RO-AI55614? CONFIDENCE INTERYALS USIlNG THE REGENERTIYE METHOD FOR 1/1 SIMULATION OUTPUT.. (U) STANFORD UNJY CA DEPT OF 1"'7 1OPERTIONS RESEARCH P W GLYNN ET AL. SEP 84 TR-3 UNCLSSIFIED RO-2927. 3-MA DAA029-84-K-930 F/G 12/1 NL
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CONFIDENCE INTERVALS USING THE REGENERATIVE METHOD FOR SIMULATION OUTPUT ANALYSIS by Peter W. Glynn and Donald L. Iglehart TECHNICAL REPORT NO. 3 September 1984 In Prepared under the Auspices of U.S. Army Research Contract* DAAG29-84-K-0030 Approved for public release: distribution unlimited. Reproduction in Whole or in Part is Permitted for any Purpose of the United States Government. DEPARTMENT OF OPERATIONS RESEARCH,. STANFORD UNIVERSITY STANFORD, CALIFORNIA -~ *This research was also partially supported under National Science Foundation Grant MCS-8203483."... 85 01 29 113
1. I UTU 'IM O" The regenerative clthod is a mathematically rigorous procedure for obtaining confidence intervals for steady state parameters. In order to properly assess the regenerative method, it is necessary to discuss those characteristics that make a confidence interval "good." 2. QULITAhfVK STCTU OF CONFlWUDIgK INTKiALS Given a parameter p, a confidence interval for p is generally based on a limit theorem of the form (2.1) (r t - O/V t L" as t.a where L is a finite r.v. with a continuous distribution function; the parameter t measures the simulation effort required to obtain r t and v. The processes r t and vt will be called a point estimate (for 1) and a normalizing process, respectively; we shall always assume v is positive. To obtain an approximate 100 (1 - a)z confidence interval for 4, select Z,, z2 such that P(z I. L z2} - I a. Then, for large t, (2.2) Ir z 2 vt, rr z- V]- contains I with probability I - a * The following hierarchy of properties largely determines the quality of the confidence interval... a.) consistency of rl: if r t is not consistent, v does not tend to t t zero, and confidence interval half-length does not shrink to zero with Increasing t. b.) asymptotic mean square error of r : in general, r t is asymptotically normal. Then, there exists a non-negative a such that.'. sc -'
(2.3) t 1/2(rt -L) O a(o,1) tt is as small as possible. c.) expected half-width of confidence interval: by (2.2), the expected half-width of the confidence interval in (22z z ) Ev:. In general, when asymptotic normality holds, (22 2)Evt 1 (Z (2 - z I)v/t1 1 for some v ;the goal is to minimize v. d.) variability of half-width of confidence interval: the variance of the half -width is given by (z - z )2var v Under quite general. conditions, (2 2 - z 1) 2 var,t _(Z2 - z )2a/ the goal is to minimize a e.) approximation error: Let At I P(zxl (rt P )/vt Y ~ -P{ZL < L < Yz be the coverage error for the confidence interval. Berry-Esseen considerations suggest that, in general, at 1/2i minimization of ~lis desirable. 3. TU UGKAII h Loosely speaking, a regenerative process is one which looks like a sequence of Independent ad Identically distributed (i.i.d.) r.v.'s, when viewed on -n appropriate random time scale. More precisely, X (1(t) : t > 0) is a regenerative process with regeneration times 0 To1 < T 1 <. if (1~k, X(s) : Tk_1. a < Tk) is a sequence of i.i.d. random elements, where rk-t k For examples of such processes, sea 2... -7-7
o. Crane and Lemoine (1977). Given a reel-valued function defined on the state - space of I, "-. (3.1) rt fi f(xls))ds +r s.s. 0 - under mild assumptions on X and f. The goal of a steady state simulation Is to produce confidence Intervals for r Accesi nn For if T' N( t) - max k> 0 T t) and ay I " f(x(s))ds T 1 then (3.2) r~ t N(t)/iN(t) where Yn' n are the sample man& of the Y' anid 's, respectively. Regenerative structure ensures that (( T ) 1-1) is a sequence -- of i.l.d, random vectors, so that (3.1) and (3.2) together suggest that I r EYI/ I Then, by (3.2),.. (t t) 122 where ZKeYk - r has mean zero. Standard central limit theory arguments prove that 1/2 t (r- r) o1(0,1) 3
V.o S a(zi)/ui 1 If 2 + T * Furthermore, -hr 1 2 a Z 2. -t d( t 8.3., u t SNl N it) ii 82 1 (y /'C,, -C We conclude that (r t - r)/v t N(O,1) where v. ft/t, 2 is the normalizing process for the regenerative method. The qualitative structure of the regenerative confidence interval can be summarized as follows: a.) rt is consistent for r b.) MSE(r) - (a 2 (ZI)/BTI)/t (note that any confidence interval method using the sample mean r- as a point estimate vill have the same HSE). C.) (z 2 -z 1 )Evt - MET1I et)1/2 where ;(a) solves P(N(o,t) _ z(a)) - I - a/2 d.) t(s 2 - a) 2 var vl * 0 (in fact, (z2-zl)2var vt a 2 see GLYNN and IGLEHART (1984)), e.) p is currently unknown. Note that is a reflection of approximation error due to the bias of rt and skewnese/kurtosis effects. It is to be anticipated that the i.i.d. 4
structure associated with the regenerative viewpoint can be used to reduce these errors. For exaxple, NEKETON and HEIDELBERGER (1982) developed a point estimate which Is asymptotically equivalent to r but which significantly reduces bias. Also, GLYNN (1982) proposed a procedure for reducing 0 in the closely related problem of estimating r on the time scale of regenerative cycles. As discussed above, the regenerative method is a theoretically sound procedure for the steady state confidence interval problem. The main advantages of the method are: I.) its good asymptotic properties (for example, a2(v 0/t ) indicates the accurate "variance constant estimatisn possible with. the regenerative method), ii.) iii.) iv.) the ability to make small-sample corrections to reduce approximation error, the i.i.d. structure allows one to develop procedures for a host of other estimation problems (e.g. comparison of stochastic systems; see HEIDELBERGER and IGLEHART (1979)), no prior parameters are needed as input for the method, other than run length. The main disadvantages of the method are: i.) the requirement to identify regeneration times means that the method is hard to "black box", ii.) the method may behave unsatisfactorily if the expected time between regenerations i long. (Estimation of parameters for such simulations is likely to be difficult using any method.) Preparation of this paper was supported by the U.S. Army Research Office under Contract DAAG29-SI-K-0030 and the National Science Foundation under Grant OSF HCS-6203463.... 5
1. CRANE, M.A. and LEMOINE, A.J. (1977). An introduction to the Regenerative method for Simulation Analysis. (Lecture Notes in Control and Information sciences.) Springer-verlag: New York, Heidelberg, Berlin. 2. GLYNN, P.W. (1982). Asymptotic theory for nonparametric confidence intervals. Technical Report 19, Department of operations Research, Stanford University, Stanford, Calif. 3. GLYNN, P.W. and IGLEHART, D.L. (1984). The joint limit distribution of the sample mean and regenerative variance estimator. Forthcoming technical report, Department of operations Research, Stanford University, Stanford, Calif. 4. HEIDELBERGER, P. and IGLEHART, D.L. (1979). Comparing stochastic systems using regenerative simulations with common random numbers. Adv. Appl. Probability, n1, 804-819. I 5. MEKETON, Hf.S. and HEIDELBERSER, P. (1982). A renewal theoretic approach to bias reduction in regenerative simulations. Management Sci., 26, 173-181. 1 6 V..**.**.****.*
--.... " II UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (tme. Date Entered) REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETIG FORM 1. REPORT NUMI[R 2. GOVT ACCESSION NO- L RECIPIENT'S CATALOG NUM119R 4. TITLE (and SAuite) S. TYPE O r REPORT & PERIOD COVERED CONFIDENCE INTERVALS USING THE REGENERATIVE METHOD FOR SIMULATION OUTPUT ANALYSIS TECHNICAL REPORT i. PERFORMING ORG. REPORT NUMBERt 7. AuTNOR(s) A. CONTRACT OR GRANT NUMIERf-O) PETER W. GLYNN DONALD L. IGLEHART DAAG29-84-K-0030 $, PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK DEPARTMENT OF OPERATIONS RESEARCH STANFORD UNIVERSITY STANFORD, CA 94305 AREA A WORK UNI HUMS 11. CONTROLLING OFFICE NAME AND ADDRESS It. REPORT DATE U.S. ARMY RESEARCH OFFICE SEPTEMBER 1984 POST OFFICE BOX 12211 't3. NUMSEROf PAGES RESEARCH TRIANGLE PARK, NC 27709 6 14. MONITORING AGENCY NAME & ADoRESS(01 digerame 1"M Cm08mWIinR Office) IS. SECURITY CLASS. (of ie reportj UNCLASSIFIED ISa. OCLASSI FICATION/ DOWNGRADING SCHEDULE IS. DISTRIBUTION STATEMENT (t tie Report) APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED IT. DISTRIGUTION STATEMENT (01 86&0 chokerdt ewed OR 8lob1 to. It different Oron RePie). IS. SUPPLEENTA'RY NOTES The view, opinions, and/or findings contained in this report are those of the author(s) and should not be construed as an official Department of the Army position, policy, or decision, unless so designated by other documentation, (TO APPEAR IN THE PROCEEDINGS OF THE 1984 WINTER SIMULATION CONFERENCE, DALLAS, TX, NOVEMBER 1984.) It. KEY WORDS (Cans1nue M reverse eie It noceeepy and Identify by wek mnmbe) APPROXIMATION ERROR, REGENERATIVE SIMULATION, CONFIDENCE INTERVALS, MEAN AND' VARIANCE OF CONFIDENCE INTERVAL WIDTH. 20. AISTRACT (CmOn/hue mm ceee 0e1f0 neeeeei7 mnd idamrilf by StanJ am" ABSTRhC: The regenerative method is a mathematically rigorous method for obtaining confidence intervals for steady state parameters. In this paper the qualitative structure of asymptotic confidence intervals is discussed in general. This' structure is then specialized to confidence intervals for steady state parameters produced by the regenerative method.-,-, DO, -AM 1473 EDITION OF I NOV Go Is OsSOLETE UNCLASSIFIED....... 8ECURITY CLASSIFICATION OP TNis PAGE (Whm Dene 110led) 77.1
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