statistical geometry in quantum mechanics a statistical model m is a family of probability distributions characterised by a set of continuous parameters known as the parameter space this possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square integrable functions more precisely by consideration of the square root density function we can regard m as a submanifold of the unit sphere s in a real hilbert space h therefore h embodies the state space of the probability distributions and the geometry of the given statistical model can be described in terms of the embedding of m in s the geometry in question is characterised by a natural riemannian metric the fisher rao metric thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting in particular we focus attention on the variance lower bounds for statistical estimation and establish generalisations of the classical cramer rao and bhattacharyya inequalities described in terms of the geometry of the underlying real hilbert space as a comprehensive illustration of the utility of the geometric framework the statistical model m is then specialised to the case of a submanifold of the state space of a quantum mechanical system this is pursued by introducing a compatible complex structure on the underlying real hilbert space which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real hilbert space geometry the application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the heisenberg uncertainty relations for canonically conjugate observables

existence of positive solutions for non positive higher order bvps abstract we shall provide conditions on non positive function f t u 1 u n 1 so that the boundary value problem has at least one positive solution then we shall apply this result to establish several existence theorems which guarantee the multiple positive solutions

on learning multicategory classification with sample queries consider the pattern recognition problem of learning multicategory classification from a labeled sample for instance the problem of learning character recognition where a category corresponds to an alphanumeric letter the classical theory of pattern recognition assumes labeled examples appear according to the unknown underlying pattern class conditional probability distributions where the pattern classes are picked randomly according to their a priori probabilities in this paper we pose the following question can the learning accuracy be improved if labeled examples are independently randomly drawn according to the underlying class conditional probability distributions but the pattern classes are chosen not necessarily according to their a priori probabilities we answer this in the affirmative by showing that there exists a tuning of the sub sample proportions which minimizes a loss criterion the tuning is relative to the intrinsic complexity of the bayes classifier as this complexity depends on the underlying probability distributions which are assumed to be unknown we provide an algorithm which learns the proportions in an on line manner utilizing sample querying which asymptotically minimizes the criterion in practice this algorithm may be used to boost the performance of existing learning classification algorithms by apportioning better sub sample proportions

positive solutions of difference equations with two point right focal boundary conditions abstract we consider the following boundary value problem formula formula where 1 p n 1 is fixed using a fixed point theorem for operators on a cone we develop criteria for the existence of two positive solutions of the boundary value problem in addition upper and lower bounds for these positive solutions are established for special cases we also include several examples to dwell upon the importance of the results obtained

mean square consistency of the variance estimator in steady state simulation output analysis in steady state simulation output analysis mean square consistency of the process variance estimator is important for a number of reasons one way to construct an asymptotically valid confidence interval around a sample mean is via construction of a consistent estimator of the process variance and a central limit theorem also if an estimator is consistent in the mean square sense a mean square error analysis is theoretically justified finally batch size selection is an open research problem in steady state output analysis and a mean square error analysis approach has been proposed in the literature to be valid the process variance estimators constructed must be consistent in the mean square sense in this paper we prove mean square consistency of the process variance estimator for the methods of batch means overlapping batch means standardized time series area and spaced batch means by rigorously computing the rate of decay of the variance of the process variance estimators asymptotic results for third and higher centered moments of the batch means and area variance estimators are also given along with central limit theorems

adjusted estimates and wald statistics for the ar 1 model with constant abstract adjusted profile likelihood of cox and reid 1993 journal of the royal statistical society b 55 467 471 for the autoregressive coefficient of the ar 1 model with constant yields an estimator which can be more accurate than the maximum likelihood estimator in small samples whether the process is stationary or a random walk and in the latter case also asymptotically a related estimator is asymptotically distributed compactly and symmetrically around zero under a random walk but is not consistent in general the adjusted wald statistic can follow the asymptotic standard normal distribution much better than the non adjusted under stationarity or a drifted random walk

simulation of stopped diffusions in this work we study standard euler updates for simulating stopped diffusions as an immediate application we discuss the computation of first exit times of diffusions from a domain we focus on one dimensional situations and show how the ideas for the simulation of killed diffusions can be adapted to this problem in particular we give a fully implementable algorithm to compute the first exit time from an interval numerically the brownian motion case is treated in detail and extensions to general diffusions are given special emphasis is given to numerical experiments for every ansatz we include numerical experiments confirming the conjectured accuracy of our methods our algorithm is of order one in a weak sense comparisons with other algorithms are shown results that are superior to those obtained with other methods are presented when approximating a first hitting time distribution the results obtained with our algorithm are much better than those achieved with other methods

statistical methods for eliciting probability distributions elicitation is a key task for subjectivist bayesians although skeptics hold that elicitation cannot or perhaps should not be done in practice it brings statisticians closer to their clients and subject matter expert colleagues this article reviews the state of the art reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information in a discussion of the elicitation process the first issue to address is what it means for an elicitation to be successful that is what criteria should be used our answer is that a successful elicitation faithfully represents the opinion of the person being elicited it is not necessarily true in some objectivistic sense and cannot be judged in that way we see that elicitation as simply part of the process of statistical modeling indeed in a hierarchical model at which point the likelihood ends and the pr

single observation unbiased priors this paper studies a class of default priors which we call single observation unbiased priors soup a prior for a parameter is a soup if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter we prove that under mild regularity conditions a default prior for a convolution parameter is noninformative in the sense of yielding a posterior inference invariant under amalgamation only if it is a soup therefore when amalgamation invariance is desirable as in our motivating example of performing imputation for census undercount soup is the only possible coherent noninformative prior for bayesian predictions that will be utilized under aggregation the use of soup also mutually calibrates bayesian and frequentist inferences for aggregates of convolution parameters across many small areas we describe approaches that identify soups in many common models in particular a constructive duality method that identifies soups in pairs of distribution families we introduce o completeness a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean uniqueness of the soup is determined by the o completeness of the dual family o completeness of a natural exponential family is implied by its completeness hence the diaconis ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation is a direct consequence of the completeness of such families for most of the examples we have examined the inverse of the variance function is the soup for the mean parameter of the corresponding family suggesting that hartigan s results on asymptotic unbiasedness can be generalized to some families with discrete parameters we also discuss a possible extension of berger s result on the inadmissibility of unbiased estimators as the nonexistence of soup can be a first step in establishing such inadmissibility

frequency synchronization in a random oscillator network we study the frequency synchronization of a randomly coupled oscillators by analyzing the continuum limit we obtain a sufficient condition for the mean field type synchronization we especially find that the critical coupling constant k becomes 0 in the random scale free network p k ensuremath propto k ensuremath ensuremath gamma if 2l ensuremath gamma ensuremath leqslant 3 numerical simulations in finite networks are consistent with this analysis

a one dimensional analysis for the probability of error of linear classifiers for normally distributed classes computing the probability of error is an important problem in evaluating classifiers when dealing with normally distributed classes this problem becomes intricate due to the fact that there is no closed form expression for integrating the probability density function in this paper we derive lower and upper bounds for the probability of error for a linear classifier where the random vectors representing the underlying classes obey the multivariate normal distribution the expression of the error is derived in the one dimensional space independently of the dimensionality of the original problem based on the two bounds we propose an approximating expression for the error of a generic linear classifier in particular we derive the corresponding bounds and the expression for approximating the error of fisher s classifier our empirical results on synthetic data including up to two hundred dimensional featured samples show that the computations for the error are extremely fast and quite accurate it differs from the actual error in at most e 0 0184340683 the scheme has also been successfully tested on real life data sets drawn from the uci machine learning repository

a probabilistic model for quadrature rules the approximate formula of quadrature rules is usually indicated as a bf x email protected i 1nw if x i where x i are the integration nodes and w i are the corresponding coefficients during last decades various cases of above type formula such as gauss quadrature rules newton cotes rules and so on have been introduced to somehow estimate x i and w i in this paper we propose a quadrature model based on a probabilistic approach to estimate the related nodes and coefficients for this purpose we apply the normal distribution as a special case and present an error analysis for the obtained probabilistic formulas some numerical examples are given in this way to show the efficiency of the proposed model

parametric simultaneous confidence bands for cumulative distributions from censored data this article describes existing methods and develops new methods for constructing simultaneous confidence bands for a cumulative distribution function our results are built on extensions of previous work by cheng and iles for two sided and one sided bands respectively cheng and iles used wald statistics with expected fisher information we consider three alternatives wald statistics with observed fisher information wald statistics with local information and likelihood ratio statistics we compare standard large sample approximate methods with simulation or bootstrap calibrated versions of the same methods for log location scale distributions with complete or failure type ii censoring the bootstrap methods have the correct coverage probability a simulation for the weibull distribution and time censored type i data shows that bootstrap methods provide coverage probabilities that are closer to nominal than those based on the usual large sample approximations we illustrate the methods with ex

triple positive solutions and dependence on higher order derivatives abstract in this paper we consider the lidstone boundary value problem y 2 m t f y t y 2 j t y 2 m 1 t 0 t 1 y 2 i 0 0 y 2 i 1 0 i m 1 where 1 m f 0 growth conditions are imposed on f and inequalities involving an associated green s function are employed which enable us to apply the leggett williams fixed point theorem to cones in ordered banach spaces this in turn yields the existence of at least three positive symmetric concave solutions the emphasis here is that f depends on higher order derivatives

performance limits of alphabet diversities for fir siso channel identification finite impulse responses fir of single input single output siso channels can be blindly identified from second order statistics of transformed data for instance when the channel is excited by binary phase shift keying bpsk minimum shift keying msk or quadrature phase shift keying qpsk inputs identifiability conditions are derived by considering that noncircularity induces diversity theoretical performance issues are addressed to evaluate the robustness of standard subspace based estimators with respect to these identifiability conditions then benchmarks such as asymptotically minimum variance amv bounds based on various statistics are presented some illustrative examples are eventually given where monte carlo experiments are compared to theoretical performances these comparisons allow to quantify limits to the use of the alphabet diversities for the identification of fir siso channels and to demonstrate the robustness of algorithms based on high order statistics

blind separation of independent sources for virtually any source probability density function the blind source separation bss problem consists of the recovery of a set of statistically independent source signals from a set of measurements that are mixtures of the sources when nothing is known about the sources and the mixture structure in the bss scenario of two noiseless real valued instantaneous linear mixtures of two sources an approximate maximum likelihood ml approach has been suggested in the literature which is only valid under certain constraints on the probability density function pdf of the sources in the present paper the expression for this ml estimator is reviewed and generalized to include virtually any source distribution an intuitive geometrical interpretation of the new estimator is also given in terms of the scatter plots of the signals involved an asymptotic performance analysis is then carried out yielding a closed form expression for the estimator asymptotic pdf simulations illustrate the behavior of the suggested estimator and show the accuracy of the asymptotic analysis in addition an extension of the method to the general bss scenario of more than two sources and two sensors is successfully implemented

diagnosis and impacts of non gaussianity of innovations in data assimilation most of the atmospheric and oceanic data assimilation da schemes rely on the best linear unbiased estimator blue which is sub optimal if errors of assimilated data are non gaussian thus calling for a full bayesian data assimilation this paper contributes to the study of the non gaussianity of errors in the observational space possible sources of non gaussianity range from the inherent statistical skewness and positiveness of some physical observables e g moisture chemical species the nonlinearity both of the data assimilation models and of the observation operators among others deviations from gaussianity can be justified from a priori hypotheses or inferred from statistical diagnostics of innovations observation minus background leading to consistency relationships between the error statistics from samples of observations and backgrounds as well as their specified error variances we evaluate some measures of the innovation non gaussianity such as the skewness kurtosis and negentropy under the assumption of additive errors and by relating statistical moments from both data errors and innovations we identify potential sources of the innovation non gaussianity these sources range from 1 univariate error non gaussianity 2 nonlinear correlations between errors 3 spatio temporal variability of error variances heteroscedasticity and 4 multiplicative noise observational and background errors are often assumed independent this leads to variance dependent bounds for the skewness and the kurtosis of errors from innovation statistics we assess the potential da impact of some scenarios of non gaussian errors this impact is measured through the mean square difference between the blue and the minimum variance unbiased estimator mvue obtained with univariate observations and background estimates in order to accomplish this we compute maximum entropy probability density functions pdfs of the errors constrained by the first four order moments these pdfs are then used to compute the bayesian posterior pdf and the mvue the referred impact is studied for a large range of statistical moments being higher for skewed innovations and growing in average with the skewness of data errors specially if the skewnesses have the same sign an application has been performed to the quality accepted ecmwf innovations of brightness temperatures of a set of high resolution infrared sounder hirs channels in this context the mvue has led in some extreme cases to a potential reduction of 20 60 of the posterior error variance as compared to the blue specially for extreme values of the innovations

controller design for discrete time stochastic processes with nonquadratic loss the problem of optimal controller synthesis for discrete time stochastic nonlinear processes is examined when the objective for design is specified as the expected value of a nonsymmetric nonquadratic loss functional it is necessary to relate the closed loop process dynamics to the stationary probability density function pdf because explicit relationships are available for only a few special cases a general solution to the problem is not possible in this work a technique for developing approximations of the optimal control law by application of a dynamics pdf approximation is presented numerical simulations illustrate application of the proposed technique

positive solutions of nonlinear functional difference equations abstract in this paper we apply a cone theoretic fixed point theorem and obtain sufficient conditions for the existence of positive solutions to some boundary value problems for a class of functional difference equations we consider analogues of sublinear or superlinear growth in the nonlinear terms

model selection through a statistical analysis of the global minimum of a weighted nonlinear least squares cost function this paper presents a model selection algorithm for the identification of parametric models that are linear in the measurements it is based on the mean and variance expressions of the global minimum of a weighted nonlinear least squares cost function the method requires the knowledge of the noise covariance matrix but does not assume that the true model belongs to the model set unlike the traditional order estimation methods available in literature the presented technique allows to detect undermodeling the theory is illustrated by simulations on signal modeling and system identification problems and by one real measurement example

spectral theory of second order vector difference equations abstract this paper is concerned with spectral problems of second order vector difference equations with two point boundary value conditions where the matrix valued coefficient of the leading term may be singular a concept of self adjointness of the boundary value conditions is introduced the self adjointness of the corresponding difference operator is discussed on a suitable admissible function space and fundamental spectral results are obtained the dual orthogonality of eigenfunctions is shown in a special case rayleigh s principles and the minimax theorems in two linear spaces are given as an application a comparison theorem for eigenvalues of two sturm liouville problems is presented

non parametric unconditional quantile estimation for efficiency analysis with an application to federal reserve check processing operations abstract this paper examines the technical efficiency of us federal reserve check processing offices over 1980 2003 we extend results from park et al park b simar l weiner c 2000 fdh efficiency scores from a stochastic point of view econometric theory 16 855 877 and daouia and simar daouia a simar l 2007 nonparametric efficiency analysis a multivariate conditional quantile approach journal of econometrics 140 375 400 to develop an unconditional hyperbolic quantile estimator of efficiency our new estimator is fully non parametric and robust with respect to outliers when used to estimate distance to quantiles lying close to the full frontier it is strongly consistent and converges at rate root n thus avoiding the curse of dimensionality that plagues data envelopment analysis dea estimators our methods could be used by policymakers to compare inefficiency levels across offices or by managers of individual offices to identify peer offices

bayesian inference in models with euclidean structures abstract when a parameter space is endowed with a euclidean structure there is a euclidean measure called the structural or geometric prior which is an obvious candidate to represent ignorance concerning an unknown parameter the rationale for using this prior is strengthened when the unknown parameter is the canonical parameter of an exponential family then the structural prior is the uniform prior for the canonical parameter it is shown also that euclidean structures can be combined in different ways with group structures the structural or geometric prior is then the product of the invariant prior for the group structure and the structural or geometric prior for the euclidean structure

sensitivity of the resilience of congested random networks to rolloff and offset in truncated power law degree distributions abstract random networks were generated with the random configuration model with prescribed truncated power law degree distributions parameterized by an exponent an offset and an exponential rolloff as a model of an attack each network had exactly one of its highest degree nodes removed with the result that in some cases one or more remaining nodes became congested with the reassignment of the load the congested nodes were then removed and the cascade failure process continued until all nodes were uncongested the ratio of the number of nodes of the largest remaining cluster to the number of nodes in the original network was taken to be a measure of the network s resiliency to highest degree node removal we found that the resiliency is sensitive to both rolloff and offset but not to cutoff in the degree distribution and that rolloff tends to decrease resiliency while offset tends to increase it

bootstrapping in controlled calibration experiments we consider the determination of an unknown quantity for example the concentration of a particular chemical in a given sample or samples using controlled calibration here several samples are prepared with concentrations chosen to cover a required range and these are used to establish the relationship between concentration and the measured response to an assay method this relationship is then used to estimate the concentration in the unknown samples from their measured responses confidence intervals for the estimated concentrations can usually be calculated by inverting a prediction interval but in some situations this method becomes intractable we explore the use of the bootstrap as an alternative in linear nonlinear and multivariate controlled calibration using both simulation and real datasets from the field of immunoassay we also discuss the alternatives afforded by replication of the design points the bootstrap is found to be comparable to the standard method in simple situations and is ea

technical communique estimation of the risk for an unstable behaviour of feedback systems in the presence of nonlinear distortions a criterion to verify experimentally the stability of a nonlinear system captured in a feedback loop is proposed the basic idea is to split the output power in coherent linearly related to the input and noncoherent power the remaining power a nonlinear power gain measuring the sensitivity of the noncoherent output power to input variations is introduced using the small gain theorem it is possible to check the local stability of the feedback for the actual class of excitation signals the risk is estimated that the feedback system would become unstable for another realization of the experiment using extreme value statistics

regression to the mean for nonnormal populations summary regression to the mean is the term used to describe the effect by which individuals selected on the basis of an initial measurement which is particularly high or low compared with other members of the population have upon remeasurement values nearer to the population mean for the normal distribution simple expressions for estimates of the magnitude of this effect are available many biological variables however are not normally distributed estimates for regression to the mean for nonnormal data are given based on edgeworth series and saddlepoint approximations

multiple solutions for higher order difference equations abstract this paper establishes the existence of twin nonnegative solutions to i conjugate and ii n p higher order discrete boundary value problems

triple positive solutions of conjugate boundary value problems ii we consider the following boundary value problems 1 n py n a t h y n 2 t 0 1 y i 0 0 0 i p 1 y i 1 0 0 i n p 1 and 1 n p ny f k y y n 1y n 2 0 k m iy 0 0 i p 1 iy m p 1 0 0 i n p 1 where 1 p n 1 is fixed by employing fixed point theorems for operators on a cone existence criteria are developed for multiple at least three positive solutions of the boundary value problems as an application we also establish the existence of radial solutions of certain partial difference equations several examples are included to dwell upon the importance of the results obtained

expansion of moments of bernstein polynomials abstract in this paper we give expansions of moments and absolute moments of the bernstein polynomials the results are applied to derive a new expansion of the bernstein polynomials in increasing powers of 1 n and a voronovskaya type theorem

eigenvalues and eigenfunctions of discrete conjugate boundary value problems abstract we consider the following boundary value problem 1 n p n y f k y y n 1 y n 2 0 k m i y 0 0 0 i p 1 i y m n i 0 0 i n p 1 where 1 p n 1 is fixed and 0 a characterization of the values of is carried out so that the boundary value problem has a positive solution next for 1 criteria are developed for the existence of two positive solutions of the boundary value problem in addition for particular cases we also offer upper and lower bounds for these positive solutions several examples are included to dwell upon the importance of the results obtained

kuramoto model with frequency degree correlations on complex networks we study the kuramoto model on complex networks in which natural frequencies of phase oscillators and the vertex degrees are correlated using the annealed network approximation and numerical simulations we explore a special case in which the natural frequencies of the oscillators and the vertex degrees are linearly coupled we find that in uncorrelated scale free networks with the degree distribution exponent 2l ensuremath gamma l3 the model undergoes a first order phase transition while the transition becomes second order at ensuremath gamma g3 if ensuremath gamma 3 the phase synchronization emerges as a result of a hybrid phase transition that combines an abrupt emergence of synchronization as in first order phase transitions and a critical singularity as in second order phase transitions the critical fluctuations manifest themselves as avalanches in the synchronization process comparing our analytical calculations with numerical simulations for erd ifmmode mbox h o else h o fi s r enyi and scale free networks we demonstrate that the annealed network approach is accurate if the mean degree and size of the network are sufficiently large we also study analytically and numerically the kuramoto model on star graphs and find that if the natural frequency of the central oscillator is sufficiently large in comparison to the average frequency of its neighbors then synchronization emerges as a result of a first order phase transition this shows that oscillators sitting at hubs in a network may generate a discontinuous synchronization transition

generating dithering noise for maximum likelihood estimation from quantized data the quantization theorem i qt i implies that the likelihood function can be reconstructed from quantized sensor observations given that appropriate dithering noise is added before quantization we present constructive algorithms to generate such dithering noise the application to maximum likelihood estimation mle is studied in particular in short dithering has the same role for amplitude quantization as an anti alias filter has for sampling in that it enables perfect reconstruction of the dithered but unquantized signal s likelihood function without dithering the likelihood function suffers from a kind of aliasing expressed as a counterpart to poisson s summation formula which makes the exact mle intractable to compute with dithering it is demonstrated that standard mle algorithms can be re used on a smoothed likelihood function of the original signal and statistically efficiency is obtained the implication of dithering to the cramer rao lower bound crlb is studied and illustrative examples are provided

correcting for omitted variables and measurement error bias in regression with an application to the effect of lead on iq abstract ordinary least squares ols regression estimates are biased in general when relevant variables are omitted from the regression equation or when included variables are measured with error the errors in variables bias can be corrected using auxiliary information about unobservable measurement errors in this article we demonstrate how auxiliary information can also be used to correct for omitted variables bias we illustrate our methods with an application to four published studies of the effect on iq of childhood exposure to lead each of the published studies used ols methods or equivalent none of the studies includes a father iq variable and none accounts for the biasing effect of measurement error in the right side variables for each of the studies we demonstrate that bias corrected estimates of the effect of lead on iq are much reduced in size and are not significantly different from 0 our methods can be used in other applications involving omitted variables or errors of measurement

edge detection in ultrasound imagery using the instantaneous coefficient of variation the instantaneous coefficient of variation icov edge detector based on normalized gradient and laplacian operators has been proposed for edge detection in ultrasound images in this paper the edge detection and localization performance of the icov squared icovs detector are examined first a simplified version of the icovs detector the normalized gradient magnitude squared is scrutinized in order to reveal the statistical performance of edge detection and localization in speckled ultrasound imagery both the probability of detection and the probability of false alarm are evaluated for the detector edge localization is characterized by the position of the peak and the 3 db width of the detector response then the speckle edge response of the icovs as applied to a realistic edge model is studied through theoretical analysis we reveal the compensatory effects of the normalized laplacian operator in the icov edge detector for edge localization error an icov based edge detection algorithm is implemented in which the icov detector is embedded in a diffusion coefficient in an anisotropic diffusion process experiments with real ultrasound images have shown that the proposed algorithm is effective in extracting edges in the presence of speckle quantitatively the icovs provides a lower localization error and qualitatively a dramatic improvement in edge detection performance over an existing edge detection method for speckled imagery

discrete polynomial interpolation green s functions maximum principles error bounds and boundary value problems abstract we construct discrete interpolating polynomials provide explicit representations of discrete green s functions give several identities and inequalities for these green s functions use the explicit forms of the interpolating polynomials and that of green s functions to establish several maximum principles further we obtain error bounds in discrete polynomial interpolation and use them to study existence and uniqueness of the discrete boundary value problems these bounds are also used to provide sufficient conditions for the convergence of the picard s method the approximate picard s method quasilinearization and the approximate quasilinearization the monotone convergence of the picard s iterative method is also analysed

the analysis of ring recovery data using random effects we show how random terms describing both yearly variation and overdispersion can easily be incorporated into models for mark recovery data through the use of bayesian methods for recovery data on lapwings we show that the incorporation of the random terms greatly improves the goodness of fit omitting the random terms can lead to overestimation of the significance of weather on survival and overoptimistic prediction intervals in simulations of future population behavior random effects models provide a natural way of modeling overdispersion which is more satisfactory than the standard classical approach of scaling up all standard errors by a uniform inflation factor we compare models by means of bayesian p values and the deviance information criterion dic

error bounds for numerical inversion of a probability characteristic function given the characteristic function of a continuous real random variable it is possible to construct a finite fourier series that approximates the density over any specified finite interval this series converges in mean square if the density decays faster than x 1 and the characteristic function decays faster than t 1 2 it converges in minimax if the density decays faster than x 1 and the characteristic function decays faster than t 1 it is also possible to construct a finite fourier series for the distribution function that has arbitrarily small maximum error over any finite interval provided that the characteristic function decays as t beta for some beta 0 and that either the distribution or the density decays to 0 or 1 as appropriate as x alpha for some alpha 1 these conditions are weak enough to include most continuous distributions of practical interest explicit error bounds can be obtained directly from the characteristic function without prior knowledge of either the distribution or density making these series useful for numerical inversion

on the method of logarithmic cumulants for parametric probability density function estimation parameter estimation of probability density functions is one of the major steps in the area of statistical image and signal processing in this paper we explore several properties and limitations of the recently proposed method of logarithmic cumulants molc parameter estimation approach which is an alternative to the classical maximum likelihood ml and method of moments mom approaches we derive the general sufficient condition for a strong consistency of the molc estimates which represents an important asymptotic property of any statistical estimator this result enables the demonstration of the strong consistency of molc estimates for a selection of widely used distribution families originating from but not restricted to synthetic aperture radar image processing we then derive the analytical conditions of applicability of molc to samples for the distribution families in our selection finally we conduct various synthetic and real data experiments to assess the comparative properties applicability and small sample performance of molc notably for the generalized gamma and k families of distributions supervised image classification experiments are considered for medical ultrasound and remote sensing sar imagery the obtained results suggest that molc is a feasible and computationally fast yet not universally applicable alternative to mom molc becomes especially useful when the direct ml approach turns out to be unfeasible

explosive synchronization enhanced by time delayed coupling this paper deals with the emergence of explosive synchronization in scale free networks by considering the kuramoto model of coupled phase oscillators the natural frequencies of oscillators are assumed to be correlated with their degrees and a time delay is included in the system this assumption allows enhancing the explosive transition to reach a synchronous state we provide an analytical treatment developed in a star graph which reproduces results obtained in scale free networks our findings have important implications in understanding the synchronization of complex networks since the time delay is present in most real world complex systems due to the finite speed of the signal transmission over a distance

explosive synchronization with partial degree frequency correlation networks of kuramoto oscillators with a positive correlation between the oscillators frequencies and the degree of their corresponding vertices exhibit so called explosive synchronization behavior which is now under intensive investigation here we study and discuss explosive synchronization in a situation that has not yet been considered namely when only a part typically a small part of the vertices is subjected to a degree frequency correlation our results show that in order to have explosive synchronization it suffices to have degree frequency correlations only for the hubs the vertices with the highest degrees moreover we show that a partial degree frequency correlation does not only promotes but also allows explosive synchronization to happen in networks for which a full degree frequency correlation would not allow it we perform a mean field analysis and our conclusions were corroborated by exhaustive numerical experiments for synthetic networks and also for the undirected and unweighed version of a typical benchmark biological network namely the neural network of the worm caenorhabditis elegans the latter is an explicit example where partial degree frequency correlation leads to explosive synchronization with hysteresis in contrast with the fully correlated case for which no explosive synchronization is observed

existence of positive solutions of bvps for second order nonlinear difference systems this paper is concerned with the following system d 2u i k f i k u 1 k u 2 k u n k 0 k 0 t i 1 2 nwith the dirichlet boundary conditionu i 0 u i t 2 0 i 1 2 n some new results are obtained for the existence and multiplicity of positive solutions to the above system by using nonlinear alternative of leray schauder type krasnosel skii s fixed point theorem in a cone and leggett williams fixed point theorem in particular it proves that the above system has n positive solutions under suitable conditions where n is an arbitrary integer

locally weighted polynomial regression parameter choice and application to forecasts of the great salt lake 1 relationships between hydrologic variables are often nonlinear usually the functional form of such a relationship is not known a priori a multivariate nonparametric regression methodology is provided here for approximating the underlying regression function using locally weighted polynomials locally weighted polynomials consider the approximation of the target function through a taylor series expansion of the function in the neighborhood of the point of estimate cross validatory procedures for the selection of the size of the neighborhood over which this approximation should take place and for the order of the local polynomial to use are provided and shown for some simple situations the utility of this nonparametric regression approach is demonstrated through an application to nonparametric short term forecasts of the biweekly great salt lake volume blind forecasts up to 1 year in the future using the 1847 2004 time series of the great salt lake are presented

the variation of non parametric estimates in closed loop it is known that non parametric transfer function estimates in closed loop often have infinite variance we characterise the probability density function of such estimates under the assumption that the corresponding closed loop system estimate has complex normal distribution in the frequency domain the probability density function can be described as a horseshoe encircling the inverse of the controller with a global maximum on the line between the true value and the inverse of the controller the expected value of the absolute value of such estimates is finite and we propose it as a measure of variation we also derive and discuss new expressions for the variance when an exclusion zone is introduced around the singularity

adaptive threshold estimation via extreme value theory determining a detection threshold to automatically maintain a low false alarm rate is a challenging problem in a number of different applications the underlying parametric assumptions of most automatic target detection algorithms are invalid therefore thresholds derived using these incorrect distribution assumptions do not produce desirable results when applied to real sensor data monte carlo methods for threshold determination work well but tend to perform poorly when targets are present in order to mitigate these effects we propose an algorithm using extreme value theory through the use of the generalized pareto distribution gpd and a kolmogorov smirnov statistical test unlike previous work based on gpd estimates this algorithm incorporates a way to adaptively maintain low false alarm rates in the presence of targets both synthetic and real world detection results demonstrate the usefulness of this algorithm

fractional bayes factors for model comparison bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves however there are difficulties with this approach when prior information about the parameters of the various models is weak partial bayes factors offer a resoiution of the problem by setting aside part of the data as a training sampie the training sampie is used to obtain an initiai informative posterior distribution of the parameters in each model model comparison is then based on a bayes factor calculated from the remaining data properties of partial bayes factors are discussed particularly in the context of weak prior information and they are found to have advantages over other proposed methods of model comparison a new variant of the partial bayes factor the fractional bayes factor is advocated on grounds of consistency simplicity robustness and coherence

multiple positive solutions of a discrete difference system this paper is concerned with the following system d 2u 1 k f 1 k u 1 k u 2 k 0 k 0 t d 2u 2 k f 2 k u 1 k u 2 k 0 k 0 t u 1 0 u 1 t 2 0 u 2 0 u 2 t 2 by using leggett williams fixed point theorem sufficient conditions are obtained for the existence of three positive solutions to the above system

double positive solutions of n p boundary value problems for higher order difference equations abstract we shall provide existence criteria for double positive solutions of the n p boundary value problem n y f k y y n 2 y g k y y n 1 y n 1 k n i y 0 0 0 i n 2 p y n n p 0 where n 2 and 0 p n 1 is fixed upper and lower bounds for the two positive solutions are also established for a particular boundary value problem when n 2 several examples are included to dwell upon the importance of the results obtained

a sinusoidal contrast function for the blind separation of statistically independent sources the authors propose a new solution to the blind separation of sources bss based on statistical independence in the two dimensional 2 d case we prove that under the whiteness constraint the fourth order moment based approximation of the marginal entropy me cost function yields a sinusoidal objective function therefore we can minimize it by simply estimating its phase we prove that this estimator is consistent for any source distribution in addition such results are useful for interpreting other algorithms such as the cumulant based independent component analysis cubica and the weighted approximate maximum likelihood waml or weighted estimator we based on the waml we provide a general unifying form for several previous approximations to the me contrast the bias and the variance of this estimator have been included finally simulations illustrate the good consistency convergence and accuracy of the proposed method

inference in semiparametric dynamic models for binary longitudinal data this article deals with the analysis of a hierarchical semiparametric model for dynamic binary longitudinal responses the main complicating components of the model are an unknown covariate function and serial correlation in the errors existing estimation methods for models with these features are of o n3 where n is the total number of observations in the sample therefore nonparametric estimation is largely infeasible when the sample size is large as in typical in the longitudinal setting here we propose a new o n markov chain monte carlo based algorithm for estimation of the nonparametric function when the errors are correlated thus contributing to the growing literature on semiparametric and nonparametric mixed effects models for binary data in addition we address the problem of model choice to enable the formal comparison of our semiparametric model with competing parametric and semiparametric specifications the performance of the methods is illustrated with detailed studies involving simul

the slope of a straight line a phony estimator plus comments by m werman this note discusses an estimator for the slope of a straight line as proposed by werman and geyzel fitting a second degree curve in the presence of error ibid vol 17 no 2 p 207 11 1995 by computing its frequency distribution the authors demonstrate that the estimator is a unreliable it possesses neither a first nor a second moment application leads therefore to completely erratic outcomes werman replies that the problem is well known and describes the methods taken to overcome it

positive solutions and nonlinear eigenvalue problems for third order difference equations first solutions that lie in a cone intersected with an annular type region are obtained for the third order difference equation 3u t a t f u t 0 2 t t 2 satisfying the boundary conditions u 0 u 1 u t 3 0 in the cases that f is either superlinear or sublinear the second part is devoted to determining values of for which there exist positive solutions of 3u t a t f u t 0 2 t t 2 satisfying the boundary conditions u 0 u 1 u t 3 0 when both limx 0 f x x and limx f x x exist as positive real numbers

conditional power with fisher s least significant difference procedure summary conditional power calculations are very useful for assessing the likelihood of an ultimately statistically significant result given the current data we show how to calculate conditional power in a multi armed trial using the classic fisher least significant difference lsd procedure we obtain upper bounds on the probabilities of type 1 and type 2 errors analogous to those in two armed trials and show that these bounds are equalities under continuous monitoring

asymptotic properties of u processes under long range dependence let x i i geq 1 be a stationary mean zero gaussian process with covariances rho k pe x 1 x k 1 satisfying rho 0 1 and rho k k d l k where d is in 0 1 and l is slowly varying at infinity consider the u process u n r r in i defined as u n r frac 1 n n 1 sum 1 leq i neq j leq n 1 g x i x j leq r where i is an interval included in rset and g is a symmetric function in this paper we provide central and non central limit theorems for u n they are used to derive the asymptotic behavior of the hodges lehmann estimator the wilcoxon signed rank statistic the sample correlation integral and an associated scale estimator the limiting distributions are expressed through multiple wiener ito integrals

transition to synchrony in degree frequency correlated sakaguchi kuramoto model we investigate transition to synchrony in degree frequency correlated sakaguchi kuramoto sk model on complex networks both analytically and numerically we analytically derive self consistent equations for group angular velocity and order parameter for the model in the thermodynamic limit using the self consistent equations we investigate transition to synchronization in sk model on uncorrelated scale free sf and erd s renyi er networks in detail depending on the degree distribution exponent of sf networks and phase frustration parameter the population undergoes from first order transition explosive synchronization es to second order transition and vice versa in er networks transition is always second order irrespective of the values of the phase lag parameter we observe that the critical coupling strength for the onset of synchronization is decreased by phase frustration parameter in case of sf network where as in er network the phase frustration delays the onset of synchronization extensive numerical simulations using sf and er networks are performed to validate the analytical results an analytical expression of critical coupling strength for the onset of synchronization is also derived from the self consistent equations considering the vanishing order parameter limit

finding an unknown number of multivariate outliers we use the forward search to provide robust mahalanobis distances to detect the presence of outliers in a sample of multivariate normal data theoretical results on order statistics and on estimation in truncated samples provide the distribution of our test statistic we also introduce several new robust distances with associated distributional results comparisons of our procedure with tests using other robust mahalanobis distances show the good size and high power of our procedure we also provide a unification of results on correction factors for estimation from truncated samples

voxel based morphometry should not be used with imperfectly registered images abstract john ashburner and karl friston 2000 introduced a standardized method of voxel based morphometry vbm for comparisons of local concentrations of gray matter between two groups of subjects segmented images of gray matter from grossly normalized high resolution images are smoothed and their group differences analyzed by the now conventional voxelwise worsley approach to gaussian random fields of differences this comment concerns an unfortunate interaction between the algorithm s spatial normalization and voxelwise comparison steps whereby several obvious quantitative confounds are injected at the core of the inference engine the authors put forward specifically the statistics of the resulting voxelwise comparisons are uninformative about group differences wherever the spatial normalization algorithm has failed to register on any robustly appearing image gradient the method of ashburner and friston is defensible only far from all image gradients

invariant bayesian inference in regression models that is robust against the jeffreys lindley s paradox we obtain the prior and posterior probability of a nested regression model as the hausdorff integral of the prior and posterior on the parameters of an encompassing linear regression model over a lower dimensional set that represents the nested model the hausdorff integral is invariant and therefore avoids the borel kolmogorov paradox basing priors and prior probabilities of nested regression models on the prior on the parameters of an encompassing linear regression model reduces the discrepancies between classical and bayesian inference like the jeffreys lindley s paradox we illustrate the analysis with examples of linear restrictions i e a linear regression model and non linear restrictions i e a cointegration and an autoregressive moving average model on the parameters of an encompassing linear regression model

high order wavelet packets and cumulant field analysis in many applications it is necessary to characterize the statistical properties of the wavelet wavelet packet coefficients of a stationary random signal in particular in a stationary non gaussian noise scenario it may be useful to determine the high order statistics of the wavelet packet coefficients in this work we prove that this task may be performed through multidimensional filter banks in particular we show how the cumulants of the m band wavelet packet coefficients of a strictly stationary signal are derived from those of the signal and we provide scale recursive decomposition and reconstruction formulae to compute these cumulants high order wavelet packets associated with these multidimensional filter banks are presented along with some of their properties it is proved that under some conditions these high order wavelet packets allow us to define frame multiresolution analyses finally the asymptotic normality of the coefficients is studied by showing the geometric decay of their polyspectra cumulants of order greater than two with respect to the resolution level

some new tests of the power law process the power law process plp is often used to model failure data from repairable systems and its adequacy is tested with goodness of fit tests such as the cramer von mises test no single test however can be sensitive to all possible deviations from the plp exact tests that are powerful against some departures from the plp of particular interest to practitioners are derived heuristically the computation of their p values is described and their use is illustrated with examples one test is a modified version of the laplace test for trend another a modification of greenwood s test for clustering it is shown that these tests can be derived as score tests which ensures that they are powerful for large samples

on a random area variable arising in discrete time queues and compact directed percolation a well known discrete time single server queueing system with mean arrival rate and mean departure rate is considered from the perspective of the area a swept out by the queue occupation process during a busy period we determine the exact form of the tail of the distribution pr a x in particular we show that pr a x cx 1 4 exp dx1 2 for all 1 where and expressions for c and d are given for the critical case 1 we show that pr a x c x 1 3 with c also given a simple mapping used in the derivation establishes a connection with compact directed percolation on a square lattice as a corollary therefore we are also able to specify the large area asymptotic behaviour of this model at all points in the phase diagram this extends previous scaling results which are only valid close to the percolation threshold

monte carlo state space likelihoods by weighted posterior kernel density estimation maximum likelihood estimation and likelihood ratio tests for nonlinear non gaussian state space models require numerical integration for likelihood calculations several methods including monte carlo mc expectation maximization mc likelihood ratios direct mc integration and particle filter likelihoods are inefficient for the motivating problem of stage structured population dynamics models in experimental settings an mc kernel likelihood mckl method is presented that estimates classical likelihoods up to a constant by weighted kernel density estimates of bayesian posteriors mckl is derived by using bayesian posteriors as importance sampling densities for unnormalized kernel smoothing integrals mc error and mode bias due to kernel smoothing are discussed and two methods for reducing mode bias are proposed zooming in on the maximum likelihood parameters using a focused prior based on an initial estimate and using a posterior cumulant based approximation of mode bias a simulated example show

posterior probability intervals for wavelet thresholding summary we use cumulants to derive bayesian credible intervals for wavelet regression estimates the first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions these powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale hence a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set johnson transformations then yield the credible intervals themselves simulations show that these intervals have good coverage rates even when the underlying function is inhomogeneous where standard methods fail in the case where the curve is smooth the performance of our intervals remains competitive with established nonparametric regression methods

fitting a second degree curve in the presence of error this correspondence presents a statistically sound simple and fast method to estimate the parameters of a second degree curve from a set of noisy points that originated from the curve

information geometric measure for neural spikes this study introduces information geometric measures to analyze neural firing patterns by taking not only the second order but also higher order interactions among neurons into account information geometry provides useful tools and concepts for this purpose including the orthogonality of coordinate parameters and the pythagoras relation in the kullback leibler divergence based on this orthogonality we show a novel method for analyzing spike firing patterns by decomposing the interactions of neurons of various orders as a result purely pairwise triple wise and higher order interactions are singled out we also demonstrate the benefits of our proposal by using several examples

sensitivity of mrqap tests to collinearity and autocorrelation conditions multiple regression quadratic assignment procedures mrqap tests are permutation tests for multiple linear regression model coefficients for data organized in square matrices of relatedness among n objects such a data structure is typical in social network studies where variables indicate some type of relation between a given set of actors we present a new permutation method called double semi partialing or dsp that complements the family of extant approaches to mrqap tests we assess the statistical bias type i error rate and statistical power of the set of five methods including dsp across a variety of conditions of network autocorrelation of spuriousness size of confounder effect and of skewness in the data these conditions are explored across three assumed data distributions normal gamma and negative binomial we find that the freedman lane method and the dsp method are the most robust against a wide array of these conditions we also find that all five methods perform better if the test statistic is pivotal finally we find limitations of usefulness for mrqap tests all tests degrade under simultaneous conditions of extreme skewness and high spuriousness for gamma and negative binomial distributions

further results on fixed sign solutions for a system of higher order difference equations abstract we consider the following system of n th order boundary value problems download full size image where u i k u i k u i k n 2 u i k 1 i m criteria are offered for the existence of single and double fixed sign solutions of the system we also include examples to illustrate the importance of the results obtained

information theoretic active polygons for unsupervised texture segmentation curve evolution models used in image segmentation and based on image region information usually utilize simple statistics such as means and variances hence can not account for higher order nature of the textural characteristics of image regions in addition the object delineation by active contour methods results in a contour representation which still requires a substantial amount of data to be stored for subsequent multimedia applications such as visual information retrieval from databases polygonal approximations of the extracted continuous curves are required to reduce the amount of data since polygons are powerful approximators of shapes for use in later recognition stages such as shape matching and coding the key contribution of this paper is the development of a new active contour model which nicely ties the desirable polygonal representation of an object directly to the image segmentation process this model can robustly capture texture boundaries by way of higher order statistics of the data and using an information theoretic measure and with its nature of the ordinary differential equations this new variational texture segmentation model is unsupervised since no prior knowledge on the textural properties of image regions is used another contribution in this sequel is a new polygon regularizer algorithm which uses electrostatics principles this is a global regularizer and is more consistent than a local polygon regularization in preserving local features such as corners

group comparison of eigenvalues and eigenvectors of diffusion tensors diffusion tensor imaging dti data differ from most medical images in that values at each voxel are not scalars but 3 3 symmetric positive definite matrices called diffusion tensors dts the anatomic characteristics of the tissue at each voxel are reflected by the dt eigenvalues and eigenvectors in this article we consider the problem of testing whether the means of two groups of dt images are equal at each voxel in terms of the dt s eigenvalues eigenvectors or both because eigendecompositions are highly nonlinear existing likelihood ratio statistics lrts for testing differences in eigenvalues or eigenvectors of means of gaussian symmetric matrices assume an orthogonally invariant covariance structure between the matrix entries while retaining the form of the lrts we derive new approximations to their true distributions when the covariance between the dt entries is arbitrary and possibly different between the two groups the approximate distributions are those of similar lrt statistics comput