SELECTING SMALL QUANTILES

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1 Proceedings of the 200 Winter Simulation Conference B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yücesan, eds. SELECTING SMALL QUANTILES Raghu Pasupathy Industrial and Systems Engineering Virginia Tech Blacksburg, VA 2406, USA Roberto Szechtman Operations Research Department Naval Postgraduate School Monterey, CA 93943, USA Enver Yücesan Technology and Operations Management Area INSEAD Fontainebleau, FRANCE ABSTRACT Ranking and selection (R&S techniques are statistical methods developed to select the best system, or a subset of systems from among a set of alternative system designs. R&S via simulatios particularly appealing as it combines modeling flexibility of simulation with the efficiency of statistical techniques for effective decision making. The overwhelming majority of the R&S research, however, focuses on the expected performance of competing designs. Alternatively, quantiles, which provide additional information about the distribution of the performance measure of interest, may serve as better risk measures than the usual expected value. In stochastic systems, quantiles indicate the level of system performance that can be delivered with a specified probability. In this paper, we address the problem of ranking and selection based on quantiles. In particular, we formulate the problem and characterize the optimal budget allocation scheme using the large deviations theory. INTRODUCTION Ranking and selection (R&S techniques are statistical methods developed to select the best system, or a subset of systems from among a set of alternative system designs. R&S via simulatios particularly appealing as it combines the modeling flexibility of simulation with the efficiency of statistical techniques for effective decision making. Furthermore, simulation experiments also allow for multi-stage sampling as required by some R&S methods. Due to randomness in output data, however, comparing a number of simulated systems requires care. If the precision requirement is high and if the total number of designs in a decision problem is large, then the total simulation cost may be prohibitively high, iting the utility of simulation for R&S problems. The effective deployment of the simulation budget in R&S is therefore crucial. The overwhelming majority of the R&S research focuses on the expected performance of competing designs. Alternatively, quantiles, which provide additional information about the distribution of the performance measure of interest, may serve as better risk measures than the usual expected value. In stochastic systems, quantiles indicate the level of system performance that can be delivered with a specified probability. For example, in the financial services industry, Value at Risk (VaR, a quantile of a portfolio s profit or loss over a period of time, is a standard tool to assess the risk of that portfolio. Similarly, in the service industry (e.g., health care or telecommunications, quantiles are used as an indicator for the quality of service. In project management, stochastic activity networks are used to represent complex projects. In such an environment, planners may wish to compute an upper /0/$ IEEE 2762

2 Report Documentation Page Form Approved OMB No Public reporting burden for the collection of informatios estimated to average hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 25 Jefferson Davis Highway, Suite 204, Arlington VA Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of informatiof it does not display a currently valid OMB control number.. REPORT DATE DEC TITLE AND SUBTITLE Selecting Small Quantiles 2. REPORT TYPE 3. DATES COVERED to a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S AND ADDRESS(ES Naval Postgraduate School,Operations Research Department,Monterey,CA, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S AND ADDRESS(ES 0. SPONSOR/MONITOR S ACRONYM(S 2. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution united. SPONSOR/MONITOR S REPORT NUMBER(S 3. SUPPLEMENTARY NOTES Presented at the Proceedings of the 200 Winter Simulation Conference, 5-8 Dec, Baltimore, MD 4. ABSTRACT Ranking and selection (R&S techniques are statistical methods developed to select the best system, or a subset of systems from among a set of alternative system designs. R&S via simulatios particularly appealing as it combines modeling flexibility of simulation with the efficiency of statistical techniques for effective decision making. The overwhelming majority of the R&S research, however, focuses on the expected performance of competing designs. Alternatively, quantiles, which provide additional information about the distribution of the performance measure of interest, may serve as better risk measures than the usual expected value. In stochastic systems, quantiles indicate the level of system performance that can be delivered with a specified probability. In this paper, we address the problem of ranking and selection based on quantiles. In particular, we formulate the problem and characterize the optimal budget allocation scheme using the large deviations theory. 5. SUBJECT TERMS 6. SECURITY CLASSIFICATION OF: 7. LIMITATION OF ABSTRACT a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Same as Report (SAR 8. NUMBER OF PAGES 9 9a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev Prescribed by ANSI Std Z39-8

3 bound on the completion time of the project that would hold with high probability. Similarly, in a newsvendor setting, where a procurement or production quantity must be determined before the market uncertainties are resolved, the optimal quantity, the one that maximizes expected profit, is given by the quantile driven by the demand-supply mismatch costs. Finally, in simulation analysis (or, more generally, in statistics, the critical values for test statistics, confidence intervals, and sequential sampling procedures are expressed as quantiles. The estimation of quantiles, however, differs considerably from that of expectations. A thorough review of quantile estimation for independent and identically distributed (IID data is given by Serfling (980. To improve quantile estimation, authors such as Hsu and Nelson (990 and Hesterberg and Nelson (998 apply control variates, while Glynn (996 uses importance sampling, and Avramidis and Wilson (998 deploy correlation-induction strategies to obtain variance reduction in simulation-based quantile estimation. Closer to our work, Jin, Fu, and Xiong (2003 provide probabilistic error bounds for simulation quantile estimators using large deviations techniques. Hong (2009 develops an estimator based onfinitesimal perturbation analysis while Liu and Hong (2007 develop kernel estimators for assessing quantile sensitivities. Batur and Choobineh (2009 have recently introduced approaches for quantile-based system selection. In this paper, we address the problem of identifying the populations that correspond to the m smallest quantiles by sampling independently from d populations. By using the large deviations framework, we characterize the optimal sampling (or budget allocation scheme that minimizes the probability of incorrect selection given a fixed sampling budget. The remainder of the paper is organized as follows: in the next section, we formally define the problem. We then characterize the budget allocation scheme. As this characterization leads to a difficult, nested optimization problem, we turn our focus to a special case, where we wish to identify those populations whose quantiles exceed a threshold value. We conclude the paper with a number of simple illustrations. 2 PROBLEM DEFINITION Suppose we have d populations from which we candependently sample. Let X i be a random variable sampled from populatio with distribution function F i (. Let q i be the α i -quantile of populatio; that is q i = inf{k : F i (k α i }. Throughout we assume that (F i ( : i =,...,d and (q i : i =,...,d are unknown, and that 0 < α i <. The goal is to determine the populations that correspond to the m smallest quantiles, where the m th smallest quantile is different than the m + st smallest quantile. Hence, without loss of generality, we suppose that q q 2 q m < q m+ q d. The simulation budget is n, p = (p,..., p d is the vector of fractional allocations, and = [np i ] is the sample size of populatio. Let (X i,k : k =,..., be a collection of IID random samples drawn from F i, and X i,:ni X i,ni : the ordered samples of populatio. The α i -quantile estimator is X i, αi :, where is the ceiling operator. To simplify the notation define the sets A = {,...,m} and B = {m+,...,d}. Ancorrect selection (IS occurs when max i A X i, αi : min j B X j, α j n j :n j. A lower bound for P(IS is max P(X i, α i A, j B i : X j, α j n j :n j P(IS, 2763

4 and an upper bound for P(IS is P(IS = P( i A, j B X i, αi : X j, α j n j :n j A B max P(X i, α i A, j B i : X j, α j n j :n j. Hence, if n logp(x i, α i : X j, α j n j :n j G i, j (p i, p j as n for some rate function G i, j, we have that n logp(is min G i, j(p i, p j. i A, j B as n. The rate functions G i, j (p i, p j depend on the large deviations of X i, αi :, which are treated next. In preparation, 3 CONTINUOUS CASE If X i has density f i (, thet can be shown (Serfling (980, pp.85 that X i, αi : has the density ( ni f i,ni (t = [F i (t] α i [ F i (t] α i f i (t. α i For < θ < and t in the support of X i define ( ( Fi (t Fi (t g i (t = θt + α i log +( α i log, ( α i α i and Λ i,ni (θ = loge exp(θx i, αi :. When g i ( is strictly concave and twice differentiable, it has a unique global maximizer τ i (θ satisfying g i (t = 0. Observe that if 0 < α i < then 0 < F(τ i (θ <, for otherwise g(τ i (θ = and we know that g i (q i = θq i is feasible and greater than. Let Λ i (θ = g i (τ i (θ. (2 Proposition. If populatio has a density f i (t with bounded first derivative and the function g i ( is twice differentiable with supg (t < 0, then Λ i,ni ( θ = Λ i (θ. 2764

5 Proof. From the definition of f i,ni we have Λ i,ni ( θ ( ( ni = log α i ( ( ni = log α i + log + log exp( θt[f i (t] α i [ F i (t] α i f i (tdt exp( (g i (t+α i log(α i +( α i log( α i R i (t f i (tdt, where R i (t = [F i (t] α i α i [ F i (t] α i α i. Since g i (t is twice differentiable everywhere, Taylor s Theorem (see Serfling (980 for g i (t around its global minimum τ i (θ yields g i (t = g i (τ i (θ+ (t τ i(θ 2 2 (3 g i (ξ, (4 where ξ lies between τ i (θ and t. Plugging (4 in (3 and dividing through by, we have Λ i,ni ( θ = ( ( ni log + g i (τ i (θ+α i log(α i +( α i log( α i + α i ( ni log exp 2 (t τ i(θ 2 g (ξ R i (t f i (tdt. (5 The binomial term on the right-hand side of (5 becomes ( ni! = α i ( α i!( α i!, and Stirling s formula leads to ( log Changing variables yields log! ( α i!( α i! ( ni exp 2 (t τ i(θ 2 g (ξ R(t f i (tdt = n 3/2 log i Expanding R( and f i ( about τ i (θ results in = ( α i log( α i α i log(α i. (6 ( t 2 exp 2 g (ξ R(τ i (θ+tn /2 f i (τ i (θ+tn /2 d R(τ i (θ+tn /2 = R(τ i (θ+ t n R (η (7 and f i (τ i (θ+tn /2 = f i (τ i (θ+ t n f i(η 2, for η and η 2 between τ i (θ and τ i (θ+tn /2. We saw earlier that 0 < F(τ i (θ <, which results in R(τ i (θ and R (τ i (θ finite in (7. The 2765

6 two assumptions then lead to n 3/2 i = log 0, n 3/2 i ( t 2 exp 2 g (ξ R(τ i (θ+tn /2 f i (τ i (θ+tn /2 dt ( t 2 ( R(τ i (θ f i (τ i (θ+o( log exp 2 g (ξ t n /2 i dt (8 as. We conclude from (5, (6, and (8 that n n i Λ i,ni ( θ = Λ i (θ. 4 DISCRETE CASE Towards stating an analogous result for the discrete case, let us use a narrower definition of a quantile. Let q i be the α i -quantile of populatio, meaning that F i (q i = α i. Suppose X i is supported on the countable set L. Then (ignoring issues due to non-integral α i, it is seen that X i, αi : has the probability mass function ( ni Pr{X i, αi : = t} = ([F i (t] α i [F i (t ] α i [ F i (t] α i, t L α i where F i (t = Pr{X i < t}. Before we state the main result for the discrete context, we note the following simple proposition without proof. Proposition 2. Let {a,n },{a 2,n },... be a finite number of positive-valued sequences with Then, n n loga j,n = a j. ( n n log a j,n = Max j {a j }. j We are now ready to state the main result in the discrete context. Proposition 3. Suppose X i has finite support L, and satisfies Pr{X i = t} > 0 for each t L. Furthermore, suppose that the function g i ( has a unique maximum at τ i (θ and that g i (t is strictly increasing (decreasing for t < τ i (θ (resp., t > τ i (θ. Then Λ i,ni ( θ = Λ i (θ. 2766

7 Proof. Denote p i (t = Pr{X i, αi : = t}, t m = max{t : t L }, and L = L \ {t m }. (Since p i (t m = 0, we see that g i (t m = and hence τ i (θ t m. We have ( Λ i,ni ( θ = log p i (texp{ θt}. (9 t L We will show that log(p i (texp{ θt} = g i (t t L. (0 The assertion of the theorem then follows from applying Proposition 2 to (9 and (0. To show (0, we notice that log(p i (texp{ θt} = θt + log p i (t = θt + ( ni log + log(f i (t α i F i (t α i + ( α i log( F i (t α i = θt + ( ni log + logf i (t niα + α i log( F i(t niαi F i (t niαi F i (t n + α i log( F iα i i (t. ( Now, through an application of Stirling s formula we see that the second term appearing on the right-hand side of ( satisfies ( ( ni! log = log α i ( α i! α i! = ( α i log( α i α i log(α i = H(α i. (2 Next, we see that since F i(t α i F i (t α i is arbitrarily close to for large enough n F i (t α i i, the fourth term appearing on the right-hand side of ( satisfies F i (t niαi F i (t niαi F i (t n = 0. (3 iα i Finally, the third and fifth terms appearing on the right-hand side of ( satisfy logf i (t niα + α i Using (2, (3, and (4 in (, we get log( F i (t=h(α i +α i log( F i(t +( α i log( F i(t. α i α i (4 log(p i (texp{ θt} = θt H(α i +H(α i +α i log( F i(t +( α i log( F i(t α i α i = g i (t, 2767

8 and thus (0 holds. 5 QUANTILE SELECTION Propositions and 3 can be used to obtain an expression for the exponential decay rate of the incorrect selection probability, in terms of the sampling budget allocation. Let I k (x = sup θ {θx Λ k (θ} be the rate function corresponding to population k. In the continuous setting, for x such that 0 < F k (x <, Proposition leads to so that x = Λ k (θ = τ k (θ+τ k (θ [θ + = τ k (θ, ] α k F k (τ k (θ f α k k(τ k (θ F k (τ k (θ f k(τ k (θ ( ( αk αk I k (x = α k log +( α k log. (5 F k (x F k (x A similar argument for the discrete case shows that Eq. (5 is valid there as well. Let Z n = (X i, αi :,X j, α j n j :n j. Then, as shown Glynn and Juneja (2004, the rate function of (Z n : n 0 is given by p i I i (x i + p j I j (x j, and applying the Gärtner-Ellis Theorem results in G i, j (p i, p j = inf x i x j {p i I i (x i + p j I j (x j }. If F i (q m <, i A and F j (q > 0, j B then the rate functions G i, j (p i, p j are finite for any feasible allocation p i, p j. Furthermore, since I k (x is strictly decreasing for x < q k and strictly increasing for x > q k, we must have G i, j (p i, p j = inf x {p i I i (x+ p j I j (x}. An optimal allocation p maximizes mi A, j B G i, j (p i, p j, which is the same as s.t. and maxζ ζ G i, j (p i, p j 0, i A and j B d p i, p i 0. i= The first-order conditions are necessary for optimality (same argument as in Glynn and Juneja (2004. They are G i, j (p i, p j λ i, j = β i A, i A p j G i, j (p i, p j λ i, j = β j B, j B p i 2768

9 and λ i, j =, i A, j B λ i, j (ζ G i, j (p i, p j = 0 where λ i, j 0 for all i A, j B, and β 0. It can be shown that and that for some ζ > Crossing a threshold i A, j B min G i, j(p i, p j = ζ i A, j B min i A G i, j(p i, p j = ζ j B. Getting insights about the optimal allocation appears very difficult because we have a nested optimization problem. It is easier, however, to characterize the allocation that minimizes the probability of crossing a threshold c [q m,q m+ ]. Let IS c be the event ( i A X i, αi : > c ( j B X j, α j n j :n j < c. Then we have P(IS P(IS c P(X i, αi : > c+ P(X j, α j n j :n j < c. i A j B Using an argument similar to the one presented in Szechtman and Yücesan (2008, we get n log( P(X i, αi : > c+ P(X j, α j n j :n j < c min{p I (c,..., p d I d (c} i A j B as n. Following Szechtman and Yücesan (2008, the optimal allocations are leading to p k = sup n n logp(is ( i A Ik (c i A Ii (c+ j B I j (c, Ii (c+ I j (c. j B That is, the optimal threshold is the one that minimizes i A Ii (c+ j B I j (c over c [q m,q m+ ]. 6 CONCLUDING REMARKS In this paper, we addressed the problem of identifying the populations that correspond to the m smallest quantiles by sampling independently from d populations. Using a large deviations framework, we characterized the optimal sampling (or budget allocation scheme that minimizes the probability of incorrect selection given a sampling budget that grows to infinity. In particular, the optimal budget allocation arises as the solution of a 3-layer nested optimization problem. The threshold crossing 2769

10 problem, where we wish to identify those populations whose quantiles exceed a threshold value, leads to more tractable budget allocations. REFERENCES Avramidis, A. N., and J. R. Wilson Correlationduction techniques for estimating quantiles in simulation experiments. Operations Research 46: Batur, D., and F. Choobineh A quantile-based approach to system selection. European Journal of Operational Research 99: Glynn, P. W Importance sampling for Monte Carlo estimation of quantiles. In Proceedings of the Second International Workshop on Mathematical Methods in Simulation and Experimental Design, Saint Petersburg, Florida. Glynn, P. W., and S. Juneja A large deviations perspective on ordinal optimization. In Proceedings of the Winter Simulation Conference, ed. R. G. Ingalls, M. D. Rossetti, J. S. Smith, and B. A. Peters, Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc. Hesterberg, T. C., and B. L. Nelson Control variates for probability and quantile estimation. Management Science 44: Hong, L. J Estimating quantile sensitivities. Operations Research 57:8 30. Hsu, J. C., and B. L. Nelson Control variates for quantile estimation. Management Science 36: Jin, X., M. C. Fu, and X. Xiong Probabilistic error bounds for simulation quantile estimators. Management Science 49: Liu, G., and L. J. Hong Kernel estimation for quantile sensitivities. In Proceedings of the 2007 Winter Simulation Conference, ed. S. G. Henderson, B. Biller, M.-H. Hsieh, J. Shortle, J. D. Tew, and R. R. Barton, Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc. Serfling, R. J Approximation theorems of mathematical statistics. New York: Wiley. Szechtman, R., and E. Yücesan A new perspective on feasibility determination. In Proceedings of the Winter Simulation Conference, ed. R. G. Ingalls, M. D. Rossetti, J. S. Smith, and B. A. Peters, Piscataway, New Jersey: Institute of Electrical and Electronics Engineers, Inc. AUTHOR BIOGRAPHIES RAGHU PASUPATHY is an Assistant Professor in the Industrial and Systems Engineering Department at Virginia Tech. His research interests lie broadly in Monte Carlo methods with a specific focus on simulation optimization and stochastic root finding. He is a member of INFORMS, IIE, and ASA, and serves as an Associate Editor for ACM TOMACS and INFORMS Journal on Computing. His web and e- mail addresses are < and <pasupath@vt.edu>. ROBERTO SZECHTMAN is an Associate Professor in the Operations Research Department at the Naval Postgraduate School. He received his Ph.D. from Stanford University, after which he joined Oracle Corp.( His research interests include applied probability and military operations research. He is Associate Editor for IIE Transactions, Naval Research Logistics, and Operations Research. His address is <rszechtm@nps.edu>. ENVER YÜCESAN is a Professor of Operations Management at INSEAD in Fontainebleau, France. His research interests include ranking and selection as well as optimization via simulation. He is an Area Editor for IIE Transactions. He is also serving as the Program Chair for WSC 200. His address is <enver.yucesan@insead.edu>. 2770