fixed width multiple comparison procedures using common random numbers for steady state simulations abstract suppose that there are k 2 different systems i e stochastic processes where each system has an unknown steady state mean performance we consider the problem of running a two stage simulation using common random numbers to construct fixed width confidence intervals for two multiple comparison problems under the assumptions that the stochastic processes representing the simulation output of the different systems satisfy a functional central limit theorem and that the asymptotic covariance matrix satisfies a condition known as sphericity we prove that our confidence intervals are asymptotically valid as the desired half width of the confidence intervals tend to zero we develop both absolute and relative width confidence intervals empirical results are presented indicating the procedures robustness to violations of the sphericity assumption

strong consistency of the variance estimator in steady state simulation output analysis in this paper we study the problem of variance estimation in steady state simulation output analysis a confidence interval for the mean of an output sequence may be constructed using a weakly consistent estimator of the time average variance constant of the process moreover a strongly consistent estimator of the variance constant is required for fixed width confidence interval estimation procedures to be asymptotically valid here we establish sufficient conditions to ensure strong consistency of the estimator of the variance constant obtained by several widely used methods of steady state simulation output analysis namely batch means overlapping batch means spaced batch means and standardized time series area estimator we also introduce the overlapping area estimator of the variance constant the established sufficient conditions provide qualitative insight into the relation between the correlation of the process and certain parameters e g batch size of the output method used

the fabry ehrenpreis gap theorem and linear differential equations of infinite order pour comprendre les theoremes de saut de type fabry dans un cadre general on utilise des operateurs de convolution determines par des hyperfonctions supportees par l origine

simple zeros of the zeta function of a quadratic number field i let k be a fixed quadratic extension of q and write k s for the dedekind zeta function of k where s it it is wellknown and easy to prove that the number nk t of zeros of k s in the region 0 1 0 t t satisfies text n text k left text t right frac text t pi log t as t on the other hand not much is known about the number of text n text k text left text t right that are simple

on the bezout problem and area of interpolating varieties in cn ii we give an estimate on the growth of the volume of zeros of entire holomorphic maps from c n to c n the methods of li taylor for the estimates for certain holomorphic maps are used to provide the better estimate which applies to general entire holomorphic maps and is shown to be precise by an example

on lojasiewicz type inequalities for exponential polynomials abstract let f z n n 1 f n z e i n z be an exponential polynomial that is the coefficients f n z are polynomials in one complex variable we discuss the question of how close distinct roots of the equation f z 0 can get to each other

multiple comparison procedures for steady state simulations suppose that there are k 2 different systems i e stochastic processes where each system has an unknown steady state mean performance and unknown asymptotic variance we allow for the asymptotic variances to be unequal and for the distributions of the k systems to be different we consider the problem of running independent single stage simulations to make multiple comparisons of the steady state means of the different systems we derive asymptotically valid as the run lengths of the simulations of the systems tend to infinity simultaneous confidence intervals for each of the following problems all pairwise comparisons of means all contrasts multiple comparisons with a control and multiple comparisons with the best our confidence intervals are based on standardized time series methods and we establish the asymptotic validity of each under the sole assumption that the stochastic processes representing the simulation output of the different systems satisfy a functional central limit theorem although simulation is the context of this paper the results naturally apply to asymptotically stationary time series

simultaneous confidence intervals for all distances from the best in practice comparisons with the best are often the ones of primary interest in this paper parametric and nonparametric simultaneous upper confidence intervals for all distances from the best are derived under the location model their improvement upon the results of bechhofer 1954 gupta 1956 1965 fabian 1962 and desu 1970 in the parametric case is discussed in the nonparametric case no comparable confidence statements were available previously

poles of artin l functions and the strong artin conjecture we show that if the l function of an irreducible 2 dimensional complex galois representation over q is not automorphic then it has infinitely many poles in particular the artin conjecture for a single representation implies the corresponding strong artin conjecture

batch size effects in the analysis of simulation output batching is a commonly used method for calculating confidence intervals on the mean of a sequence of correlated observations arising from a simulation experiment several recent papers have considered the effect of using batch sizes too small to satisfy assumptions of normality and or independence and the resulting incorrect probabilities of the confidence interval covering the mean this paper quantifies the effects of using batch sizes larger than necessary to satisfy normality and independence assumptions these effects include 1 correct probability of covering the mean 2 an increase in expected half length 3 an increase in the standard deviation and coefficient of variation of the half length and 4 an increase in the probability of covering points not equal to the mean for any sample size and independent and normal batch means the results are 1 the effects of less than 10 batches are large and the effects of more than 30 batches small and 2 additional batches have lesser effects on

the use of subseries values for estimating the variance of a general statistic from a stationary sequence let z i infty i infty be a strictly stationary infty mixing sequence without specifying the dependence model giving rise to z i and without specifying the marginal distribution of z i we address the question of variance estimation for a general statistic t n t n z 1 z n for estimating var t n from just the available data z 1 z n we propose computing subseries values t m z i 1 z i 2 z i m 0 leq i i m leq n these subseries values are used as replicates to model the sampling variability of t n in particular we use adjacent nonoverlapping subseries of length m m n with m rightarrow infty and m n n rigtharrow 0 our variance estimator is just the usual sample variance computed amongst these subseries values after appropriate standardization this estimator is shown to be consistent under mild integrability conditions we present optimal i e minimum m s e choices of m n for the special case where t n overset z n and z i is a normal ar 1 sequence a simulation study is conducted showing that those same choices of m n are effective when t n is a robust estimator of location and z i is subject to contamination

simple zeros of the ramanujan τ dirichlet series it is known that l s has all of its non real zeros in the strip l l 2 t 6 t l simple zeros in the region 0 t t the methods of these papers however will probably not give any results for l s the levinson method requires a mean value theorem for the square of the modulus of s multiplied by a dirichlet polynomial the length of this polynomial is critical and it seems unlikely that such a theorem with a long enough polynomial could be proven at this time in the case of l as far as the method of 1 is concerned we relied on a factorization of the dedekind zeta function as a product of the

tests of sphericity of normal distributions and the analysis of repeated measures designs the locally best invariant test statistic for testing sphericity of normal distributions is shown to be a simple function of the box geisser greenhouse degrees of freedom correction factor in a repeated measures design because of this relationship it provides a more intuitively appealing test of the necessary and sufficient conditions for valid f tests in repeated measures analysis of variance than the likelihood ratio test the properties of the two tests are compared and tables of the critical values of the box geisser greenhouse correction factor are given

multiple comparisons with the best treatment abstract let 1 2 k be k 2 sources of observations treatments populations and suppose the goodness of treatment i is characterized by the size of an unknown real valued parameter i let k max1 i k i if i is preferred to j when i j the parameters i k i i 1 2 k reflect in an inverse sense the goodness of each treatment relative to the best treatment a general technique for obtaining simultaneous confidence intervals on the i is demonstrated with several examples this technique can be applied in any setting where comparison with control intervals can be computed regarding any j as the control these results have special importance in ranking and selection problems in that the process of generating upper bounds on the i generates traditional confidence statements of both the indifference zone and the subset selection schools simultaneously as established by hsu 1981

constrained simultaneous confidence intervals for multiple comparisons with the best for comparing k treatment effects 01 02 ok often the parameters of primary interest are 0i maxeioi0 i 1 k in this article we develop constrained 100p two sided simultaneous confidence intervals for 0i maxj i0j which we refer to as constrained mcb intervals it turns out that the lower bounds of the intervals imply indifference zone selection inference and the upper bounds of these intervals imply subset selection inference each given at the same confidence level 100p as mcb we also extend our

simulation output analysis using standardized time series the method of standardized time series sts was proposed by schruben as an approach for constructing asymptotic confidence intervals for the steady state mean from a single simulation run the sts method cancels out the variance constant while other methods attempt to consistently estimate the variance constant our goal in this paper is to generalize the sts method and to study some of its basic properties starting from a functional central limit theorem fclt for the sample mean of the simulated process a class of mappings of c 0 1 to is identified each of which leads to a sts confidence interval one of these mappings leads to the batch means method a lower bound is obtained for the expected length of the asymptotic as the run size becomes large sts confidence intervals this lower bound is not attained but can be approached arbitrarily closely by sts confidence intervals methods that consistently estimate the variance constant do realize this lower bound the variance of the length of a sts confidence interval is of larger order in the run length than is that for the regenerative method

simulation run length control in the presence of an initial transient this paper studies the estimation of the steady state mean of an output sequence from a discrete event simulation it considers the problem of the automatic generation of a confidence interval of prespecified width when there is an initial transient present it explores a procedure based on schruben s brownian bridge model for the detection of nonstationarity and a spectral method for estimating the variance of the sample mean the procedure is evaluated empirically for a variety of output sequences the performance measures considered are bias confidence interval coverage mean confidence interval width mean run length and mean amount of deleted data if the output sequence contains a strong transient then inclusion of a test for stationarity in the run length control procedure results in point estimates with lower bias narrower confidence intervals and shorter run lengths than when no check for stationarity is performed if the output sequence contains no initial transient then the performance measures of the procedure with a stationarity test are only slightly degraded from those of the procedure without such a test if the run length is short relative to the extent of the initial transient the stationarity tests may not be powerful enough to detect the transient resulting in a procedure with unreliable point and interval estimates

an investigation of finite sample behavior of confidence interval estimators we investigate the small sample behavior and convergence properties of confidence interval estimators cies for the mean of a stationary discrete process we consider cies arising from nonoverlapping batch means overlapping batch means and standardized time series all of which are commonly used in discrete event simulation the performance measures of interest are the coverage probability and the expected value and variance of the half length we use empirical and analytical methods to make detailed comparisons regarding the behavior of the cies for a variety of stochastic processes all the cies under study are asymptotically valid however they are usually invalid for small sample sizes we find that for small samples the bias of the variance parameter estimator figures significantly in cie coverage performance the less bias the better a secondary role is played by the marginal distribution of the stationary process we also point out that some cies require fewer observations before manifesting

on a theorem of polya and levinson abstract in the present paper we prove a variation of the polya levinson interpolation result in the class of entire functions of infraexponential type for the sequences n n 1 satisfying limn n n and n m c n m that is without any condition on the asymptotic directions of the sequence

average growth estimates for hyperplane sections of entire analytic sets suppose x is an analytic subvariety of c a basic problem in value distribution theory is to relate the growth of the volume of x to that of intersections of x with linear subspaces p of with dimx dim p n in contrast to the case where x is algebraic as described by the bezout theorem for a fixed subspace p there is no relationship between the counting functions of x and xnp it is easy to find examples where x n p is empty while x is large on the other hand an example given in 6 shows that x n p can be of infinite order even if x is of order 0 however average transcendental bezout estimates have been given by carlson 1 5 gruman 8 9 and stoll 18 gruman showed that for almost all linear subspaces p x n p has the same order of growth as x and is transcendental ifx is transcendental

two stage stopping procedures based on standardized time series we propose some new two stage stopping procedures to construct absolute width and relative width confidence intervals for a simulation estimator of the steady state mean of a stochastic process the procedures are based on the method of standardized time series proposed by schruben and on stein s two stage sampling scheme we prove that our two stage procedures give rise to asymptotically valid confidence intervals as the prescribed length of the confidence interval approaches zero and the size of the first stage grows to infinity the sole assumption required is that the stochastic process satisfy a functional central limit theorem

confidence interval estimation using standardized time series observations of a stationary stochastic process can be transformed into a standardized time series this paper presents a lemma giving the asymptotic properties of this standardized series under quite general conditions in particular the conditions are satisfied by stationary discrete event simulations confidence intervals can be constructed using this lemma for illustration we develop two easily computed interval estimators for the process mean when independent replications of the series are available such as in computer simulation experiments these interval estimators may be combined with the classical confidence interval estimator these interval estimators also tend to compensate for simulation initialization bias if the sign of the bias is known in an empirical study using three elementary simulated processes the interval estimators presented here compare favorably with the classical interval estimator in a recent paper goldsman and schruben goldsman d l schruben 1982 asymptotic properties of some confidence interval estimators tech rep 544 school of o r i e cornell university ithaca ny show that the asymptotic properties of the confidence intervals presented in this paper strictly dominate those of classical confidence intervals

planning queueing simulations simple heuristic formulas are developed to estimate the simulation run lengths required to achieve desired statistical precision in queueing simulations the formulas are intended to help in the early planning stages before any data have been collected the queueing simulations considered are single replications one long run conducted to estimate steady state characteristics such as expected equilibrium queue lengths the formulas can be applied to design simulation experiments to develop and evaluate queueing approximations in fact this work was motivated by efforts to develop approximations for packet communication networks with multiple classes of traffic having different service characteristics and bursty arrival processes in addition to indicating the approximate simulation run length required in each case of a designed experiment the formulas can help determine what cases to consider what statistical precision to aim for and even whether to conduct the experiment at all the formulas are bas

additivizations of nonadditive measures the concept of an additivization of a nonadditive probability measure is introduced and studied the relation to cooperative game theory is explored and it is shown that in many cases of interest additivizations may be considered as a special type of semivalues

using common random numbers for indifference zone selection and multiple comparisons in simulation we present a general recipe for constructing experiment design and analysis procedures that simultaneously provide indifference zone selection and multiple comparison inference for choosing the best among k simulated systems we then exhibit two such procedures that exploit the variance reduction technique of common random numbers to reduce the sample size required to attain a fixed precision one procedure is based on the bonferroni inequality and is guaranteed to be statistically conservative the other procedure is exact under a specific dependence structure but may be slightly liberal otherwise both are easy to apply requiring only simple calculations and tabled constants we illustrate the procedures with a numerical example

achieving a confidence interval for parameters estimated by simulation this paper presents a procedure for determining the number of simulation observations required to achieve a preassigned confidence interval for means estimated by simulation this procedure which is simple to implement and efficient to use is compared with two other methods for determining the required sample size in a simulation run the empirical results show that this procedure gives good results in the precision of estimated means and in sample size requirement

interpolating varieties for entire functions of minimal type abstract we obtain both necessary and sufficient conditions for a discrete variety in c n to be an interpolating variety for entire functions of minimal type

an invariance principle for certain dependent sequences let x 1 x 2 cdots be a strictly stationary second order sequence which is associated i e is such that any two coordinatewise nondecreasing functions of the x i s of finite variance are nonnegatively correlated if sum j operatorname cov x 1 x j infty then the partial sum processes w n t defined in the usual way so that w n m n x 1 cdots x m me x 1 sqrt n for m 1 2 cdots converge in distribution on c lbrack 0 t rbrack to a wiener process this result is based on two general theorems concerning associated random variables which are of independent interest

asymptotic properties of some confidence interval estimators for simulation output the classical confidence interval estimator commonly used in simulation is compared with four new estimators based on standardization of a time series presented in a previous paper these new interval estimators are shown to have asymptotic properties that strictly dominate the classical estimator when used with data from independent simulation replications or means of batched observations from a single simulation run two of the new estimators also can be used with a single unbatched replication of a simulation program a situation where the classical estimator is not defined