Student t-tests and compound tests to detect transients in simulated time series

Given a stationary stochastic process, with unknown mean μ, interest often lies in developing estimates and confidence intervals for μ. Unfortunately, if the process is actually only asymptotically stationary, the transient behavior can affect the validity of these estimates and confidence intervals. We present a test for detecting transients in simulation output based on the familiar Student t-test. We also demonstrate and exploit the independence of this t-test and certain other initialization bias tests to construct “compound” initialization bias tests. We demonstrate and evaluate our tests using Monte Carlo simulation.     

A Generalized Smoother for Linear Ordinary Differential Equations

Ordinary differential equations (ODEs) are equalities involving a function and its derivatives that define the evolution of the function over a prespecified domain. The applications of ODEs range from simulation and prediction to control and diagnosis in diverse fields such as engineering, physics, medicine, and finance. Parameter estimation is often required to calibrate these theoretical models to data. While there are many methods for estimating ODE parameters from partially observed data, they are invariably subject to several problems including high computational cost, complex estimation procedures, biased estimates, and large sampling variance. We propose a method that overcomes these issues and produces estimates of the ODE parameters that have less bias, a smaller sampling variance, and a 10-fold improvement in computational efficiency. The package GenPen containing the Matlab code to perform the methods described in this article is available online.     

A simple test for the parametric form of the variance function in nonparametric regression

In this paper a new test for the parametric form of the variance function in the common nonparametric regression model is proposed which is applicable under very weak smoothness assumptions. The new test is based on an empirical process formed from pseudo residuals, for which weak convergence to a Gaussian process can be established. In the special case of testing for homoscedasticity the limiting process is essentially a Brownian bridge, such that critical values are easily available. The new procedure has three main advantages. First, in contrast to many other methods proposed in the literature, it does not depend directly on a smoothing parameter. Secondly, it can detect local alternatives converging to the null hypothesis at a rate n ^−1/2. Thirdly, in contrast to most of the currently available tests, it does not require strong smoothness assumptions regarding the regression and variance function. We also present a simulation study and compare the tests with the procedures that are currently available for this problem and require the same minimal assumptions.     

Improving estimation in speckled imagery

We propose an analytical bias correction for the maximum likelihood estimators of theG ₁ ⁰ distribution. This distribution is a very powerful tool for speckled imagery analysis, since it is capable of describing a wide range of target roughness. We compare the performance of the corrected estimators with the corresponding original version using Monte Carlo simulation. This second-order bias correction leads to estimators which are better from both the bias and mean square error criteria.     

Testing for serial correlation of unknown form in cointegrated time series models

Portmanteau test statistics are useful for checking the adequacy of many time series models. Here we generalized the omnibus procedure proposed by Duchesne and Roy (2004,Journal of Multivariate Analysis,89, 148–180) for multivariate stationary autoregressive models with exogenous variables (VARX) to the case of cointegrated (or partially nonstationary) VARX models. We show that for cointegrated VARX time series, the test statistic obtained by comparing the spectral density of the errors under the null hypothesis of non-correlation with a kernel-based spectral density estimator, is asymptotically standard normal. The parameters of the model can be estimated by conditional maximum likelihood or by asymptotically equivalent estimation procedures. The procedure relies on a truncation point or a smoothing parameter. We state conditions under which the asymptotic distribution of the test statistic is unaffected by a data-dependent method. The finite sample properties of the test statistics are studied via a small simulation study.     

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

Approximating nonstationaryPh(t)/M(t)/s/c queueing systems

Nonstationary phase processes are defined and a surrogate distribution approximation (SDA) method for analyzing transient and nonstationary queueing systems with nonstationary phase arrival processes is presented. Regardless of system capacityc, the SDA method requires the numerical solution of only 6K differential equations, whereK is the number of phases in the arrival process, compared to theK(c+1) Kolmogorov forward equations required for the classical method of solution. Time-dependent approximations of mean and variance of the number of entities in the system and the number of busy servers are obtained. Empirical test results over a wide range of systems indicate the SDA is quite accurate.This research was partially funded by National Science Foundation grant ECS-8404409.     

Evaluating alternative system configurations using simulation: A nonparametric approach

The real utility of simulation lies in comparing different alternatives that might represent competing system designs. Conventional statistical techniques are not directly applicable to the analysis of simulation output data in the evaluation of competing alternatives since the usual assumptions of normality and common variance are difficult to justify in simulation experiments. This paper revisits a known nonparametric test whose application has recently become feasible due to considerable increases in computing power:randomization tests assess the significance of the observed value of the test statistic by evaluating different permutations of the data. The procedure only requires invariance of the data under all permutations.     

Second derivative estimation using harmonic analysis

Simulation sensitivity analysis is an important problem for simulation practitioners analyzing complex systems. The significance of this problem has resulted in the development of various gradient estimators that can be used to address this issue. Although higher derivative estimators have been discussed concurrently, less attention has been given to assess the efficiency and feasibility of computing such estimators. In this paper, two second derivative estimators are presented. The first estimators, called the HFD estimators, combine harmonic gradient estimators with finite differences second derivative estimators. The resulting hybrid estimators requireO(p) fewer simulation runs to implement compared to the straightforward finite differences approach, wherep is the number of input parameters in the simulation model. The second estimators, called the HA estimators, incorporate harmonic analysis directly, requiring one or two simulation runs to implement, depending on whether a control variate simulation run is made. Expressions for the bias and the variance of the HFD and the HA estimators (with and without variance reduction techniques) are derived. Optimal mean squared error convergence rates are also discussed. In particular, the convergence rates for both these estimators are shown to be the same, though the computational performance of the HFD estimators is better than that for the HA estimators on anM/M/1 queue simulation model. Computational results for the HFD estimators on an (s, S) inventory system simulation model are also included.     

Estimating and Visualizing Conditional Densities

Abstract

We consider the kernel estimator of conditional density and derive its asymptotic bias, variance, and mean-square error. Optimal bandwidths (with respect to integrated mean-square error) are found and it is shown that the convergence rate of the density estimator is order n ^–2/3. We also note that the conditional mean function obtained from the estimator is equivalent to a kernel smoother. Given the undesirable bias properties of kernel smoothers, we seek a modified conditional density estimator that has mean equivalent to some other nonparametric regression smoother with better bias properties. It is also shown that our modified estimator has smaller mean square error than the standard estimator in some commonly occurring situations. Finally, three graphical methods for visualizing conditional density estimators are discussed and applied to a data set consisting of maximum daily temperatures in Melbourne, Australia.     

The Hamiltonians of Pseudorelativistic Atoms with Finite-Mass Nuclei: The Structure of the Discrete Spectrum

We study the structure of the discrete spectrum of pseudorelativistic Hamiltonians H for atoms and positive ions with finite-mass nuclei and with n electrons, where n 1 is arbitrary. The center-of-mass motion cannot be separated, and hence we study the spectrum of the restriction H _P of H to the subspace of states with given value P of the total momentum of the system. For the operators H _P we discover a) two-sided estimates for the counting function of the discrete spectrum _d (H _P ) of H _P in terms of the counting functions of some effective two-particle operators; b) the leading term of the spectral asymptotics of _d (H _P ) near the lower bound inf _ess(H _P ) of the essential spectrum of H _P . The structure of the discrete spectrum of such systems was known earlier only for n=1.     

Spectra of Hamiltonians of molecule pseudorelativistic electrons in spaces of functions with permutational and point symmetry

We study the spectral properties of the Hamiltonian H _n of n pseudorelativistic electrons in the Coulomb field of k fixed nuclei in spaces of functions having arbitrary given types of permutational and point symmetry. For this operator, we establish the location of the essential spectrum, obtain two-sided estimates of the discrete spectrum counting function in terms of the counting functions of the discrete spectrum of some two-particle nonrelativistic operators, and find the leading term of the spectral asymptotics.     

Order statistics applications to queueing and scheduling problems

We prove several basic combinatorial identities and use them in two applications: the queue inference engine (QIE) and earliest due date rule (EDD) scheduling. Larson (1990) introduced the QIE. His objective was to deduce the behavior of a multiserver queueing system without observing the queue. With only a Poisson arrival assumption, he analyzed the performance during a busy period. Such a period starts once all servers are busy with the queue empty, and it ends as soon as a server becomes idle. We generalize the standard order statistics result for Poisson processes, and show how to sample a busy period in the M/M/c system. We derive simple expressions for the variance of the total waiting time in the M/M/c and M/D/1 queues given that n Poisson arrivals and departures occur during a busy period. We also perform a probabilistic analysis of the EDD for a one-machine scheduling problem with earliness and tardiness penalties. The schedule is without preemption and with no inserted idle time. The jobs are independent and each may have a different due date. For large n, we show that the variance of the total penalty costs of the EDD is linear in n. The mean of the total penalty costs of the EDD is known to be proportional to the square root of n (see Harel (1993)). This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on accelerated sequential estimation of the mean of NEF-PVF distributions

The minimum risk point estimation for the mean is addressed for a natural exponential family (NEF) that also has a power variance function (PVF) under a loss function given by the squared error plus linear cost. An appropriate accelerated version of the full purely sequential methodology of Bose and Boukai (1993b, submitted) is proposed along the lines of Mukhopadhyay (1993a, Tech. Report, No. 93-27, Department of Statistics, University of Connecticut) in order to achieve operational savings. The main result provides the asymptotic second-order expansion of the regret function associated with the accelerated sequential estimator of the population mean.     

Bias and variance reduction techniques for bootstrap information criteria

We discuss the problem of constructing information criteria by applying the bootstrap methods. Various bias and variance reduction methods are presented for improving the bootstrap bias correction term in computing the bootstrap information criterion. The properties of these methods are investigated both in theoretical and numerical aspects, for which we use a statistical functional approach. It is shown that the bootstrap method automatically achieves the second-order bias correction if the bias of the first-order bias correction term is properly removed. We also show that the variance associated with bootstrapping can be considerably reduced for various model estimation procedures without any analytical argument. Monte Carlo experiments are conducted to investigate the performance of the bootstrap bias and variance reduction techniques.     

Vertices of given degree in series‐parallel graphs

We show that the numbers of vertices of a given degree k ≥ 1 in several kinds of series‐parallel labeled graphs of size n satisfy a central limit theorem with mean and variance proportional to n, and quadratic exponential tail estimates. We further prove a corresponding theorem for the number of nodes of degree two in labeled planar graphs. The proof method is based on generating functions and singularity analysis. In particular, we need systems of equations for multivariate generating functions and transfer results for singular representations of analytic functions. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Empirical likelihood ratio tests for multivariate regression models

This paper proposes some diagnostic tools for checking the adequacy of multivariate regression models including classical regression and time series autoregression. In statistical inference, the empirical likelihood ratio method has been well known to be a powerful tool for constructing test and confidence region. For model checking, however, the naive empirical likelihood (EL) based tests are not of Wilks’ phenomenon. Hence, we make use of bias correction to construct the EL-based score tests and derive a nonparametric version of Wilks’ theorem. Moreover, by the advantages of both the EL and score test method, the EL-based score tests share many desirable features as follows: They are self-scale invariant and can detect the alternatives that converge to the null at rate n ^−1/2, the possibly fastest rate for lack-of-fit testing; they involve weight functions, which provides us with the flexibility to choose scores for improving power performance, especially under directional alternatives. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of possible alternatives. A simulation study is carried out and an application for a real dataset is analyzed.      

Nonnegatively realizable spectra with two positive eigenvalues

Boyle and Handelman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316.] characterized all lists of n complex numbers that can be the nonzero spectrum of a nonnegative matrix. This article presents a constructive proof of this result in the special case when the lists are real and contain two positive numbers and n ? 2 negative numbers. A bound for the number of zeros that needs to be added to the list to achieve a nonnegative realization is presented in this case.     

Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients

Summary. General error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperature-dependent coefficients. These variable coefficients turn the diffusion and the buoyancy terms to be nonlinear, which increases the nonlinearity of the problems. An argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions. Error estimates are also provided for schemes modified by approximate coefficients, which are used conveniently in practical computations.Mathematics Subject Classification (2000): 65M12, 65M60, 76M10     

On some representations of sample variance

The usual formula for variance depending on rounding off the sample mean lacks precision, especially when computer programs are used for the calculation. The well-known simplification of the total sums of squares does not always give benefit. Since the variance of two observations is easily calculated without the use of a sample mean, and the variance of a sample of n observations is the average of the variances of observations based on n(n-1)/2 distinct subsets of units of size 2 from the sample, it is argued that this sense of pairing may result in precision. Some other forms of variance are presented which provide some insight into it. The contribution of a new observation of variance is highlighted, which is important in sequential sampling. Notions are illustrated with examples.