on l_p norms of multivariate density estimators central limit theorems are proven for l p norms 1 leq p infty of multivariate density estimators

transformations to reduce boundary bias in kernel density estimation we consider kernel estimation of a univariate density whose support is a compact interval if the density is non zero at either boundary then the usual kernel estimator can be seriously biased reflection at a boundary removes some bias but unless the first derivative of the density is 0 at the boundary the estimator with reflection can still be much more severely biased at the boundary than in the interior we propose to transform the data to a density that has its first derivative equal to 0 at both boundaries the density of the transformed data is estimated and an estimate of the density of the original data is obtained by a change of variables the transformation is selected from a parametric family which is allowed to be quite general in our theoretical study we propose algorithms where the transformation is either a quartic polynomial a b cumulative density function cdf or a linear combination of a polynomial and a b cdf the last two types of transformation are designed to accommodate possible poles at the boundaries the first two algorithms are tested on simulated data and compared with an adjusted kernel method of rice we find that our proposal performs similarly to rice s for densities with one sided derivatives at the boundaries unlike rice s method our proposal is guaranteed to produce non negative estimates this can be a distinct advantage when the density is 0 at either boundary our algorithm for densities with poles outperforms the rice adjustment when the density has a pole at a boundary

on the measurement of polarization suppose that a population of individuals may be grouped according to some vector of characteristics into clusters such that each cluster is very similar in terms of the attributes of its members but different clusters have members with very dissimilar attributes in that case we say that the society is polarized our purpose is to study polarization and to provide a theory of its measurement our contention is that polarization as conceptualized here is closely related to the generation of social tensions to the possibilities of revolution and revolt and to the existence of social unrest in general we take special care to distinguish our theory from the theory of inequality measurement we derive measures of polarization that are easily applicable to distributions of characteristics such as income and wealth

nonparametric tests of stochastic dominance in income distributions tests for stochastic dominance based upon extensions of the goodness of fit test to the nonparametric comparison of income distributions are proposed implemented and compared with indirect tests of second order stochastic dominance currently utilized in income distribution studies

the bm l _ mathbf 1 norm density estimator process the notion of an l1 norm density estimator process indexed by a class of kernels is introduced then a functional central limit theorem and a glivenko cantelli theorem are established for this process while assembling the necessary machinery to prove these results a body of poissonization techniques and restricted chaining methods is developed which is useful for studying weak convergence of general processes indexed by a class of functions none of the theorems imposes any condition at all on the underlying lebesgue density f also somewhat unexpectedly the distribution of the limiting gaussian process does not depend on f

a new solution algorithm for improving performance of ant colony optimization this study proposes an improved solution algorithm using ant colony optimization aco for finding global optimum for any given test functions the procedure of the aco algorithms simulates the decision making processes of ant colonies as they forage for food and is similar to other artificial intelligent techniques such as tabu search simulated annealing and genetic algorithms aco algorithms can be used as a tool for optimizing continuous and discrete mathematical functions the proposed algorithm is based on each ant searches only around the best solution of the previous iteration with b the proposed algorithm is called as acorses an abbreviation of aco reduced search space b is proposed for improving aco s solution performance to reach global optimum fairly quickly the acorses is tested on fourteen mathematical test functions taken from literature and encouraging results were obtained the performance of acorses is compared with other optimization methods the results showed that the acorses performs better than other optimization algorithms available in literature in terms of minimum values of objective functions and number of iterations

on the estimation of pareto fronts from the point of view of copula theory a probabilistic framework is applied to pareto front estimation a link between pareto front and level lines of a multivariate distribution is given parametric approximations of pareto front are provided for archimedean copulas consequences and limits of the archimedean hypothesis are discussed the full estimation methodology is analyzed on several examples given a first set of observations from a design of experiments sampled randomly in the design space the corresponding set of non dominated points usually does not give a good approximation of the pareto front we propose here to study this problem from the point of view of multivariate analysis introducing a probabilistic framework with the use of copulas this approach enables the expression of level lines in the objective space giving an estimation of the position of the pareto front when the level tends to zero in particular when it is possible to use archimedean copulas analytical expressions for pareto front estimators are available several case studies illustrate the interest of the approach which can be used at the beginning of the optimization when sampling randomly in the design space

statistical inference for stochastic dominance and for the measurement of poverty and inequality we derive the asymptotic sampling distribution of various estimators frequently used to order distributions in terms of poverty welfare and inequality this includes estimators of most of the poverty indices currently in use as well as estimators of the curves used to infer stochastic dominance of any order these curves can be used to determine whether poverty inequality or social welfare is greater in one distribution than in another for general classes of indices and for ranges of possible poverty lines we also derive the sampling distribution of the maximal poverty lines up to which we may confidently assert that poverty is greater in one distribution than in another the sampling distribution of convenient dual estimators for the measurement of poverty is also established the statistical results are established for deterministic or stochastic poverty lines as well as for paired or independent samples of incomes our results are briefly illustrated using data for four countries drawn from the luxembourg income study data bases

polarization concepts measurement estimation distributions can be described using density functions the main theorem uniquely characterizes a class of polarization measures that fits into what we call the identityalienation framework and simultanously satisfies a set of axioms second we provide sample estimators of population polarization indices that can be used to compare polarization across time or entities distribution free statistical inference results are also used in order to ensure that the orderings of polarization across entities are not simply due to sampling noise an illustration of the use of these tools using data from 21 countries shows that polarization and inequality orderings can often differ in practice

global optimizations and tabu search based on memory based on the idea of tabu search that glover et al put forward a new tabu search named memory tabu search mts is proposed for solving the multiple minima problem of continuous functions two convergence theorems which show that mts asymptotically converges to the global optimal solutions in probability one under suitable conditions are given numerical results illustrate that this algorithm is efficient robust and easy to implement

the asymptotic distribution of sums of extreme values from a regularly varying distribution on donne une description complete de la distribution asymptotique de la somme des k n valeurs extremes superieures x n 1 kd n n x n 2 kd n n x n n

on the estimation of the extreme value index and large quantile estimation l article se constitue de deux parties on donne une preuve simple de la convergence faible de l estimation de pickand pour le parametre principal d une loi de valeurs extremes on donne d autres conditions naturelles pour la convergence forte et pour la normalite asymptotique de l estimation ensuite une grande quantile d une loi est estimee par une combinaison de statistiques d ordre intermediaires ou extremes ceci mene a un intervalle de confiance asymptotique

economic distance and overlapping of distributions abstract stratification and segmentation describe markets and societies the aim of this paper is to further develop the index of the degree of overlapping between distributions as a measure of segmentation and stratification the similarities and differences between the concepts of overlapping and economic distance are investigated

a statistical distribution with an unbounded hazard function and its application to a theory from demography we introduce a statistical distribution with hazard function mu t proportional to psi t beta 1 for 0 less than or equal to t less than psi and shape parameter beta satisfying 0 less than beta less than 1 this hazard function is suggested by a theory of aging in demography we discuss properties of this distribution and maximum likelihood estimates of its parameters mixture distributions are considered to account for pooling across dissimilar populations this model is compared with the gompertz and generalized pareto distributions and used to estimate a finite limit on human lifespan based on the survival of a group of female centenarians

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

comparison of location models of weibull type samples and extreme value processes four different location parameter models are compared within the sufficiency and deficiency concept the starting is a location model of a weibull type sample with shape parameter 1 a 1 here our basic inequality concerns the approximate sufficiency of the k lower extremes in addition the lower extremes are approximately equal in distribution to left s m 1 1 a t right m leqq k where s m is the sum of m i i d standard exponential random variables and t is the location parameter the final step leads us to the model of extreme value processes left s m 1 1 a t right m 1 2 3

testing inequality constraints in linear econometric models abstract this paper develops three asymptotically equivalent tests for examining the validity of imposing linear inequality restrictions on the parameters of linear econometric models first we consider the model y x e where e is n 0 and the hypothesis test h r r versus k rk later we generalize this testing framework to the linear simultaneous equations model we show that the joint asymptotic distribution of these test statistics and the test statistics from the hypothesis test h r r versus k r r is a weighted sum of two sets of independent 2 distributions we also derive a useful duality relation between the multivariate inequality constraints test developed here and the multivariate one sided hypothesis test in small samples these three test statistics satisfy inequalities similar to those derived by berndt and savin 1977 for the case of equality constraints the paper also contains an illustrative application of this testing technique

the dip test of unimodality the dip test measures multimodality in a sample by the maximum difference over all sample points between the empirical distribution function and the unimodal distribution function that minimizes that maximum difference the uniform distribution is the asymptotically least favorable unimodal distribution and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform

a parametric solution for simple stress strength model of failure with an application abstract estimation of p r s is considered for the simple stress strength model of failure using the pareto and power distributions together with their combined form a useful parametric solution is obtained and is illustrated numerically it is shown that these models are also applicable when only the tails of distributions for r and s are considered an application to the failure study concerning the fractures is also included

improving genetic algorithms performance by local search for continuous function optimization abstract the genetic algorithms gas can be used as a global optimization tool for continuous and discrete functions problems however a simple ga may suffer from slow convergence and instability of results gas problem solution power can be increased by local searching in this study a new local random search algorithm based on gas is suggested in order to reach a quick and closer result to the optimum solution

bayesian likelihood methods for estimating the end point of a distribution summary we consider maximum likelihood methods for estimating the end point of a distribution the likelihood function is modified by a prior distribution that is imposed on the location parameter the prior is explicit and meaningful and has a general form that adapts itself to different settings results on convergence rates and limiting distributions are given in particular it is shown that the limiting distribution is non normal in non regular cases parametric bootstrap techniques are suggested for quantifying the accuracy of the estimator we illustrate performance by applying the method to multiparameter weibull and gamma distributions

a plug in approach to support estimation we suggest a new approach based on the use of density estimators for the problem of estimating the compact support of a multivariate density this subject motivated in terms of pattern analysis by grenander has interesting connections with detection and clustering a natural class of density based estimators is defined universal consistency results and convergence rates are established for these estimators with respect to the usual measure based metric d between sets further convergence rates with respect to both d and the hausdorff metric d h are also obtained under some fairly intuitive shape restrictions

variants of some of the brauer fowler theorems abstract brauer and fowler noted restrictions on the structure of a finite group g in terms of c g t for an involution t g we consider variants of these themes we first note that for an arbitrary finite group g of even order we have g k f c g t 4 for each involution t g where f denotes the fitting subgroup of g and k f denotes the number of conjugacy classes of f in particular for such a group g we have g f g c g t 4 for each involution t g this result requires the classification of the finite simple groups the groups sl 2 2 n illustrate that the above exponent 4 cannot be replaced by any exponent less than 3 we do not know at present whether the exponent 4 can be improved in general though we note that the exponent 3 suffices when g is almost simple we are however able to prove that every finite group g of even order contains an involution u with g f g c g u 3 the proof of this fact reduces to proving two residual cases one in which g is almost simple where the classification of the finite simple groups is needed and one when g has a sylow 2 subgroup of order 2 for the latter result the classification of finite simple groups is not needed though the feit thompson odd order theorem is we also prove a very general result on fixed point spaces of involutions in finite irreducible linear groups which does not make use of the classification of the finite simple groups and some other results on the existence of non central elements not necessarily involutions with large centralizers in general finite groups lastly we prove without using the classification of finite simple groups that if g is a finite group and t g is an involution then all prime divisors of g f g are less than or equal to c g t 1

almost sure behaviour of f valued random fields letx n n d be a field of independent random variables taking values in a semi normed measurable vector spacef for a broad class of fields n d of positive numbers the almost sure behaviour of k nxk n n d is studied the main result allows us to deduce some new and well known theorems for fields of independentf random variables from related results for fields of independent real random variables

testing functional inequalities abstract this paper develops tests for inequality constraints of nonparametric regression functions the test statistics involve a one sided version of l p type functionals of kernel estimators 1 p drawing on the approach of poissonization this paper establishes that the tests are asymptotically distribution free admitting asymptotic normal approximation in particular the tests using the standard normal critical values have asymptotically correct size and are consistent against general fixed alternatives furthermore we establish conditions under which the tests have nontrivial local power against pitman local alternatives some results from monte carlo simulations are presented

wald criteria for jointly testing equality and inequality restrictions on montre comment le test de wald s applique a des ensembles de restrictions d egalite et d inegalite et on presente la distribution pour de grands echantillons du test sous l hypothese nulle

a heuristic approach for finding the global minimum adaptive random search technique in this paper a new random search technique which facilitates the determination of the global minimum is presented this method called adaptive random search technique arset is experimented on test problems and successful results are obtained arset algorithm outcome of which is observed to be relatively better is also compared with other methods in addition applicability of the algorithm on artificial neural network training is tested with xor problem

inference based on conditional moment inequalities in this paper we propose an instrumental variable approach to constructing confidence sets cs s for the true parameter in models defined by conditional moment inequalities equalities we show that by properly choosing instrument functions one can transform conditional moment inequalities equalities into unconditional ones without losing identification power based on the unconditional moment inequalities equalities we construct cs s by inverting cramer von mises type or kolmogorov smirnov type tests critical values are obtained using generalized moment selection gms procedures we show that the proposed cs s have correct uniform asymptotic coverage probabilities new methods are required to establish these results because an infinite dimensional nuisance parameter affects the asymptotic distributions we show that the tests considered are consistent against all fixed alternatives and typically have power against n 1 2 local alternatives to some but not all sequences of distributions in the null hypothesis monte carlo simulations for five different models show that the methods perform well in finite samples

ant colony optimization for finding the global minimum the ant colony optimization aco algorithms are multi agent systems in which the behaviour of each ant is inspired by the foraging behaviour of real ants to solve optimization problem this paper presents the aco based algorithm to find global minimum algorithm is based on that each ant searches only around the best solution of the previous iteration this algorithm was experimented on test problems and successful results were obtained the algorithm was compared with other methods which had been experimented on the same test problems and observed to be better

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

nonparametric estimation and inference about the overlap of two distributions this paper develops methodology for nonparametric estimation of a measure of the overlap of two distributions based on kernel estimation techniques this quantity has been proposed as a measure of economic polarization between two groups anderson 2004 and anderson et al 2010 in ecology it has been used to measure the overlap of species we give the asymptotic distribution theory of our estimator which in some cases of practical relevance is nonstandard due to a boundary value problem we also propose a method for conducting inference based on estimation of unknown quantities in the limiting distribution and show that our method yields consistent inference in all cases we consider we investigate the finite sample properties of our methods by simulation methods we give an application to the study of polarization within china in recent years using household survey data from two provinces taken in 1987 and 2001 we find a big increase in polarization between 1987 and 2001 according to monetary outcomes but less change in terms of living space

tests of stochastic monotonicity with improved power abstract we develop improved statistical procedures for testing stochastic monotonicity while existing tests use a fixed critical value to set the limiting rejection rate equal to nominal size at the least favorable case we use a bootstrap procedure to raise the limiting rejection rate to nominal size over much of the null this improves power against relevant local alternatives to show the validity of our approach we draw on recent results on the directional differentiability of the least concave majorant operator and on bootstrap inference when smoothness conditions sufficient to apply the functional delta method for the bootstrap are not satisfied

toward an empirical analysis of polarization stochastic dominance conditions may be interpreted as a means of defining the degree and nature of separateness of two distributions by describing polarization states in terms of stochastic dominance conditions this paper presents a taxonomy of and tests for polarization both between and within population distributions four examples from the poverty growth and development wage distribution and assortative pairing literatures are used to illustrate how the techniques may be used in empirical applications

estimating tails of probability distributions we study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold a key idea is the use of pickands generalised pareto distribution and its fitting in most cases by the method of maximum likelihood the results cover all three limiting types of extreme value theory we propose a new estimator for an index of regular variation and show that it often performs better than hill s estimator we give new results for estimating the endpoint of a distribution extending earlier work by hall and by smith and weissman finally we give detailed results for the domain of attraction of exp e x and show that in most cases our proposed estimator is more efficient than two others one based on the exponential distribution and the other due to davis and resnick we also touch briefly on the problem of large deviations from a statistical point of view the results make extensive use of existing work on rates of convergence

maximum likelihood estimation of the lower tail of a probability distribution summary we consider the estimation of the three parameters of the lower tail of a distribution function based on the k smallest out of n observations the likelihood function has a singularity but it is argued that a local maximum when it exists should be taken as the m l e asymptotic results as k oo n oo k n 0 show that such a local maximum does exist and provides consistent estimators whenever the shape parameter is greater than one otherwise there is no local maximum and likelihood inference fails we also discuss interval estimation and propose a test to distinguish between the type i and weibull limit laws

a heuristic approach to find the global optimum of function this paper presents the modified ant colony optimization maco based algorithm to find global optimum algorithm is based on that solution space of problem is restricted by the best solution of the previous iteration furthermore the proposed algorithm is that variables of problem are optimized concurrently this algorithm was tested on some standard test functions and successful results were obtained its performance was compared with the other algorithms and observed to be better

optimum bounds for the distributions of martingales in banach spaces a general device is proposed which provides for extension of exponential inequalities for sums of independent real valued random variables to those for martingales in the 2 smooth banach spaces this is used to obtain optimum bounds of the rosenthal burkholder and chung types on moments of the martingales in 2 smooth banach spaces in turn it leads to best order bounds on moments of sums of independent random vectors in any separable banach spaces although the emphasis is put on infinite dimensional martingales most of the results seem to be new even for one dimensional martingales moreover the bounds on moments of the rosenthal burkholder type seem to be to a certain extent new even for sums of independent real valued random variables analogous inequalities for one dimensional supermartingales are given

statistical properties of measures of between group income differentials abstract a variety of statistical measures have been used to analyze interdistributional income inequality two recently introduced measures indicate a much larger secular change in the black white income differential than the currently used measures in order to understand this phenomenon both the theoretical properties of and empiric results obtained from five measures are given it is shown that the new measures are more sensitive to the type of change that actually occured than the usual measures which were designed to detect a shift in location

consistent tests for stochastic dominance methods are proposed for testing stochastic dominance of any pre specified order with primary interest in the distributions of income we consider consistent tests that are similar to kolmogorov smirnov tests of the complete set of restrictions that relate to the various forms of stochastic dominance for such tests in the case of tests for stochastic dominance beyond first order we propose and justify a variety of approaches to inference based on simulation and the bootstrap we compare these approaches to one another and to alternative approaches based on multiple comparisons in the context of a monte carlo experiment and an empirical example