weak convergence of linear and quadratic forms and related statements on lp approximability abstract in this paper we obtain central limit theorems for quadratic forms of non causal short memory linear processes with independent identically distributed innovations nabeya and tanaka 1988 suggested the format which links the asymptotic distribution to integral operators in their approach integral operators had to have continuous symmetric kernels mynbaev 2001 employed the theory of approximations to get rid of the continuity requirement here we go one step further by lifting the kernel symmetry condition also we establish l p approximability of the special sequences which arise in the theory of regressions with slowly varying regressors

convexity of sample path performance and strong consistency of infinitesimal perturbation analysis estimates a strong consistency of infinitesimal perturbation analysis ipa estimates for a class of stochastic processes satisfying certain convex properties is established the approach is based on convex analysis of sample paths applications of the results to tandem and cyclic queueing systems are discussed possible extensions to more general systems are also illustrated

a trend resistant test for structural change based on ols residuals abstract we discuss the null distribution and local power of an ols based version of the cusum test and show that this contrary to other tests for structural change like e g the ploberger kontrus kramer fluctuation test can also be applied to trending data

confidence sets for the date of a single break in linear time series regressions this paper considers the problem of constructing confidence sets for the date of a single break in a linear time series regression we establish analytically and by small sample simulation that the current standard method in econometrics for constructing such confidence intervals has a coverage rate far below nominal levels when breaks are of moderate magnitude given that breaks of moderate magnitude are a theoretically and empirically relevant phenomenon we proceed to develop an appropriate alternative we suggest constructing confidence sets by inverting a sequence of tests each of the tests maintains a specific break date under the null hypothesis and rejects when a break occurs elsewhere by inverting a certain variant of a locally best invariant test we ensure that the asymptotic critical value does not depend on the maintained break date a valid confidence set can hence be obtained by assessing which of the sequence of test statistics exceeds a single number

asymptotic theory of a test for the constancy of regression coefficients against the random walk alternative one component of the coefficients follows a random walk process we discuss the limiting null behavior of the test statistic without assuming normality under two situations where the initial value of the random walk process is known or unknown the limiting distribution is that of a quadratic functional of brownian motion and the characteristic function is obtained from the fredhohm determinant associated with a certain integral equation the limiting distribution is then computed by numerical inversion of the characteristic function

production control of manufacturing systems with production rate dependent failure rates it is known that for single part type production systems with homogeneous markovian machine failure rates special single threshold feedback policies called hedging point policies are optimal and the stationary probability distribution of the part type surplus for given tentative hedging point values can be obtained analytically this approach is extended to multiple threshold policies with production rate dependent machine failure rates it is shown that the stationary distribution of the part type surplus can be obtained under the extended policy and in the presence of production rate dependent failure rates the advantage of multiple threshold policies is that they can provide a piecewise constant step function approximation of any feedback policy it is observed that hedging point policies are not always optimal and in fact feedback policies are not always optimal either

median unbiased estimation of coefficient variance in a time varying parameter model abstract this article considers inference about the variance of coefficients in time varying parameter models with stationary regressors the gaussian maximum likelihood estimator mle has a large point mass at 0 we thus develop asymptotically median unbiased estimators and asymptotically valid confidence intervals by inverting quantile functions of regression based parameter stability test statistics computed under the constant parameter null these estimators have good asymptotic relative efficiencies for small to moderate amounts of parameter variability we apply these results to an unobserved components model of trend growth in postwar u s per capita gross domestic product the mle implies that there has been no change in the trend growth rate whereas the upper range of the median unbiased point estimates imply that the annual trend growth rate has fallen by 0 9 per annum since the 1950s

high order operator splitting methods based on an integral deferred correction framework integral deferred correction idc methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties in this paper we construct high order operator splitting schemes using the idc procedure to solve initial value problems ivps we present analysis to show that the idc methods can correct for both the splitting and numerical errors lifting the order of accuracy by r with each correction where r is the order of accuracy of the method used to solve the correction equation we further apply this framework to solve partial differential equations pdes numerical examples in two dimensions of linear and nonlinear initial boundary value problems are presented to demonstrate the performance of the proposed idc approach

ode solvers using band limited approximations we use generalized gaussian quadratures for exponentials to develop a new ode solver nodes and weights of these quadratures are computed for a given bandlimit c and user selected accuracy e so that they integrate functions e i b x for all b

perturbation analysis of discrete event systems concepts algorithms and applications abstract perturbation analysis of discrete event systems is a technique that provides estimates of performance sensitivities with respect to system parameters by analyzing a single sample path of a discrete event system this technique saves a great amount of computation in simulation and can be applied to on line performance optimization of real world systems where changing the values of parameters is infeasible this paper gives an overview of the main concepts general methodologies applications and further directions of the area

testing for a slowly changing level with special reference to stochastic volatility abstract a test for the presence of a stationary first order autoregressive process embedded in white noise is constructed so as to be relatively powerful when the autoregressive parameter is close to one the test statistic is shown to have a cramer von mises distribution in large samples a comparison is made with some standard tests for serial correlation in the context of a stochastic volatility model and the proposed test is shown to have relatively high power for the parameter values typically found with daily financial time series it is recommended that tests on returns be carried out using absolute values

testing for stationarity with a break in this paper we investigate a test for the null hypothesis of trend stationarity with a structural change against a unit root we derive the limiting distribution of an lagrange multiplier lm test statistic and its characteristic function under a sequence of local alternatives the local limiting power of the lm test depends on the persistence of the stationary component of the process and the more persistent the process the less powerful is the test statistic we also propose a test statistic that does not depend on the fraction of the pre break points to the sample size under the null hypothesis which we call the ps test though it is convenient for the critical point not to depend on the break point the ps test is found to be less powerful than the lm test under the alternative close to the null hypothesis finite sample simulations show that when the break point is known the lm test tends to be oversized when the process is rather persistent while the size distortion of the ps test is not so pronounced on the other hand the empirical sizes of both tests are close to the nominal one when the break point is estimated by the least squares method though the power decreases compared with the known break point case

the effect of serial correlation on tests for parameter change at unknown time it is shown that serial correlation can produce striking effects in distributions of change point statistics failure to account for these effects is shown to invalidate change point tests either through increases in the type 1 error rates if low frequency spectral mass predominates in the spectrum of the noise process or through diminution of the power of the tests when high frequency mass predominates these effects are characterized by the expression 2i f o fjr f a da where f is the spectral density of the noise process in sample survey work this is known as the design effect or deff simple precise adjustments to change point test statistics which account for serial correlation are provided the same adjustment applies to all commonly used regression models residual processes are derived for both stationary time series satisfying a moment condition and for general linear regression models with stationary error structure 1 introduction stochastic models for time sequenced data are generally characterized by several unknown parameters these parameters may change over time and if the changes when they occur do so unannounced and at unknown time points then the associated inferential problem is referred to as the change point problem various important application areas of statistics involve change detection in a central way two of these areas are quality assurance and environmental monitoring most of the statistics commonly applied to the change point problem involve cumulative sums or partial sums of regression residuals the distribution theory for these statistics has been computed under the assumption that the error process for the regression model is white noise in this paper we consider linear regression of a random variable against general nonstochastic functions of time but with error variables that form a serially correlated time series we then examine the large sample properties of the stochastic processes defined by the partial sums of the regression residuals large sample distribution theory for fixed sample size statistics used to detect changes in regression parameters is usually derived by computing the distributions of various functionals on

a highly accurate solver for stiff ordinary differential equations we introduce a solver for stiff ordinary differential equations that is based on the deferred correction scheme for the corresponding picard integral equation our solver relies on the assumption that the solution can be accurately represented by a combination of carefully selected complex exponentials the solver s accuracy and stability rely on the computation of highly accurate quadrature weights for the integration of the selected exponentials on equidistant nodes we analyze our solver stability and accuracy regions and demonstrate its fast convergence on stiff problems the solver is combined with an adaptive step size scheme employing interpolation formulas for the exponentially fitted solution

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

optimal average cost manufacturing flow controllers convexity and differentiability the authors consider the control of a production facility consisting of a single workstation with multiple failure modes and part types using a continuous flow control model technical issues concerning the convexity and differentiability of the differential cost function are investigated it is proven that under an optimal control policy the differential cost is c sup 1 on attractive control switching boundaries

structural conditions for perturbation analysis derivative estimation finite time performance indices in recent years there has been a surge of research into methods for estimating derivatives of performance measures from sample paths of stochastic systems in the case of queueing systems typical performance measures are mean queue lengths throughputs etc and the derivatives estimated are with respect to system parameters such as parameters of service and interarrival time distributions derivative estimates potentially offer a general means of optimizing performance and are useful in sensitivity analysis this paper concerns one approach to derivative estimation known as infinitesimal perturbation analysis we first develop a general framework for these types of estimates then give simple sufficient conditions for them to be unbiased the key to our results is identifying conditions under which certain finite horizon performance measures are almost surely continuous functions of the parameter of differentiation throughout an interval the sufficient conditions we introduce are formulated in the setting of generalized semi markov processes but translate into readily verifiable conditions for queueing systems these results substantially extend the domain of problems in which infinitesimal perturbation analysis is provably applicable

rank tests for short memory stationarity abstract we propose a rank test of the null hypothesis of short memory stationarity possibly after linear detrending for the level stationarity hypothesis the test statistic we propose is a modified version of the popular kpss statistic in which ranks substitute the original observations we prove that the rank kpss statistic shares the same limiting distribution as the standard kpss statistic under the null and diverges under i 1 alternatives for the trend stationarity hypothesis we apply the same rank kpss statistic to the residuals of a theil sen regression on a linear trend we derive the asymptotic distribution of the theil sen estimator under short memory errors and prove that the theil sen detrended rank kpss statistic shares the same weak limit as the least squares detrended kpss we study the asymptotic relative efficiency of our test compared to the kpss and prove that it may have unbounded efficiency gains under fat tailed distributions compensated by very moderate efficiency losses under thin tailed distributions for this and other reasons discussed in the body of the article our rank kpss test turns out to be a valuable alternative to the kpss for most real world economic and financial applications the weak convergence results and asymptotic representations proved in this article should interest a wider audience than that concerned with stationarity testing as they extend to ranks analogous invariance principles widely used in unit root econometrics

a manufacturing system with general stationary failure process stability and ipa of hedging control policies this paper concerns a new methodology for the adaptive optimization of piecewise deterministic non markovian systems via a simple example of interest in manufacturing this methodology takes into account the fact that piecewise deterministic systems are rarely markovian and that classical control theory based on dynamic programming of markovian systems cannot provide quantitative answers in most realistic situations the example considered is that of a single machine single part production system from the point of view of the control theory by kimemia and gershwin 1983 we obtain stochastic gradient estimates via the perturbation analysis of ho and cao 1991 by the regenerative method of konstantopoulos and zazanis 1992 in view of stochastic optimization its mathematical justification requires a careful study of the regenerative structure of the process in particular we give the necessary and sufficient conditions of stability of this system via the method of loynes 1962

maximum likelihood estimation of regression models with stochastic trend components abstract this article is concerned with the estimation of a regression model with a stochastic trend component it is shown that the probability of estimating the trend to be deterministic is very sensitive to the type of likelihood function used as the basis of inference

optimal control of production rate in a failure prone manufacturing system we address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost where certain cost rates are specified for both positive and negative inventories and there is a constant demand rate for the commodity produced the underlying theoretical problem is the optimal control of a continuous time system with jump markov disturbances with an infinite horizon discounted cost criterion we use two complementary approaches first proceeding informally and using a combination of stochastic coupling linear system arguments stable and unstable eigenspaces renewal theory parametric optimization etc we arrive at a conjecture for the optimal policy then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides singularity of the optimal control problem smoothness and validity of the dynamic programming equation etc to give a rigorous proof of optimality of the conjectured policy it is hoped that both approaches will find uses in other such problems also we obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number which we call the optimal inventory level if the current inventory level exceeds the optimal one should not produce at all if less one should produce at the maximum rate while if exactly equal one should produce exactly enough to meet demand we also give a simple explicit formula for the optimal inventory level

perturbation analysis for the design of flexible manufacturing system flow controllers dynamic allocation of stochastic capacity among competing activities in a just in time manufacturing environment is addressed by optimal flow control optimal policies are characterized by generally intractable bellman equations a near optimal controller design technique is proposed it provides an approximate numerical solution to the bellman equation a tight lower bound for the optimality gap of tractable near optimal controller designs and a building block for improved near optimal controller designs that rely on the decomposition of a multiple part type problem to smaller two or three part type problems computational experience is reported for two and three part type problems

production control of a manufacturing system with multiple machine states the production control of a single product manufacturing system with arbitrary number of machine states failure modes is discussed the objective is to find a production policy that would meet the demand for the product with minimum average inventory or backlog cost the optimal production policy has a special structure and is called a hedging point policy if the hedging points are known the optimal production rate is readily specified assuming a set of tentative hedging points the simple structure of the optimal policy is utilized to find the steady state probability distribution of the surplus inventory or backlog once this function is determined the average surplus cost is easily calculated in terms of the values of the hedging points the average cost is then minimized to find the optimum hedging points

deciding between i 1 and i 0 abstract a class of procedures that consistently classify the stochastic component of a time series as being integrated either of order zero i 1 or one i 1 are proposed for general i 0 or i 1 processes and polynomial or piecewise linear detrending large sample bayesian inference is free of nuisance parameters describing short run dynamics and requires specifying priors only on the point hypotheses i 1 and i 1 thereby avoiding problematic choices of parametric priors over roots and nuisance parameters applied to the nelson plosser 1982 data with linear detrending these procedures largely support nelson and plosser s original inferences with piecewise linear detrending these data are typically uninformative producing bayes ratios close to one

on the compression of low rank matrices a procedure is reported for the compression of rank deficient matrices a matrix a of rank k is represented in the form a u circ b circ v where b is a k times k submatrix of a and u v are well conditioned matrices that each contain a k times k identity submatrix this property enables such compression schemes to be used in certain situations where the singular value decomposition svd cannot be used efficiently numerical examples are presented

dynamic linear models for time series components abstract this paper describes a general state space approach for the modelling of the unobserved components trend cycle seasonality trading day variations and irregulars of a time series and the calculation of the mean square errors of the estimated unobserved components the unobserved components models are presented in a state space form the kalman filter and fixed interval smoother are applied on the observed series to obtain the estimates of the unobserved components and their corresponding variances implementation problems related to the estimation of the initial conditions and the initial values for the variances of both the observation noise and noise processes of the unobserved components are solved using the estimates of the unobserved components from x 11 arima dagum 1980 the estimation of the signal to noise ratio is made by maximum likelihood using the method of scoring the mle of the noise variance is obtained analytically conditional on the estimates of the signal to noise ratios statistical tests to distinguish between alternative models are also provided

production control of manufacturing systems a multiple time scale approach in this note a stochastic production model containing processes with different time scales is developed it is shown that if the time scales of the processes are very different some hierarchical algorithms that are much more efficient than the standard policy iteration method can be developed to find the optimal production control moreover if the time scales fall far apart the optimal control of a deterministic limiting problem depending only on the mean characteristics of the processes can be used to approximate the optimal control of the original problem the limiting problem has much lower dimension than its original counterpart and thus is much easier to solve a numerical example is used to illustrate the potential of the proposed approach

a numerical method to approximate optimal production and maintenance plan in a flexible manufacturing system abstract the simultaneous planning of the production and the maintenance in a flexible manufacturing system is considered in this paper the manufacturing system is composed of one machine that produces a single product there is a preventive maintenance plan to reduce the failure rate of the machine this paper is different from the previous researches in this area in two separate ways first the failure rate of the machine is supposed to be a function of its age second we assume that the demand of the manufacturing product is time dependent and its rate depends on the level of advertisement on that product the objective is to maximize the expected discounted total profit of the firm over an infinite time horizon in the process of finding a solution to the problem we first characterize an optimal control by introducing a set of hamilton jacobi bellman partial differential equations then we realize that under practical assumptions this set of equations can not be solved analytically thus to find a suboptimal control we approximate the original stochastic optimal control model by a discrete time deterministic optimal control problem then proposing a numerical method to solve the steady state riccati equation we approximate a suboptimal solution to the problem

integral deferred correction methods constructed with high order runge kutta integrators spectral deferred correction sdc methods for solving ordinary differential equations odes were introduced by dutt greengard and rokhlin 2000 it was shown in that paper that sdc methods can achieve arbitrary high order accuracy and possess nice stability properties their sdc methods are constructed with low order integrators such as forward euler or backward euler and are able to handle stiff and non stiff terms in the odes in this paper we use high order runge kutta rk integrators to construct a family of related methods which we refer to as integral deferred correction idc methods the distribution of quadrature nodes is assumed to be uniform and the corresponding local error analysis is given the smoothness of the error vector associated with an idc method measured by the discrete sobolev norm is a crucial tool in our analysis the expected order of accuracy is demonstrated through several numerical examples superior numerical stability and accuracy regions are observed when high order rk integrators are used to construct idc methods

stochastic dynamic job shops and hierarchical production planning this paper presents an asymptotic analysis of hierarchical production planning in a general manufacturing system consisting of a network of unreliable machines producing a variety of products the concept of a dynamic job shop is introduced by interpreting the system as a directed graph and the structure of the system dynamics is characterized for its use in the asymptotic analysis the optimal control problem for the system is a state constrained problem since the number of parts in any buffer between any two machines must remain nonnegative a limiting problem is introduced in which the stochastic machine capacities are replaced by corresponding equilibrium mean capacities as the rate of change in machine states approaches infinity the value function of the original problem is shown to converge to that of the limiting problem and the convergence rate is obtained furthermore near optimal controls for the original problem are constructed from near optimal controls of the limiting problem and an error estimate is obtained on the near optimality of the constructed controls