bandwidth selection for the smoothing of distribution functions summary several approaches can be made to the choice of bandwidth in the kernel smoothing of distribution functions recent proposals by sarda 1993 and by altman leger 1995 are analogues of the leave one out and plug in methods which have been widely used in density estimation in contrast a method of crossvalidation appropriate to the smoothing of distribution functions is proposed selection of the bandwidth parameter is based on unbiased estimation of a mean integrated squared error curve whose minimising value defines an optimal smoothing parameter this procedure is shown to lead to asymptotically optimal bandwidth choice not just in the usual first order sense but also in the second order sense in which kernel methods improve on the standard empirical distribution function some general theory on the performance of optimal data based methods of bandwidth choice is also provided leading to results which do not have analogues in the context of density estimation the numerical performances of all the methods discussed in the paper are compared a bandwidth based on a simple reference distribution is also included simulations suggest that the crossvalidatory proposal works well although the simple reference bandwidth is also quite effective

inference in components of variance models with low replication in components of variance models the data are viewed as arising through a sum of two random variables representing between and within group variation respectively the former is generally interpreted as a group effect and the latter as error it is assumed that these variables are stochastically independent and that the distributions of the group effect and the error do not vary from one instance to another if each group effect can be replicated a large number of times then standard methods can be used to estimate the distributions of both the group effect and the error this cannot be achieved without replication however how feasible is distribution estimation if it is not possible to replicate prolifically can the distributions of random effects and errors be estimated consistently from a small number of replications of each of a large number of noisy group effects for example in a nonparametric setting often extensive replication is practically infeasible in particular if inherently small numbers of individuals exhibit any given group effect yet it is quite unclear how to conduct inference in this case we show that inference is possible even if the number of replications is as small as 2 two methods are proposed both based on fourier inversion one which is substantially more computer intensive than the other exhibits better performance in numerical experiments

nonparametric confidence intervals for receiver operating characteristic curves summary we study methods for constructing confidence intervals and confidence bands for estimators of receiver operating characteristics particular emphasis is placed on the way in which smoothing should be implemented when estimating either the characteristic itself or its variance we show that substantial undersmoothing is necessary if coverage properties are not to be impaired a theoretical analysis of the problem suggests an empirical plug in rule for bandwidth choice optimising the coverage accuracy of interval estimators the performance of this approach is explored our preferred technique is based on asymptotic approximation rather than a more sophisticated approach using the bootstrap since the latter requires a multiplicity of smoothing parameters all of which must be chosen in nonstandard ways it is shown that the asymptotic method can give very good performance

trace class operators and commutators abstract we identify the linear span of commutators ab ba where a is a trace class operator and b is any bounded operator on a separable hilbert space or where a b are both hilbert schmidt operators thus answering questions raised by work of anderson vaserstein and weiss

empirical likelihood for estimating equations with missing values we consider an empirical likelihood inference for parameters defined by general estimating equations when some components of the random observations are subject to missingness as the nature of the estimating equations is wide ranging we propose a nonparametric imputation of the missing values from a kernel estimator of the conditional distribution of the missing variable given the always observable variable the empirical likelihood is used to construct a profile likelihood for the parameter of interest we demonstrate that the proposed nonparametric imputation can remove the selection bias in the missingness and the empirical likelihood leads to more efficient parameter estimation the proposed method is further evaluated by simulation and an empirical study on a genetic dataset on recombinant inbred mice

convergence in competition models with small diffusion coefficients abstract it is well known that for reaction diffusion 2 species lotka volterra competition models with spatially independent reaction terms global stability of an equilibrium for the reaction system implies global stability for the reaction diffusion system this is not in general true for spatially inhomogeneous models we show here that for an important range of such models for small enough diffusion coefficients global convergence to an equilibrium holds for the reaction diffusion system if for each point in space the reaction system has a globally attracting hyperbolic equilibrium this work is planned as an initial step towards understanding the connection between the asymptotics of reaction diffusion systems with small diffusion coefficients and that of the corresponding reaction systems

which traces are spectral abstract among ideals of compact operators on a hilbert space we identify a subclass of those closed with respect to the logarithmic submajorization within this subclass we answer the questions asked by pietsch 22 and by dykema figiel weiss and wodzicki 7 in the first case we show that lidskii type formulae hold for every trace on such ideal in the second case we provide the description of the commutator subspace associated with a given ideal finally we prove that a positive trace on an arbitrary ideal is spectral if and only if it is monotone with respect to the logarithmic submajorization

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

the action functional in non commutative geometry we establish the equality between the restriction of the adler manin wodzicki residue or non commutative residue to pseudodifferential operators of order n on ann dimensional compact manifoldm with the trace which j dixmier constructed on the macaev ideal we then use the latter trace to recover the yang mills interaction in the context of non commutative differential geometry

data based optimal smoothing of orthogonal series density estimates let f be a density possessing some smoothness properties and let x 1 cdots x n be independent observations from f some desirable properties of orthogonal series density estimates f n m lambda of f of the form f n m lambda t sum n nu 1 frac hat f nu 1 lambda nu 2m phi nu t where phi nu is an orthonormal sequence and hat f nu 1 n sum n j 1 phi nu x j is an estimate of f nu int phi nu t f t dt are discussed the parameter lambda plays the role of a bandwidth or smoothing parameter and m controls a shape factor the major novel result of this note is a simple method for estimating lambda and m from the data in an objective manner to minimize integrated mean square error the results extend to multivariate estimates

spatial heterogeneity of resources versus lotka volterra dynamics abstract the problem is motivated by a consideration of two phenotypes of a species in a strongly heterogenous spatial environment the phenotypes vary in their diffusion rates and interspecific interactions the aim is to investigate the relative strengths of the diffusion and reaction effects the problem is thus of competing species type but there are many questions which arise which are not covered by standard theory we investigate in particular the stability of the equilibria and the existence of coexistence solutions with emphasis on cases where the spatial variation of the environment becomes large we discuss briefly the implications of our results for the principle of competitive exclusions and for the question of the evolution of diffusion discussed in dockery et al j math biol 37 1998 61 83

traces of compact operators and the noncommutative residue abstract we extend the noncommutative residue of m wodzicki on compactly supported classical pseudo differential operators of order d and generalise a connes trace theorem which states that the residue can be calculated using a singular trace on compact operators contrary to the role of the noncommutative residue for the classical pseudo differential operators a corollary is that the pseudo differential operators of order d do not have a unique trace pseudo differential operators can be non measurable in connes sense other corollaries are given clarifying the role of dixmier traces in noncommutative geometry including the definitive statement of connes original theorem

global dynamics of a classical lotka volterra competition diffusion advection system abstract in this paper we study a classical two species lotka volterra competition diffusion advection system where the diffusion and advection rates of two competitors are supposed to be proportional by employing the principal spectral theory we first establish a key a priori estimate on the co existence positive steady state which is a powerful tool to link the local and global dynamics we then further present a complete classification on all possible long time dynamical behaviors by appealing to the theory of monotone dynamical systems lastly we apply these results to a special situation where two species are competing for the same resources and obtain a sharp criteria in term of certain variable parameters for all kinds of global dynamics this work gives a positive answer to the conjecture proposed by lou et al in 34 by considering a more general model under certain conditions and also can be seen as a further development of he and ni 19 for competition diffusion system where we bring new ingredients in the arguments to overcome the difficulty caused by the involvement of advection

uniqueness and complete dynamics in heterogeneous competition diffusion systems in this paper we study the interactions between diffusion and heterogeneity of the environment in the classical diffusive lotka volterra competition systems in the weak competition case we establish the uniqueness hence the global asymptotic stability of coexistence steady states under various circumstances and thereby we obtain a complete understanding of the change in dynamics when one of the interspecific competition coefficients is small

dynamics of a diffusive competition model in spatially heterogeneous environment abstract we discuss the effects of dispersal rate and spatial heterogeneity on population dynamics via a diffusive lotka volterra competition model under homogeneous dirichlet boundary conditions by the theory of monotone dynamical system we show that in the weak competition case for some ranges of dispersal rate of competing species 1 if dispersal rate of competing species 2 is sufficiently small there exists a unique coexistence steady state of the model which is globally asymptotically stable moreover for certain ranges of dispersal rate of competing species 1 we derive the limit behavior of the coexistence steady state as dispersal rate of competing species 2 tends to zero on the other hand we demonstrate that for some larger scopes of dispersal rate of competing species 1 if dispersal rate of competing species 2 is small enough there is no coexistence steady state whereas one of the semi trivial steady states of the model is globally asymptotically stable

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

the amount of noise inherent in bandwidth selection for a kernel density estimator abstract any practical method of constructing a bandwidth must depend only on a statistical sample and should produce some sort of estimate of this bandwidth the purpose of this paper is to show that there a well defined limits to the accuracy of all data driven bandwidth estimates put another way there is an unbridgeable gap between the minimum integrated square error attained using a optimal bandwidth and the minimum achievable integrated square error using a data driven bandwidth estimate additional keywords stochastic processes cross validation and random variables

recovery of a quantile function from moments the problem of recovering a quantile function of a positive random variable via the values of moments or given the values of its laplace transform is studied two new approximations as well as two new estimates of a quantile function given the sample from underlying distribution are proposed the uniform and l 1 upper bounds of proposed estimates are derived the plots illustrate the behavior of the recovered approximants for the moderate and large sample sizes

comparison of bandwidth selection methods for kernel smoothing of roc curves in this paper we compared four non parametric kernel smoothing methods for estimating an roc curve based on a continuous scale test all four methods produced a smooth roc curve of the test the difference in these four methods lay with the way they chose their bandwidth parameters to assess the relative performance of the four bandwidth selection methods we conducted a simulation study using different underlying distributions along with varied sample sizes the results from our simulation study suggested that the kernel smoothing method originally proposed by altman and leger for estimation of the distribution function was the best choice for estimation of an roc curve we illustrated these methods with a real example

global dynamics of the lotka volterra competition diffusion system diffusion and spatial heterogeneity i in the first part of this series of three papers we investigate the combined effects of diffusion spatial variation and competition ability on the global dynamics of a classical lotka volterra competition diffusion system we establish the main results that determine the global asymptotic stability of semitrivial as well as coexistence steady states hence a complete understanding of the change in dynamics is obtained immediately our results indicate confirm that when spatial heterogeneity is included in the model diffusion driven exclusion could take place when the diffusion rates and competition coefficients of both species are chosen appropriately 2015 wiley periodicals inc

a smooth nonparametric conditional quantile frontier estimator abstract traditional estimators for nonparametric frontier models dea fdh are very sensitive to extreme values outliers recently aragon et al 2005 nonparametric frontier estimation a conditional quantile based approach econometric theory 21 358 389 proposed a nonparametric frontier model and estimator based on a suitably defined conditional quantile which is more robust to extreme values outliers their estimator is based on a nonsmooth empirical conditional distribution in this paper we propose a new smooth nonparametric conditional quantile estimator for the frontier model our estimator is a kernel based conditional quantile estimator that builds on early work of azzalini 1981 a note on the estimation of a distribution function and quantiles by a kernel method biometrika 68 326 328 it is computationally simple resistant to outliers and extreme values and smooth in addition the estimator is shown to be consistent and n asymptotically normal under mild regularity conditions we also show that our estimator s variance is smaller than that of the estimator proposed by aragon et al a simulation study confirms the asymptotic theory predictions and contrasts our estimator with that of aragon et al

global dynamics of a competition model with nonlocal dispersal ii the full system abstract in this paper we study logistic models with non local dispersal for two competing species in the weak competition case the global asymptotic behavior of solutions is completely classified provided one of the following conditions is valid one diffusion rate is small one diffusion rate is large two diffusion rates are close this is a continuation of the work 16 our results illustrate the effects of non local dispersals and interspecific competition coefficients in the competition system with spatially heterogeneous environments

sums of commutators in non commutative banach function spaces let m be a ii factor and denote by its normal faithful semi finite trace for any rearrangement invariant kothe function space x on 0 let x m be the associated non commutative banach function space this paper is concerned with ideals in m of the form ix m m x m that are contained in lp m for some p 0 it is proved that an element t in ix m is a finite sum of commutators of the form a b with a ix m and b m if and only if the function t 1t t t d t belongs to x where t is the brown spectral measure of t and t t t is the non increasing rearrangement of the function with respect to t this extends to general banach function spaces a result obtained by kalton for quasi banach ideals of compact operators and implies that the dixmier s trace of a quasi nilpotent element in l1 m is always zero

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

the effects of diffusion and spatial variation in lotka volterra competition diffusion system ii the general case abstract it is well known that the interactions between diffusion and spatial heterogeneity could create very interesting phenomena in this series of two papers using the classical lotka volterra competition system we will illustrate the combined effects of dispersal and spatial variations on the outcome of the competition in part i with the total resources being fixed at exactly the same level we first show that a heterogeneous distribution of resources is usually superior to its homogeneous counterpart in the presence of diffusion then we study the more general case when both species have heterogeneous carrying capacities but still with the same total resources limiting behaviors of co existence steady states as the dispersal rates tend to 0 or are also obtained in part ii we continue our investigation but under much broader situations including different strengths and distributions of the resources and with different competition abilities

a note on the estimation of a distribution function and quantiles by a kernel method there has recently been extensive work on the estimation by kernel methods of probability densities and their derivatives for a review see wertz 1978 for estimating the cumulative distribution function the empirical distribution function f is in a sense already quite smooth although it is known that further smoothing can be an advantage the object of the present note is to report briefly an investigation of kernel estimation applied to distribution function and quantiles further details are available from the author for independent and identically distributed random variables x1 x the usual kernel estimator of the density f is

commutator structure of operator ideals commutator structure of operator ideals ken dykema 1 tadeusz figiel gary weiss and mariusz wodzicki department of mathematics texas a m university college station tx 77843 3368 usa b instytut matematyczny polskiej akademii nauk 81 825 sopot poland department of mathematical sciences university of cincinnati cincinnati oh 45221 0025 usa department of mathematics university of california berkeley ca 94720 3840 usa

global stability and pattern formation in a nonlocal diffusive lotka volterra competition model abstract a diffusive lotka volterra competition model with nonlocal intraspecific and interspecific competition between species is formulated and analyzed the nonlocal competition strength is assumed to be determined by a diffusion kernel function to model the movement pattern of the biological species it is shown that when there is no nonlocal intraspecific competition the dynamics properties of nonlocal diffusive competition problem are similar to those of classical diffusive lotka volterra competition model regardless of the strength of nonlocal interspecific competition global stability of nonnegative constant equilibria are proved using lyapunov or upper lower solution methods on the other hand strong nonlocal intraspecific competition increases the system spatiotemporal dynamic complexity for the weak competition case the nonlocal diffusive competition model may possess nonconstant positive equilibria for some suitably large nonlocal intraspecific competition coefficients

pulsating fronts and front like entire solutions for a reaction advection diffusion competition model in a periodic habitat abstract this paper is devoted to the study of pulsating fronts and pulsating front like entire solutions for a reaction advection diffusion model of two competing species in a periodic habitat under certain assumptions the competition system admits a leftward and a rightward pulsating fronts in the bistable case in this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the x axis which turn out to be unique and liapunov stable 2 dimensional manifolds of solutions furthermore the leftward and rightward pulsating fronts are on the boundary of these 2 dimensional manifolds the others behave as the two pulsating fronts propagating from one side of the x axis the faster one then invades the slower one as t these kinds of pulsating front like entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat

on a lotka volterra competition diffusion advection system homogeneity vs heterogeneity abstract as we know interactions between movement and environmental heterogeneity could create very interesting phenomena in population dynamics in this paper based on a lotka volterra competition diffusion advection system we try to reveal the combined effect of movement and spatial variations on the outcome of competition for the purpose of comparison we suppose that the total resources for two populations are fixed at the same level but one distribution is even across space while the other one not our main results suggest that the contest between heterogeneous and homogeneous distribution is very complex either one of the two competitors becomes the final single winner exclusion or both populations coexist eventually coexistence depending closely on the movement rates and the shape of the given resource

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p