optimal multinational capital budgeting under uncertainty this paper discusses the multinational capital budgeting problem when there are some candidate foreign projects which project s should the investor choose in the paper special cash flows and value sources of foreign projects are introduced regarding project parameters such as construction costs annual net operating cash flows terminal values of the projects as well as the foreign exchange rates as uncertain variables the paper proposes one new uncertain zero one integer model for optimal multinational project selection to solve the problem a hybrid intelligent algorithm integrating the 99 methods and genetic algorithm is provided as an illustration an application example is also presented

influential features pca for high dimensional clustering we consider a clustering problem where we observe feature vectors x i in r p i 1 2 ldots n from k possible classes the class labels are unknown and the main interest is to estimate them we are primarily interested in the modern regime of p gg n where classical clustering methods face challenges we propose influential features pca if pca as a new clustering procedure in if pca we select a small fraction of features with the largest kolmogorov smirnov ks scores obtain the first k 1 left singular vectors of the post selection normalized data matrix and then estimate the labels by applying the classical k means procedure to these singular vectors in this procedure the only tuning parameter is the threshold in the feature selection step we set the threshold in a data driven fashion by adapting the recent notion of higher criticism as a result if pca is a tuning free clustering method we apply if pca to 10 gene microarray data sets the method has competitive performance in clustering especially in three of the data sets the error rates of if pca are only 29 or less of the error rates by other methods we have also rediscovered a phenomenon on empirical null by efron j amer statist assoc 99 2004 96 104 on microarray data with delicate analysis especially post selection eigen analysis we derive tight probability bounds on the kolmogorov smirnov statistics and show that if pca yields clustering consistency in a broad context the clustering problem is connected to the problems of sparse pca and low rank matrix recovery but it is different in important ways we reveal an interesting phase transition phenomenon associated with these problems and identify the range of interest for each

about some new generalizations of bellman bihari results for integro functional inequalities with discontinuous functions and applications abstract in this article we present some new nonlinear bellman bihari type integral inequalities with delay for discontinuous functions integro sum inequalities impulse integral inequalities some applications of the results are included conditions of boundedness uniformly stability by lyapunov uniformly practical stability by chetaev uniformly for the solutions of impulsive differential and integro differential systems of ordinary differential equations

maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

phase transitions for high dimensional clustering and related problems consider a two class clustering problem where we observe x i ell i mu z i z i stackrel mathit i i d sim n 0 i p 1 leq i leq n the feature vector mu in r p is unknown but is presumably sparse the class labels ell i in 1 1 are also unknown and the main interest is to estimate them we are interested in the statistical limits in the two dimensional phase space calibrating the rarity and strengths of useful features we find the precise demarcation for the region of impossibility and region of possibility in the former useful features are too rare weak for successful clustering in the latter useful features are strong enough to allow successful clustering the results are extended to the case of colored noise using le cam s idea on comparison of experiments we also extend the study on statistical limits for clustering to that for signal recovery and that for global testing we compare the statistical limits for three problems and expose some interesting insight we propose classical pca and important features pca if pca for clustering for a threshold t 0 if pca clusters by applying classical pca to all columns of x with an l 2 norm larger than t we also propose two aggregation methods for any parameter in the region of possibility some of these methods yield successful clustering we discover a phase transition for if pca for any threshold t 0 let xi t be the first left singular vector of the post selection data matrix the phase space partitions into two different regions in one region there is a t such that cos xi t ell rightarrow 1 and if pca yields successful clustering in the other cos xi t ell leq c 0 0 our results require delicate analysis especially on post selection random matrix theory and on lower bound arguments

adapting to unknown sparsity by controlling the false discovery rate we attempt to recover an n dimensional vector observed in white noise where n is large and the vector is known to be sparse but the degree of sparsity is unknown we consider three different ways of defining sparsity of a vector using the fraction of nonzero terms imposing power law decay bounds on the ordered entries and controlling the lp norm for p small we obtain a procedure which is asymptotically minimax for l r loss simultaneously throughout a range of such sparsity classes the optimal procedure is a data adaptive thresholding scheme driven by control of the false discovery rate fdr fdr control is a relatively recent innovation in simultaneous testing ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections in our treatment the fdr control parameter qn also plays a determining role in asymptotic minimaxity if q lim qn 0 1 2 and also qn log n we get sharp asymptotic minimaxity simultaneously over a wide range of sparse parameter spaces and loss functions on the other hand q lim qn 1 2 1 forces the risk to exceed the minimax risk by a factor growing with q to our knowledge this relation between ideas in simultaneous inference and asymptotic decision theory is new our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors these new rules impose complexity penalization of the form 2 log potential model size actual model size we exhibit a close connection with fdr controlling procedures under stringent control of the false discovery rate

mathematical considerations for the problem of fourier transform phase retrieval from magnitude in this paper we deal with the problem of retrieving a finite extent function from the magnitude of its fourier transform this so called phase retrieval problem will first be posed under its different underlying models we will present a brief review of the main results known in this area for both discrete and continuous phase retrieval models giving special emphasis to the algebraic problem of the uniqueness of the solution several important issues which are yet unresolved will be pointed out and discussed we will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real valued sequence x from the magnitude of the output of a linear distortion hx j j 1 cdots n a number of important results will be obtained for this general setting by means of algebraic geometric techniques in particular the problems of the existence of a solution for phase retrieval number of feasible solutions stability of the essential

risk vulnerability and the tempering effect of background risk we examine in this paper a new natural restriction on utility functions namely that adding an unfair background risk to wealth makes risk averse individuals behave in a more risk averse way with respect to any other independent risk this concept is called risk vulnerability it is equivalent to the condition that an undesirable risk can never be made desirable by the presence of an independent unfair risk moreover under risk vulnerability adding an unfair background risk reduces the demand for risky assets risk vulnerability generalizes the concept of properness individually undesirable independent risks are always jointly undesirable introduced by pratt and zeckhauser 1987 it implies that the two first derivatives of the utility function are concave transformations of the original utility function under decreasing absolute risk aversion a sufficient condition for risk vulnerability is local properness i e r r r where r is the arrow pratt coefficient of absolute risk aversion

about some new integral inequalities of wendroff type for discontinuous functions abstract in this article we obtain some new nonlinear integral inequalities for discontinuous functions of two independent variables wendroff type by including also inequalities with delay we deduce new generalizations of earlier results given by r p agarwal r bellman i bihari b k bondge v lakshmikantham s leela b g pachpatte for continuous and discrete functions furthermore generalizations of some results for integro sum inequalities are obtained as well

consistent inference of probabilities for reproducible experiments the need for inducing a probability distribution from partial data and the complementary problem of the analysis of an observed distribution in terms of fewer relevant variables occur in many branches of physics for reproducible experiments consistency conditions which must be satisfied by any algorithm for inferring a discrete probability distribution with given averages are formulated the only consistent algorithm is the one leading to the distribution of maximal entropy subject to the given constraints

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

generalized information theory abstract generalized information theory is viewed in this paper as an information theory that is liberated from the boundaries of probability theory after overviewing classical probabilistic information theory the paper examines recent developments regarding nonprobabilistic measures and principles of uncertainty based information which form a nucleus of the emerging generalized information theory

investigations of the properties motion for essential nonlinear systems perturbed by impulses on some hypersurfaces abstract in this work sufficiently conditions boundedness stability asymptotical stability exponential stability attraction practical stability nondisturbed motion systems with impulse perturbations on some hypersurfaces are obtained we formulate the conditions which guarantee that no beating motion of hypersurfaces will be present estimates of cauchy matrix system of variation for approximated system are obtained for different kinds of right side shorted system these estimates are used as one parametric scale increasing functions moreover we also consider two parametric scale such as proposed by demidovich

score test variable screening variable screening has emerged as a crucial first step in the analysis of high throughput data but existing procedures can be computationally cumbersome difficult to justify theoretically or inapplicable to certain types of analyses motivated by a high dimensional censored quantile regression problem in multiple myeloma genomics this paper makes three contributions first we establish a score test based screening framework which is widely applicable extremely computationally efficient and relatively simple to justify secondly we propose a resampling based procedure for selecting the number of variables to retain after screening according to the principle of reproducibility finally we propose a new iterative score test screening method which is closely related to sparse regression in simulations we apply our methods to four different regression models and show that they can outperform existing procedures we also apply score test screening to an analysis of gene expression data from multiple myeloma patients using a censored quantile regression model to identify high risk genes

multi period portfolio selection with mental accounts and realistic constraints based on uncertainty theory abstract this paper discusses an uncertain multi period portfolio selection problem in the situation where the future security return rates are given by experts estimations instead of historical data in financial market investors may have different attitudes towards risk for different goals in order to reflect these conflicting risk attitudes for different goals mental accounts are introduced to the investment in addition investment strategies may be affected by some realistic constraints such as background risk liquidity risk transaction cost and cardinality constraint based on uncertainty theory a nonlinear multi period portfolio selection model is proposed with consideration of mental accounts and realistic constraints in the model indeterminate quantities such as security return rates and turnover rates are assumed to be uncertain variables within the framework of uncertainty theory we discuss the equivalents of the model and show that the nonlinear model can be equivalently transformed into a linear programming model finally a case study is given to illustrate the performance of the proposed model

optimal detection of heterogeneous and heteroscedastic mixtures the problem of detecting heterogeneous and heteroscedastic gaussian mixtures is considered the focus is on how the parameters of heterogeneity heteroscedasticity and proportion of non null component influence the difficulty of the problem we establish an explicit detection boundary which separates the detectable region where the likelihood ratio test is shown to reliably detect the presence of non null effect from the undetectable region where no method can do so in particular the results show that the detection boundary changes dramatically when the proportion of nonnull component shifts from the sparse regime to the dense regime furthermore it is shown that the higher criticism test which does not require the specific information of model parameters is optimally adaptive to the unknown degrees of heterogeneity and heteroscedasticity in both the sparse and dense cases

on use of the em algorithm for penalized likelihood estimation summary the em algorithm is a popular approach to maximum likelihood estimation but has not been much used for penalized likelihood or maximum a posteriori estimation this paper discusses properties of the em algorithm in such contexts concentrating on rates of conver gence and presents an alternative that is usually more practical and converges at least as quickly the em algorithm is a general approach to maximum likelihood estimation rather than a specific algorithm dempster et al 1977 discussed the method and derived basic properties demonstrating that a variety of procedures previously developed rather informally could be unified the common strand to problems where the approach is applicable is a notion of incomplete data this includes the conventional sense of missing data but is much broader than that the em algorithm demon strates its strength in situations where some hypothetical experiment yields data from which estimation is particularly convenient and economical the incomplete data actually at hand are regarded as observable functions of these complete data the resulting algorithms while usually slow to converge are often extremely simple and remain practical in large problems where no other approaches may be feasible dempster et al 1977 briefly refer to the use of the same approach to the problem of finding the posterior mode maximum a posteriori estimate in a bayesian estima

a new look at entropy for solving linear inverse problems entropy based methods are widely used for solving inverse problems particularly when the solution is known to be positive here we address linear ill posed and noisy inverse problems of the form z ax n with a general convex constraint x spl isin x where x is a convex set although projective methods are well adapted to this context we study alternative methods which rely highly on some information based criteria our goal is to clarify the role played by entropy in this field and to present a new point of view on entropy using general tools and results coming from convex analysis we present then a new and broad scheme for entropic based inversion of linear noisy inverse problems this scheme was introduced by navaza in 1985 in connection with a physical modeling for crystallographic applications and further studied by dacunha castelle and gamboa 1990 important features of this paper are i a unified presentation of many well known reconstruction criteria ii proposal of new criteria for reconstruction under various prior knowledge and with various noise statistics iii a description of practical inversion of data using the aforementioned criteria and iv a presentation of some reconstruction results

about new analogies of gronwall bellman bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems abstract in this article we present new integral gronwall bellman bihari type inequalities for discontinuous functions integro sum inequalities as applications we investigate estimated solutions for impulsive differential systems conditions of boundedness stability practical stability

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

analytical p value calculation for the higher criticism test in finite d problems the higher criticism is effective for testing a joint null hypothesis against a sparse alternative e g for testing the effect of a gene or a genetic pathway that consists of d genetic markers accurate p value calculations for the higher criticism based on the asymptotic distribution require a very large d which is not the case for the number of genetic variants in a gene or a pathway in this paper we propose an analytic method that accurately computes the p value of the higher criticism test for finite d problems unlike previous treatments this method does not rely on asymptotics in d or simulation and is exact for arbitrary d when the test statistics are normally distributed the method is particularly computationally advantageous when d is not large we illustrate the proposed method with a case control genome wide association study of lung cancer and compare its power to competing methods through simulations

about a generalization of bellman bihari type inequalities for discontinuous functions and their applications abstract in the present paper we introduce the conditions of solvability for chaplygin s problem with discontinuous functions in two independent variables satisfying integro sum inequalities the new type of nonlinear integral and wendroff s inequality for discontinuous functions are investigated as applications the conditions of boundedness solutions of partial differential equations of hyperbolic type with impulse influence on some hypersurfaces j r 2 are obtained some historical aspects of the theory of integro sum inequalities are presented

optimal classification in sparse gaussian graphic model consider a two class classification problem where the number of features is much larger than the sample size the features are masked by gaussian noise with mean zero and covariance matrix sigma where the precision matrix omega sigma 1 is unknown but is presumably sparse the useful features also unknown are sparse and each contributes weakly i e rare and weak to the classification decision by obtaining a reasonably good estimate of omega we formulate the setting as a linear regression model we propose a two stage classification method where we first select features by the method of innovated thresholding it and then use the retained features and fisher s lda for classification in this approach a crucial problem is how to set the threshold of it we approach this problem by adapting the recent innovation of higher criticism thresholding hct we find that when useful features are rare and weak the limiting behavior of hct is essentially just as good as the limiting behavior of ideal threshold the threshold one would choose if the underlying distribution of the signals is known if only somewhat surprisingly when omega is sufficiently sparse its off diagonal coordinates usually do not have a major influence over the classification decision compared to recent work in the case where omega is the identity matrix proc natl acad sci usa 105 2008 14790 14795 philos trans r soc lond ser a math phys eng sci 367 2009 4449 4470 the current setting is much more general which needs a new approach and much more sophisticated analysis one key component of the analysis is the intimate relationship between hct and fisher s separation another key component is the tight large deviation bounds for empirical processes for data with unconventional correlation structures where graph theory on vertex coloring plays an important role

tests alternative to higher criticism for high dimensional means under sparsity and column wise dependence we consider two alternative tests to the higher criticism test of donoho and jin ann statist 32 2004 962 994 for high dimensional means under the sparsity of the nonzero means for sub gaussian distributed data with unknown column wise dependence the two alternative test statistics are constructed by first thresholding l 1 and l 2 statistics based on the sample means respectively followed by maximizing over a range of thresholding levels to make the tests adaptive to the unknown signal strength and sparsity the two alternative tests can attain the same detection boundary of the higher criticism test in ann statist 32 2004 962 994 which was established for uncorrelated gaussian data it is demonstrated that the maximal l 2 thresholding test is at least as powerful as the maximal l 1 thresholding test and both the maximal l 2 and l 1 thresholding tests are at least as powerful as the higher criticism test

high dimensional classification using features annealed independence rules classification using high dimensional features arises frequently in many contemporary statistical studies such as tumor classification using microarray or other high throughput data the impact of dimensionality on classifications is largely poorly understood in a seminal paper bickel and levina 2004 show that the fisher discriminant performs poorly due to diverging spectra and they propose to use the independence rule to overcome the problem we first demonstrate that even for the independence classification rule classification using all the features can be as bad as the random guessing due to noise accumulation in estimating population centroids in high dimensional feature space in fact we demonstrate further that almost all linear discriminants can perform as bad as the random guessing thus it is paramountly important to select a subset of important features for high dimensional classification resulting in features annealed independence rules fair the conditions under which all the important features can be selected by the two sample t statistic are established the choice of the optimal number of features or equivalently the threshold value of the test statistics are proposed based on an upper bound of the classification error simulation studies and real data analysis support our theoretical results and demonstrate convincingly the advantage of our new classification procedure

mean absolute deviation portfolio optimization model and its applications to tokyo stock market the purpose of this paper is to demonstrate that a portfolio optimization model using the l1 risk mean absolute deviation risk function can remove most of the difficulties associated with the classical markowitz s model while maintaining its advantages over equilibrium models in particular the l1 risk model leads to a linear program instead of a quadratic program so that a large scale optimization problem consisting of more than 1 000 stocks may be solved on a real time basis numerical experiments using the historical data of nikkei 225 stocks show that the l1 risk model generates a portfolio quite similar to that of the markowitz s model within a fraction of time required to solve the latter

does the maximum entropy method improve sensitivity maximum entropy reconstruction has been used in several fields to produce visually striking reconstructions of positive objects images densities spectra from noisy indirect measurements in magnetic resonance spectroscopy this technique is notable for its apparent noise suppression and its avoidance of the artifacts that affect discrete fourier transform spectra of short zero extended data records in the general case where the length of the reconstructed spectrum exceeds that of the data record or where a convolution kernel is incorporated in the reconstruction no known analytical solution to the reconstruction problem exists consequently knowledge of the properties of maximum entropy reconstruction has been mainly anecdotal based on a small selection of published reconstructions however in the limiting case where the lengths of the reconstructed spectrum and the data record are the same and a convolution kernel is not applied the problem can be solved analytically the solution has a simple structure that helps explain several commonly observed features of maximum entropy reconstructions for example the biases in the recovered intensities and the fact that noise near the baseline is more successfully suppressed than is noise superimposed on broad features in the spectrum the solution also shows that the noise suppression offered by maximum entropy reconstruction could in this special case be equally well obtained by a cosmetic device simply displaying the conventional fourier transform reconstruction using a certain nonlinear plotting scale for the vertical y coordinate

on the stability of invariant sets of systems with impulse effect abstract a system of differential equations with impulse effect is considered it is assumed that this system has an invariant set m by means of the direct lyapunov method the necessary and sufficient conditions of its uniform asymptotic stability are obtained the conditions on the perturbations of right hand sides of differential equations and impulse effects under which the uniform asymptotic stability of the invariant set m of the nonperturbed system implies the uniform asymptotic stability of the invariant set of the perturbed system are obtained the stability properties of invariant sets of periodic systems are also studied

mean risk model for uncertain portfolio selection with background risk abstract when making investment decisions one will face not only the risk of financial assets within portfolio but also background risk this paper discusses an uncertain portfolio selection problem in which background risk is considered and the security returns and background assets return are given by experienced experts evaluations instead of historical data regarding the returns of the securities and background assets as uncertain variables we propose an uncertain mean risk model with background risk for portfolio selection and the crisp forms of the model are provided when security returns obey different uncertainty distributions in addition when everything else is same it is concluded that the optimal expected return of the mean risk model with background risk is usually smaller than that without background risk moreover the relationship between the optimal solution of our model and that of the model in huang s paper uncertain portfolio selection with background risk is discussed finally numerical examples are presented to illustrate the effectiveness of the model and to show the effect of background risk on investment decision

on possibilistic mean value and variance of fuzzy numbers dubois and prade introduced the mean value of a fuzzy number as a closed interval bounded by the expectations calculated from its upper and lower distribution functions in this paper introducing the notations of lower possibilistic and upper possibilistic mean values we definine the interval valued possibilistic mean and investigate its relationship to the interval valued probabilistic mean we also introduce the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous possibility distributions which are consistent with the extension principle we also show that the variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

integro sum inequalities and motion stability of systems with impulse perturbations abstract this article is devoted to the investigation of the properties of motion represented by essential nonlinear system of differential equations with impulse effect the results based on the solvability problem by s a chaplygin for discontinuous functions satisfy integro sum inequalities the conditions of boundedness stability asymptotical stability practical stability attraction motion for different kinds of nonlinearities of systems are obtained

analysis of variance why it is more important than ever analysis of variance anova is an extremely important method in exploratory and confirmatory data analysis unfortunately in complex problems e g split plot designs it is not always easy to set up an appropriate anova we propose a hierarchical analysis that automatically gives the correct anova comparisons even in complex scenarios the inferences for all means and variances are performed under a model with a separate batch of effects for each row of the anova table we connect to classical anova by working with finite sample variance components fixed and random effects models are characterized by inferences about existing levels of a factor and new levels respectively we also introduce a new graphical display showing inferences about the standard deviations of each batch of effects we illustrate with two examples from our applied data analysis first illustrating the usefulness of our hierarchical computations and displays and second showing how the ideas of anova are helpful in understanding a previously fit hierarchical model

portfolio selection based on fuzzy cross entropy in this paper the kapur cross entropy minimization model for portfolio selection problem is discussed under fuzzy environment which minimizes the divergence of the fuzzy investment return from a priori one first three mathematical models are proposed by defining divergence as cross entropy average return as expected value and risk as variance semivariance and chance of bad outcome respectively in order to solve these models under fuzzy environment a hybrid intelligent algorithm is designed by integrating numerical integration fuzzy simulation and genetic algorithm finally several numerical examples are given to illustrate the modeling idea and the effectiveness of the proposed algorithm

mean semivariance models for fuzzy portfolio selection this paper discusses portfolio selection problem in fuzzy environment in the paper semivariance is originally presented for fuzzy variable and three properties of the semivariance are proven based on the concept of semivariance of fuzzy variable two fuzzy mean semivariance models are proposed to solve the new models in general cases a fuzzy simulation based genetic algorithm is presented in the paper in addition two numerical examples are also presented to illustrate the modelling idea and the effectiveness of the designed algorithm

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

theoretical measures of relative performance of classifiers for high dimensional data with small sample sizes summary we suggest a technique related to the concept of detection boundary that was developed by ingster and by donoho and jin for comparing the theoretical performance of classifiers constructed from small training samples of very large vectors the resulting classification boundaries are obtained for a variety of distance based methods including the support vector machine distance weighted discrimination and kth nearest neighbour classifiers for thresholded forms of those methods and for techniques based on donoho and jin s higher criticism approach to signal detection assessed in these terms standard distance based methods are shown to be capable only of detecting differences between populations when those differences can be estimated consistently however the thresholded forms of distance based classifiers can do better and in particular can correctly classify data even when differences between distributions are only detectable not estimable other methods including higher criticism classifiers can on occasion perform better still but they tend to be more limited in scope requiring substantially more information about the marginal distributions moreover as tail weight becomes heavier the classification boundaries of methods designed for particular distribution types can converge to and achieve the boundary for thresholded nearest neighbour approaches for example although higher criticism has a lower classification boundary and in this sense performs better in the case of normal data the boundaries are identical for exponentially distributed data when both sample sizes equal 1

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

straight to the source detecting aggregate objects in astronomical images with proper error control the next generation of telescopes coming online in the next decade will acquire terabytes of image data each night collectively these large images will contain billions of interesting objects which astronomers call sources one critical task for astronomers is to construct from the image data a detailed source catalog that gives the sky coordinates and other properties of all detected sources the source catalog is the primary data product produced by most telescopes and serves as an important input for studies that build and test new astrophysical theories to construct an accurate catalog the sources must first be detected in the image a variety of effective source detection algorithms exist in the astronomical literature but few if any provide rigorous statistical control of error rates a variety of multiple testing procedures exist in the statistical literature that can provide rigorous error control over pixelwise errors but these do not provide control over errors at the level of sources

higher criticism thresholding optimal feature selection when useful features are rare and weak in important application fields today genomics and proteomics are examples selecting a small subset of useful features is crucial for success of linear classification analysis we study feature selection by thresholding of feature z scores and introduce a principle of threshold selection based on the notion of higher criticism hc for i 1 2 p let i denote the two sided p value associated with the ith feature z score and i denote the ith order statistic of the collection of p values the hc threshold is the absolute z score corresponding to the p value maximizing the hc objective i p i i p 1 i p we consider a rare weak rw feature model where the fraction of useful features is small and the useful features are each too weak to be of much use on their own hc thresholding hct has interesting behavior in this setting with an intimate link between maximizing the hc objective and minimizing the error rate of the designed classifier and very different behavior from popular threshold selection procedures such as false discovery rate thresholding fdrt in the most challenging rw settings hct uses an unconventionally low threshold this keeps the missed feature detection rate under better control than fdrt and yields a classifier with improved misclassification performance replacing cross validated threshold selection in the popular shrunken centroid classifier with the computationally less expensive and simpler hct reduces the variance of the selected threshold and the error rate of the constructed classifier results on standard real datasets and in asymptotic theory confirm the advantages of hct

distributions and power of optimal signal detection statistics in finite case for detecting weak and sparse signals by a set of n input p values the higher criticism hc type statistics the berk jones b j type statistics and the phi divergence statistics have the equivalent asymptotic optimality as n goes to infinity however they can have significantly different performance in practical data analysis where n is always finite and even very small to address this problem in a broader context this paper introduces a general family of goodness of fit statistics called the ggof which unifies a broad signal detection statistics including these optimal ones efficient and accurate analytical calculations for the distributions of the ggof statistics are provided under arbitrary i i d continuous models of the null and the alternative hypotheses based on that a systematic power study reveals that in finite case the number of signals is often more relevant than the signal proportion the hc and the reverse hc have advantages for relatively sparser and denser signals respectively while the b j is more robust a general framework is given to apply the ggof into data analysis based on the generalized linear models an application to the snp set based genome wide association study gwas for crohn s disease shows that these optimal statistics have a good potential for detecting novel disease genes with weak snp effects the calculations have been implemented into an r package settest and published on the cran

mean variance skewness model for portfolio selection with fuzzy returns numerous empirical studies show that portfolio returns are generally asymmetric and investors would prefer a portfolio return with larger degree of asymmetry when the mean value and variance are same in order to measure the asymmetry of fuzzy portfolio return a concept of skewness is defined as the third central moment in this paper and its mathematical properties are studied as an extension of the fuzzy mean variance model a mean variance skewness model is presented and the corresponding variations are also considered in order to solve the proposed models a genetic algorithm integrating fuzzy simulation is designed finally several numerical examples are given to illustrate the modelling idea and the effectiveness of the proposed algorithm

convergence rates for maximum entropy regularization the authors prove results about stability convergence and convergence rates for the maximum entropy method by interpreting it as a nonlinear regularization method the results are applied to a widely studied problem from physical chemistry namely the reconstruction of fluorescence lifetime distributions from the distribution of the delays between absorption and emission of photons

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

impulsive differential systems the problem of stability and practical stability abstract this article is devoted to the analysis properties of the solutions of nonlinear impulsive differential systems with perturbations on some hypersurfaces t i x the conditions of boundedness practical stability uniformly attractive stability uniformly asymptotical attraction of the solutions perturbed system by using linear approximation are obtained

partial correlation screening for estimating large precision matrices with applications to classification given n samples x1 x2 xn from n 0 we are interested in estimating the p p precision matrix 1 we assume is sparse in that each row has relatively few nonzeros we propose partial correlation screening pcs as a new row by row approach to estimate the i th row of 1 i p pcs uses a screen step and a clean step in the

on minimax estimation of a sparse normal mean vector mallows has conjectured that among distributions which are gaussian but for occasional contamination by additive noise the one having least fisher information has two sided geometric contamination a very similar problem arises in estimation of a non negative vector parameter in gaussian white noise when it is known also that most i e 1i components are zero we provide a partial asymptotic expansion of the minimax risk as 0 while the conjecture seems unlikely to be exactly true for flnite we verify it asymptotically up to the accuracy of the expansion numerical work suggests the expansion is accurate for as large as 05 the best 1 estimation rule is flrst but not second order minimax the results bear on an earlier study of maximum entropy estimation and various questions in robustness and function estimation using wavelet bases

on the necessity of irrelevant variables this work explores the effects of relevant and irrelevant boolean variables on the accuracy of classifiers the analysis uses the assumption that the variables are conditionally independent given the class and focuses on a natural family of learning algorithms for such sources when the relevant variables have a small advantage over random guessing the main result is that algorithms relying predominately on irrelevant variables have error probabilities that quickly go to 0 in situations where algorithms that limit the use of irrelevant variables have errors bounded below by a positive constant we also show that accurate learning is possible even when there are so few examples that one cannot determine with high confidence whether or not any individual variable is relevant

some new integral inequalities of bellman bihari type with delay for discontinuous functions abstract in this article we investigate some integral functional inequalities of bellman bihari type for piecewise continuous functions with some fixed points of discontinuity we also prove a new analogy and generalization of results which were obtained by bellman and bihari to integro sum inequalities with delay and discontinuities that do not belong to lipschitz s type

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed