of proof the generality of the approach allows the incorporation of nonlinear constraints such as time or space domain positivity or minimum and maximum value constraints finally the underrelaxed form of these iterations is also shown to converge even when the solution is not guaranteed to be unique

small time compactness and convergence behavior of deterministically and self normalised levy processes the first author s research was partially supported by arc grant dp0664603 the second author s research was partially supported by nsf grant dms 0503908

finding numerical solutions of diophantine equations using ant colony optimization the paper attempts to find numerical solutions of diophantine equations a challenging problem as there are no general methods to find solutions of such equations it uses the metaphor of foraging habits of real ants the ant colony optimization based procedure starts with randomly assigned locations to a fixed number of artificial ants depending upon the quality of these positions ants deposit pheromone at the nodes a successor node is selected from the topological neighbourhood of each of the nodes based on this stochastic pheromone deposit if an ant bumps into an already encountered node the pheromone is updated correspondingly a suitably defined pheromone evaporation strategy guarantees that premature convergence does not take place the experimental results which compares with those of other machine intelligence techniques validate the effectiveness of the proposed method

numerical solution of certain exponential and non linear diophantine systems of equations by using a discrete particle swarm optimization algorithm this article proposes the numerical solution of some non linear systems of diophantine equations including some exponential cases by its transformation into an optimization problem for the first example the implemented algorithm is able to find all the non negative integer roots in a given region of the domain of the objective function when compared to the analytic solution systems of different sizes are solved as an example for these their answers were found being them of excellent quality in terms of accuracy and precision even though there is no way of being certain that they are the only ones following matiyasevich theorem however and for systems of modest size the computation time increased considerably when the search space was increased