simultaneous nonparametric inference of time series we consider kernel estimation of marginal densities and regression functions of stationary processes it is shown that for a wide class of time series with proper centering and scaling the maximum deviations of kernel density and regression estimates are asymptotically gumbel our results substantially generalize earlier ones which were obtained under independence or beta mixing assumptions the asymptotic results can be applied to assess patterns of marginal densities or regression functions via the construction of simultaneous confidence bands for which one can perform goodness of fit tests as an application we construct simultaneous confidence bands for drift and volatility functions in a dynamic short term rate model for the u s treasury yield curve rates data

consecutive optimizers for a partitioning problem with applications to optimal inventory groupings for joint replenishment we consider several subclasses of the problem of grouping n items indexed 1 2 n into m subsets so as to minimize the function gs1 sm in general these problems are very difficult to solve to optimality even for the case m 2 we provide several sufficient conditions on gi that guarantee that there is an optimum partition in which each subset consists of consecutive integers or else the partition s1 sm satisfies a more general condition called semiconsecutiveness moreover by restricting attention to consecutive or semiconsecutive partitions we can solve the partition problem in polynomial time for small values of m if in addition g is symmetric then the partition problem is solvable in purely polynomial time we apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper we also relate the results to the neyman pearson lemma in statistical hypothesis testing and to a graph partitioning problem of barnes and hoffman

optimal partitioning which maximizes the sum of the weighted averages we consider an optimal partitioning problem that occurs in the assignment of computer jobs to a multiple cache and in other combinatorial optimization problems for a given set of n elements where each element i has a given frequency pi and a specific weight wi we would like to divide the elements into m mutually exclusive groups such that the sum over all the groups of the average group weight is maximal we characterize the optimal solution and present an algorithm which is polynomial in n for obtaining the optimal partitioning optimal partitioning policies for two groups has an especially simple characterization there exist two numbers and with minwi maxwi such that all the elements with weight wi satisfying wi belong to one group and all other elements belong to the other group a modification of this policy gives the optimal partitioning for an arbitrary number of groups

why instantons are monopoles it is shown that instantons are hyperbolic monopoles for the loop group with non maximal symmetry breaking at infinity

optimal partitions having disjoint convex and conic hulls leta1 an be distinctk dimensional vectors we consider the problem of partitioning these vectors intom sets so as to maximize an objective which is a quasi convex function of the sum of vectors in each set we show that there exists an optimal partition whose sets have pairwise disjoint conic hulls we also show that if the number of vectors in each of the sets is constrained then a weaker conclusion holds namely there exists an optimal partition whose sets have pairwise disjoint convex hulls the results rely on deriving necessary and sufficient conditions for a point to be an extreme point of a corresponding polytope

simultaneous confidence bands for linear regression and smoothing suppose we observe y i f x i varepsilon i i 1 ldots n we wish to find approximate 1 alpha simultaneous confidence regions for f x x in mathscr x our regions will be centered around linear estimates hat f x of nonparametric or nonparametric f x there is a large amount of previous work on this subject substantial restrictions have been usually placed on some or all of the dimensionality of x the class of functions f that can be considered the class of linear estimates hat f and the region mathscr x the method we present is an approximation to the tube formula dn can be used for multidimensional x and a wide class of linear estimates by considering the effect of bias we are able to relax assumptions on the class of functions f which are considered simultaneous and numerical computations are used to illustrate the performance

bias corrected confidence bands in nonparametric regression summary bias corrected confidence bands for general nonparametric regression models are considered we use local polynomial fitting to construct the confidence bands and combine the cross validation method and the plug in method to select the bandwidths related asymptotic results are obtained our simulations show that confidence bands constructed by local polynomial fitting have much better coverage than those constructed by using the nadaraya watson estimator the results are also applicable to nonparametric autoregressive time series models

the joint replenishment problem with general joint cost structures we consider inventory systems with several distinct items demands occur at constant item specific rates the items are interdependent because of jointly incurred fixed procurement costs the joint cost structure reflects general economies of scale merely assuming a monotonicity and concavity submodularity property under a power of two policy each item is replenished with constant reorder intervals which are power of two multiples of some fixed or variable base planning period our main results include a proof that depending upon whether the base planning period is fixed or variable the best among all power of two policies has an average cost which comes within either 6 or 2 of an easily computable lower bound for the minimum cost value we also derive two efficient algorithms to compute an optimal power of two policy the proposed algorithms generate as a by product a specific cost allocation of the joint cost structure to the individual items with this specific allocation the problem with separable costs is in fact equivalent to the original problem with nonseparable joint costs in the sense that the two problems share the same sets of optimal power of two policies with identical associated long run average costs

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

confidence bands for regression functions abstract a method for obtaining conservative confidence bands for regression functions in low dimensions is given of particular importance is the applicability of the method to nonparametric regression computations examples and comparisons indicate the effectiveness of the method

confidence bands in nonparametric regression abstract new bias corrected confidence bands are proposed for nonparametric kernal regression these bands are constructed using only a kernel estimator of the regression curve and its data selected bandwidth they are shown to have asymptotically correct coverage properties and to behave well in a small sample study one consequence of the large sample developments is that bonferroni type bands for the regression curve at the design points also have conservative asymptotic coverage behavior with no bias correction

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

a polynomial time algorithm for shaped partition problems we consider the class of shaped partition problems of partitioning n given vectors in d dimensional criteria space into p parts so as to maximize an arbitrary objective function which is convex on the sum of vectors in each part subject to arbitrary constraints on the number of elements in each part this class has broad expressive power and captures np hard problems even if either d or p is fixed in contrast we show that when both d and p are fixed the problem can be solved in strongly polynomial time our solution method relies on studying the corresponding class of shaped partition polytopes such polytopes may have exponentially many vertices and facets even when one of d or p is fixed however we show that when both d and p are fixed the number of vertices of any shaped partition polytope is o n d p choose 2 and all vertices can be produced in strongly polynomial time

inference of trends in time series summary we consider statistical inference of trends in mean non stationary models a test statistic is proposed for the existence of structural breaks in trends on the basis of a strong invariance principle of stationary processes we construct simultaneous confidence bands with asymptotically correct nominal coverage probabilities the results are applied to global warming temperature data and nile river flow data our confidence band of the trend of the global warming temperature series supports the claim that the trend is increasing over the last 150 years

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

optimal locations on a line are interleaved suppose n facilities are to be located on a fine segment so as to minimize cost function one might expect that the facilities optimal locations have the following interleaving property if one of the n facilities is removed and if the locations of the others are shifted by reoptimizing each remaining facility s location shifts toward the location of the one removed but not farther toward it than the original location of the adjacent facility this paper presents two models whose solutions have this interleaving property and four examples of such models an additive criterion is used in one model a minimax criterion in the other in the additive model the minimum cost is a convex function of n in the minimax model it is nonincreasing

confidence intervals for nonparametric curve estimates numerous nonparametric regression methods exist that yield consistent estimators of function curves often one is also interested in constructing confidence intervals for the unknown function when a function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 on average but because bias in nonparametric estimation is not uniform they do not hold the desired level uniformly at all design points most research in this area has focused on mean squared error properties of the estimator for example mise itself a global measure in addition measures like mise are one step removed from the practical issue of coverage probability recent work that focuses on coverage probability has considered only coverage in an average sense ignoring the important issue of uniformity of coverage across the design space to deal with this problem a new estimator is developed which uses a local cross validation criterion lcv to determi

modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model summary the behaviour of nonlinear deterministic vibrations has been studied by many authors and may typically include such features as jump phenomena and limit cycles nonlinear random vibrations in continuous time have also been studied and these may commonly give rise to the phenomenon of amplitude dependent frequency a discrete time series model is introduced which may be demonstrated to have properties similar to those of nonlinear random vibrations this model is of autoregressive form with amplitudedependent coefficients and may be estimated using an extension of a method for estimating linear time series models the model is fitted to the canadian lynx data and demonstrates that it may be possible to regard the periodic behaviour of this series as being generated by some underlying self exciting mechanism

structured partitioning problems in many important combinatorial optimization problems such as bin packing allocating customer classes to queueing facilities vehicle routing multi item inventory replenishment and combined routing inventory control an optimal partition into groups needs to be determined for a finite collection of objects each is characterized by a single attribute the cost is often separable in the groups and the group cost often depends on the cardinality and some aggregate measure of the attributes such as the sum or the maximum element an upper bound capacity may be specified for the cardinality of each group and the number of groups in the partition may either be fixed or variable the objects are indexed in nondecreasing order of their attribute values and within a given partition the groups are indexed in nondecreasing order of their cardinalities we identify easily verifiable analytical properties of the group cost function under which it is shown that an optimal partition exists of one of three increas

technical note a partitioning problem with additive objective with an application to optimal inventory groupings for joint replenishment we consider a problem of optimal grouping and provide conditions under which an optimal partition of an ordered set s r1 rn consists of subsets of consecutive elements we transform the problem into the problem of finding a shortest path on a directed acyclic graph with n 1 vertices for which efficient algorithms exist these results may be used to solve the problem of grouping n items in stock into subgroups with a common order cycle per group so as to minimize the resulting economic order quantity costs

nonparametric pricing of interest rate derivative securities we propose a nonparametric estimation procedure for continuous time stochastic models because prices of derivative securities depend crucially on the form of the instantaneous volatility of the underlying process we leave the volatility function unrestricted and estimate it nonparametrically only discrete data are used but the estimation procedure still does not rely on replacing the continuous time model by some discrete approximation instead the drift and volatility functions are forced to match the densities of the process we estimate the stochastic differential equation followed by the short term interest rate and compute nonparametric prices for bonds and bond options

incorporating inventory and routing costs in strategic location models we consider a supply chain design problem where the decision maker needs to decide the number and locations of the distribution centers dcs customers face random demand and each dc maintains a certain amount of safety stock in order to achieve a certain service level for the customers it serves the objective is to minimize the total cost that includes location costs and inventory costs at the dcs and distribution costs in the supply chain we show that this problem can be formulated as a nonlinear integer programming model for which we propose a lagrangian relaxation based solution algorithm by exploring the structure of the problem we find a low order polynomial algorithm for the nonlinear integer programming problem that must be solved in solving the lagrangian relaxation sub problems we present computational results for several instances of the problem with sizes ranging from 40 to 320 customers our results show the benefits of having an integrated supply chain design framework that includes location inventory and routing decisions in the same optimization model

variance function estimation in regression the effect of estimating the mean abstract the authors consider estimation of a variance function g in regression problems such estimation requires simultaneous estimation of the mean function f we obtain sharp results on the extent to which the smoothness of f influences best rates of convergence for estimating g for example in nonparametric regression with two derivatives on g classical rates of convergence are possible if and only if the unknown f satisfies a lipschitz condition of order 1 3 or more if a parametric model is known for g then g may be estimated n 1 2 consistently if and only if f is lipschitz of order 1 2 or more optimal rates of convergence are attained by kernel estimators

generalized likelihood ratio statistics and wilks phenomenon likelihood ratio theory has had tremendous success in parametric inference due to the fundamental theory of wilks yet there is no general applicable approach for nonparametric inferences based on function estimation maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting even if they exist they are hard to find and can not be optimal as shown in this paper we introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics a new s wilks phenomenon is unveiled we demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow chi 2 distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including gaussian white noise models nonparametric regression models varying coefficient models and generalized varying coefficient models we further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by ingster they can even be adaptively optimal in the sense of spokoiny by using a simple choice of adaptive smoothing parameter our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation

worst case analysis of heuristics for the bin packing problem with general cost structures we consider the famous bin packing problem where a set of items must be stored in bins of equal capacity in the classical version the objective is to minimize the number of bins used motivated by several optimization problems that occur in the context of the storage of items we study a more general cost structure where the cost of a bin is a concave function of the number of items in the bin the objective is to store the items in such a way that total cost is minimized such cost functions can greatly alter the way the items should be assigned to the bins we show that some of the best heuristics developed for the classical bin packing problem can perform poorly under the general cost structure on the other hand the so called next fit increasing heuristic has an absolute worst case bound of no more than 1 75 and an asymptotic worst case bound of 1 691 for any concave and monotone cost function our analysis also provides a new worst case bound for the well studied next tit decreasing heuristic when

optimality of monotone assemblies for coherent systems composed of series modules we consider a system with m modules as components these modules are composed of parts of finitely many types and the number of parts of each type that is needed in each of the modules is given e g module i requires nui parts of type u parts of the same type may have different reliabilities but they are functionally interchangeable a module works if and only if all of its parts work i e the internal composition of the modules has series structure an assembly of the modules consists of an assignment of each of the u i nui parts to the modules such that each module meets its specification by getting the required number of parts of each type such an assembly is called monotone if the best parts of each type go to one module the next best parts of each type go to a second module and so on until finally the last module gets the worst parts of each type we prove that for coherent systems there always exists a monotone assembly which maximizes the reliability of the system furthermore we obt

fully nonparametric estimation of scalar diffusion models we propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process we show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter the asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens we prove almost sure consistency and weak convergence to mixtures of normal laws where the mixing variates depend on the chronological local time of the underlying diffusion process that is the random time spent by the process in the vicinity of a generic spatial point the estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

sieve bootstrap for smoothing in nonstationary time series we propose a sieve bootstrap procedure for time series with a deterministic trend the sieve for constructing the bootstrap is based on nonparametric trend estimation and autoregressive approximation for some noise process the bootstrap scheme itself does i i d resampling of estimated innovations from fitted autoregressive models we show the validity and indicate second order correctness of such sieve bootstrap approximations for the limiting distribution of nonparametric linear smoothers the resampling can then be used to construct nonparametric confidence intervals for the underlying trend in particular we show asymptotic validity for constructing confidence bands which are simultaneous within a neighborhood of size of the order of the smoothing bandwidth our resampling procedure yields satisfactory results in a simulation study for finite sample sizes we also apply it to the longest series of total ozone measurements from arosa switzerland and find a significant decreasing trend

confidence bands in nonparametric time series regression we consider nonparametric estimation of mean regression and conditional variance or volatility functions in nonlinear stochastic regression models simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct the imposed dependence structure allows applications in many linear and nonlinear auto regressive processes the results are applied to the s p 500 index data

strong invariance principles for dependent random variables we establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds applications to linear and some nonlinear processes are discussed strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions