OPERATIONS RESEARCH
informs
Vol. 55, No. 6, November–December 2007, pp. 1090–1103 issn 0030-364X eissn 1526-5463 07 5506 1090
®
doi 10.1287/opre.1070.0475 © 2007 INFORMS
Overlapping Variance Estimators for Simulation Christos Alexopoulos
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332,
[email protected]
Nilay Tanık Argon
Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599,
[email protected]
David Goldsman
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332,
[email protected]
Gamze Tokol
Decision Analytics, Atlanta, Georgia 30306,
[email protected]
James R. Wilson
Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, North Carolina 27695,
[email protected]
To estimate the variance parameter (i.e., the sum of covariances at all lags) for a steady-state simulation output process, we formulate certain statistics that are computed from overlapping batches separately and then averaged over all such batches. We form overlapping versions of the area and Cramér–von Mises estimators using the method of standardized time series. For these estimators, we establish (i) their limiting distributions as the sample size increases while the ratio of the sample size to the batch size remains fixed; and (ii) their mean-square convergence to the variance parameter as both the batch size and the ratio of the sample size to the batch size increase. Compared with their counterparts computed from nonoverlapping batches, the estimators computed from overlapping batches asymptotically achieve reduced variance while maintaining the same bias as the sample size increases; moreover, the new variance estimators usually achieve similar improvements compared with the conventional variance estimators based on nonoverlapping or overlapping batch means. In follow-up work, we present several analytical and Monte Carlo examples, and we formulate efficient procedures for computing the overlapping estimators with only order-of-sample-size effort. Subject classifications: simulation: statistical analysis, steady-state variance estimation. Area of review: Simulation. History: Received January 2006; revision received June 2006; accepted July 2006.
of the variability of Yn that is useful in characterizing the asymptotic behavior of Yn , we are interested in the variance parameter, 2 ≡ limn→ n2 , provided that this limit exists. Because the outputs from a steady-state simulation are usually not independent and identically distributed (i.i.d.) random variables, the “standard” estimator of 2 given by n 2 −1 2 = n − 1 i=1 Yi − Yn can be severely biased (usually on the low side for queueing simulations); and in these situations 2 should not be used. There are many techniques in the literature for estimating the n2 n = 1 2 and 2 . In particular, the following well-known methods of steady-state simulation output analysis are discussed in popular books such as Alexopoulos et al. (2006), Bratley et al. (1987), and Law and Kelton (2000): nonoverlapping batch means (NBM); overlapping batch means (OBM); spectral analysis; regenerative analysis; autoregressive modeling; and standardized time series (STS). A common strategy used by some of the above methodologies, e.g., NBM, OBM, and STS, requires batching the observations. The concept of batching is simple—instead
1. Introduction Steady-state simulations are used to analyze a variety of complex probabilistic systems—for example, a continuously operating production facility, a telecommunications network, or a financial portfolio. As part of a complete simulation study, one should always carry out a careful statistical analysis of the simulation’s output. In the case of a steady-state simulation experiment, a good analysis might start off with, at the very least, an estimate of the unknown mean of the output process, Yi i = 1 2 , which is assumed to be stationary. Of course for a simulation-generated time series Yi i = 1 n of length n, the sample mean Yn = n−1 ni=1 Yi is the usual estimator for ; but because Yn is a random variable, the experimenter should characterize the variability of this statistic as well. In terms of the variance of the sample mean, Var Yn = E Yn − 2 , the quantities n2 ≡ nVar Yn n = 1 2 provide information about the variability of Yn and its asymptotic behavior as the sample size (simulation run length) n → . As a single measure 1090
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of considering the entire simulation-generated time series Yi i = 1 n all at once, we break up this data set into smaller batches or subseries that are composed of consecutive observations (where the batches may be disjoint or overlapping, depending on the analysis method); and then we perform the relevant analysis on each batch separately. For example, in the NBM method we split the observations into adjacent disjoint (nonoverlapping) batches; then we assume that the resulting sample (batch) means computed from each batch are approximately i.i.d. normal; and finally we apply “standard” variance estimation techniques to the batch means (as explained in §3.4). In the STS method, batched estimators also use adjacent nonoverlapping batches. The idea is to compute a separate STS estimator from each batch, assume that the resulting STS estimators are i.i.d., and then average those estimators (see §§3.2 and 3.3). In the OBM method, we form overlapping batches, with the full realization that the associated overlapping batch means are not independent (although they are identically distributed and asymptotically normal); then we compute the “standard” estimator of 2 , with the overlapping batch means taken as the individual data items used in the computation. This seemingly problematic technique exploits key results from the theory of spectral analysis to yield an estimator of the variance parameter 2 that is provably superior to the NBM variance estimator, at least asymptotically for certain types of serially correlated simulation output processes. What do we mean by a superior estimator of the variance parameter 2 ? We most often care about the bias and variance of an estimator for 2 , as well as the resulting mean-squared error (MSE), that is, the estimator’s variance plus the square of the estimator’s bias. Batching typically increases bias but decreases variance so that its net effect on MSE requires careful analysis. What is nice about the OBM estimator is that asymptotically it has the same bias as, but smaller variance than, the NBM estimator when the sample size n → . Thus, OBM gives better performance than NBM based on the same simulation-generated time series Yi i = 1 n of length n, provided that n is sufficiently large. The preceding comparison raises an interesting question: What happens if we apply the technique of overlapping batches to the STS estimators? In this paper, we answer the question, and the news is almost all good—in particular, the STS estimators computed from overlapping batches possess the same bias as, but asymptotically substantially smaller variance than, their counterparts computed from nonoverlapping batches; moreover, the overlapping STS variance estimators similarly outperform the OBM variance estimator. There has already been significant progress in the study of overlapping estimators. In their seminal article, Meketon and Schmeiser (1984) introduce the OBM methodology, including some results on performance characteristics and
computational methods. Welch (1987) relates OBM to certain spectral estimators and looks into the effects of partial overlapping. Goldsman and Meketon (1986), Song (1988), and Song and Schmeiser (1993, 1995) derive bias and variance properties of OBM estimators, among others. Song and Schmeiser (1993) also give additional insight by plotting the coefficients of the estimators’ quadratic-form representations; included in the presentation are the OBM estimators as well as overlapped versions of the STS area estimators described in §4.2 below. Additional early work on the subject is undertaken by Pedrosa and Schmeiser (1993, 1994), who establish covariance properties between OBM estimators and subsequently propose a batch-size determination algorithm. In a remarkable series of papers, Damerdji (1991, 1994, 1995) establishes consistency results (both in the strong and mean-square senses) for a variety of variance estimators, including OBM and an overlapping version of a certain STS estimator. In the spirit of Welch (1987), Damerdji also establishes a formal linkage between the spectral method and simulation analysis methods based on overlapping batches. The rest of this paper is organized as follows. We present background material in §2, where we introduce two benchmark estimators—namely, the area and Cramér–von Mises (CvM) estimators as they were originally formulated using the STS method for simulation analysis. In §3, we review the effects of grouping the observations into nonoverlapping batches, then computing an area or CvM estimator from each batch, and finally averaging the computed estimators over all batches. The main contribution of this paper is contained in §4, where we perform an analysis similar to that of §3, except that the batches are now allowed to overlap. In §4, we also examine the asymptotic performance characteristics of the new overlapping estimators, including: (i) their limiting distributions as the sample size tends to infinity while the ratio of the sample size to the batch size remains fixed; and (ii) their convergence in mean square (quadratic mean) to the variance parameter as both the batch size and the sample-size-to-batch-size ratio tend to infinity. In §5, we summarize our main findings, and we make recommendations for future work. The appendix contains most of the proofs of the key properties of the overlapping variance estimators that are stated in §4; complete proofs of all new results are contained in Alexopoulos et al. (2007a), the online supplement to this paper. In a companion paper (Alexopoulos et al. 2007b), we present several analytical and Monte Carlo examples that illustrate the properties of the overlapping estimators for finite but progressively increasing sample sizes; and we formulate efficient procedures for computing these estimators with only order-of-sample-size work. Alexopoulos et al. (2004) present a preliminary, abridged version of some of the results that are fully developed in this paper.
2. Background This section reviews results that will be needed in the sequel. In §2.1, we introduce some basics regarding the
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convergence of a stationary time series, including the standardized time series associated with a stochastic process. We define the STS weighted area and weighted CvM variance estimators in §§2.2 and 2.3, respectively. 2.1. Basics Throughout we consider a stationary stochastic process Yi i = 1 2 that is usually taken to be the output of a probabilistic simulation in steady-state operation. To begin with, we assume that the stochastic process satisfies a functional central limit theorem (FCLT). This assumption applies to a broad class of processes, and it will allow us to determine the limiting properties of the different variance estimators formulated in this paper. As explained in Glynn and Iglehart (1990), the following are examples of stochastic processes that satisfy Assumption FCLT: stationary -mixing processes; stationary strongly mixing sequences; associated stationary sequences; and regenerative processes. Assumption FCLT. The series 2 = Var Y1 + 2 k=1 Cov Y1 Y1+k converges absolutely with 2 > 0, and the sequence of random functions
nt Y nt − Xn t ≡ √ n for t ∈ 0 1 and n = 1 2
(1)
satisfies
Xn · −→ · n→
where: · is the greatest integer function; · is a stan dard Brownian motion process on 0 1 ; and −→ denotes n→
weak convergence (as n → ) in the Skorohod space D 0 1 of real-valued functions on 0 1 that are right continuous with left-hand limits. Note that if the variance parameter exists, then we have 2 = limn→ nVar Yn = limn→ n2 as discussed at the beginning of this paper. The standardized time series based on the sample Yi i = 1 n of size n is (cf. Schruben 1983)
nt Yn − Y nt Tn t ≡ for t ∈ 0 1 (2) √ n Under Assumption FCLT, it can be shown that √
n Yn − Tn · −→ 1 · n→
(3)
where · is a standard Brownian bridge process on 0 1 that is independent of 1 ; see Foley and Goldsman (1999), Glynn and Iglehart (1990), or Schruben (1983). All
finite-dimensional joint distributions of · are normal with E t = 0 and Cov s t = min s t − st for 0 s t 1. Furthermore, we can express the Brownian bridge process · in terms of the standard Brownian motion process · to which the process Xn · converges in distribution—namely, t = t ≡ t 1 − t for t ∈ 0 1 . 2.2. The Weighted Area Estimator In this subsection, we discuss the weighted area estimator for 2 ; for additional information, see Goldsman et al. (1990), Goldsman and Schruben (1990), and Schruben (1983). We define the square of the weighted area under the standardized time series (2) and its limiting functional as n 2 1 k k A f n ≡ f and Tn n k=1 n n 1
2 A f ≡ f t t dt 0
respectively, where the weighting function f · satisfies the following condition: E A f = 2
and
(4) d f
t is continuous at every t ∈ 0 1 dt 2 1 If (4) holds, then 0 f t t dt = Nor 0 1 , where = denotes equivalence in distribution and Nor 0 1 denotes a standard normal random variable; and under Assumption FCLT, the continuous mapping theorem (CMT) (see Theorem 5.5 of Billingsley 1968) implies that A f n −→ A f = 2 12 , where in general 2 denotes a 2
n→
chi-squared random variable with degrees of freedom. For this reason, we call A f n the weighted area estimator for 2 . Throughout the rest of this paper, we let Rk ≡ Cov Yi Yi+k = E Yi − Yi+k − denote the covariance at lag k (where k = 0 ±1 ±2 ) for the process Yi ; and we define the associated quantity ≡ −2
k=1
kRk
provided
k=1
kRk <
(5)
see Song and Schmeiser (1995). We will use the notation an −→ a to mean limn→ an = a; and we use the “little-oh” n→
notation p n = o q n to mean limn→ p n /q n = 0. Theorem 1 gives results on the expected value and variance of the weighted area estimator. Theorem 1 (Foley and Goldsman 1999, Goldsman et al. 1990). Suppose that Assumption FCLT and (4) hold together with the moment condition k=1
k2 Rk <
(6)
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Further suppose that the family of random variables A2 f n n = 1 2 is uniformly integrable (see Billingsley 1968 for a definition and sufficient conditions). Then, the convergence condition in (5) is satisfied so that is well defined, and we have E A f n = 2 +
F 1 − F 1 2 + F2 1 + o 1/n 2n
and Var A f n −→ Var A f = Var 2 12 = 2 4 n→
(7)
where the functions F · and F · are defined as follows: s F s ≡ f t dt for s ∈ 0 1 and 0 (8) u F u ≡ F s ds for u ∈ 0 1 0
The limiting variance in (7) does not depend on the form of the weighting function f · . Example 1. Schruben (1983) studied the√area estimator with constant weighting function f0 t ≡ 12 for all t ∈ 0 1 ; in this case, Theorem 1 implies that E A f0 n = 2 + 3 /n + o 1/n . If one chooses a weighting function f · for which F 1 = F 1 = 0, then the resulting estimator is first-order unbiased for 2 ; that is, A f n has bias of the form o 1/n . An example of a weighting function√yielding a first-order unbiased estimator for 2 is f2 t ≡ 840 3t 2 − 3t + 1/2 ; see Goldsman et al. (1990) and Goldsman and Schruben (1990). Other weighting functions that yield first-order unbiased√estimators for 2 are given by the family fcos j t = 8(j cos 2(jt j = 1 2 . Foley and Goldsman (1999) show that this “orthonormal” sequence of weights yields area estimators A fcos j n j = 1 2 that are not only firstorder unbiased, but asymptotically independent—that is, the A fcos j j = 1 2 are i.i.d. 2 12 . 2.3. The Weighted Cramér–von Mises Estimator In this subsection, we give an overview of the weighted Cramér–von Mises (CvM) estimator for 2 ; see also Goldsman et al. (1999). We begin by defining some notation for the weighted area under the square of the standardized time series (2) and its limiting functional, 2 n 1 k k C g n ≡ g and Tn n k=1 n n 1 C g ≡ g t t 2 dt 0
respectively, where g · is a normalized weighting function so that E C g = 2
and
d2 g t is continuous at every t ∈ 0 1 dt 2
(9)
If Assumption FCLT and (9) hold, then it can be shown that C g n −→ C g ; see Remark 2 of Goldsman et al. (1999). n→
We call C g n the weighted CvM estimator for 2 . Theorem 2 gives results on the expected value and variance of the weighted CvM estimator. Theorem 2 (Goldsman et al. 1999). Suppose that Assumption FCLT, (6), and (9) hold. Further suppose that the family of random variables C 2 g n n = 1 2 is uniformly integrable. Then, E C g n = 2 + G − 1 + o 1/n n 1 where G ≡ 0 g t dt and Var C g n −→ Var C g n→
= 4 4
1 0
g t 1 − t 2
t 0
g s s 2 ds dt
(10)
Example 2. Theorem 2 implies that the CvM estimator with constant weighting function g0 t ≡ 6 has E C g0 n = 2 + 5 /n + o 1/n . If one chooses a weighting function satisfying G = 1 and (9), then Theorem 2 implies that the CvM estimator C g n has bias o 1/n . An example of such a first-order unbiased weighting function is g2, t ≡ −24 + 150t − 150t 2 , where t ∈ 0 1 . The choice of weighting function g · affects the variances of C g n and C g . (The weighting function f · of §2.2 affects the variance of A f n , but not that of A f , which is always Var A f = 2 4 .) Example 3. Theorem 2 implies that Var C g0 = 4 4 /5 and Var C g2, = 121 4 /70. Although Var C g2, > Var C g0 , the estimator C g2, n is first-order unbiased for 2 , while C g0 n is not. Moreover, Goldsman et al. (1999) show that g2, t minimizes Var C g over all first-order unbiased quadratic weighting functions. Firstorder unbiased minimum-variance polynomial weights are derived in Goldsman et al. (1999) for polynomials up to degree six.
3. Estimators from Nonoverlapping Batches Up to this point, our variance estimators have been constructed directly from one long run of n observations. For the most part, in this section we examine what happens if we do the following: (i) divide the run into contiguous, nonoverlapping batches; (ii) compute an STS estimator from each batch; and (iii) take the average of the estimators. In §3.1, we establish some minor notational adjustments that subsequently will be used to define various estimators for 2 based on nonoverlapping batches of observations. In §§3.2–3.4, we deal with the STS batched area, STS batched CvM, and NBM estimators, respectively. In each case, we give results on the expected value and variance of the estimator.
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3.1. Batching Basics In this section, we will work with b contiguous, nonoverlapping batches of observations, each of length m, from the simulation-generated time series Yj j = 1 n of length n, where we assume that n = bm. Thus, the observations Y i−1 m+k k = 1 m constitute batch i for i = 1 b. Throughout this paper, we take b ≡ n/m so that b always represents the ratio of the sample size to the batch size; and when we work with nonoverlapping batches, b also equals the number of batches. To parallel the discussion in §2.1, we define the standardized time series computed from batch i as
mt Yi m − Yi mt Ti m t ≡ √ m for t ∈ 0 1 and i = 1 b
1 b are asymptotically independent as m → ; and by the remarks in §2.2, we have f b m −→ f b = m→
2 b2 /b, where f b is the average of b i.i.d. realizations of A f . Theorem 1 implies E f b m = E A1 f m = 2 +
F 1 − F 1 2 + F2 1 + o 1/m 2m
Further, if we assume that the family of random variables 2 f b m m = 1 2 is uniformly integrable, then lim bVar f b m
m→
= lim Var A1 f m = Var A f = 2 4 m→
where
for i = 1 b and j = 1 m (11) If Assumption FCLT holds, then √ √ m Y1m − m Ybm − T1m · Tbm ·
−→ Z1 Zb 0 · b−1 · (12)
The CvM estimator computed from batch i is 2 m k 1 k Ci g m ≡ Ti m g for i = 1 b m k=1 m m The batched CvM estimator for 2 is g b m ≡
m→
where (i) the Zi i = 1 b are i.i.d. standard normal random variables; (ii) the Zi i = 1 b are independent of the i · i = 0 b − 1; and (iii) s · denotes a standard Brownian bridge process on 0 1 given by s t ≡ t s + 1 − s − s + t − s (13)
so that the i · i = 0 b − 1 in (12) are independent Brownian bridge processes. 3.2. Batched Area Estimator We define the area estimator computed exclusively from batch i as 2 m k 1 k Ti m f for i = 1 b Ai f m ≡ m k=1 m m
b 1 A f m b i=1 i
Because the Ti m · i = 1 b converge to independent Brownian bridge processes as m becomes large (with fixed b), we conclude that the corresponding Ai f m i =
b 1 Ci g m −→ g b m→ b i=1
where g b is the average of b i.i.d. realizations of C g . Theorem 2 implies E g b m = E C1 g m = 2 + G − 1 + o 1/m m
(16)
As before, if we assume that the family of random variables 2 g b m m = 1 2 is uniformly integrable, then limm→ bVar g b m = limm→ Var C1 g m = Var C g , where Var C g is given by Equation (10). 3.4. NBM Estimator In Equation (11), the quantities Yi m i = 1 b are referred to as the batch means, and are often assumed to be i.i.d. normal random variables, at least for large enough batch size m; this is borne out by relation (12). The i.i.d. assumption immediately suggests that for fixed b we consider the NBM estimator for 2 ,
The batched area estimator for 2 is f b m ≡
(15)
3.3. Batched CvM Estimator
j 1 Y Yi j ≡ j k=1 i−1 m+k
for t ∈ 0 1 and s ∈ 0 b − 1
(14)
b m ≡
b 2 2 m b−1
Yi m − Yn 2 −→ m→ b − 1 b − 1 i=1
see, e.g., Glynn and Whitt (1991), Schmeiser (1982), and Steiger and Wilson (2001). The statistic b m is one of the most popular estimators for 2 , and it serves as a benchmark for comparison with the other estimators discussed
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in this paper. Under mild conditions, Chien et al. (1997), Goldsman and Meketon (1986), and Song and Schmeiser (1995) show that E b m = 2 +
b + 1 + o 1/m bm
(17)
As for the NBM estimator’s variance, Glynn and Whitt (1991) (among others) find that, for fixed b, lim b − 1 Var b m = 2 4
m→
Remark 1. An argument based on relation (12) and an extension of the work in Schruben (1983) can be used to show that, in general, f b m and g b m are each asymptotically independent of b m as m → with fixed b (although f b m and g b m are not independent of each other); and thus we can take an appropriate linear combination of the STS and NBM estimators, e.g., b f b m + b − 1 b m / 2b − 1 , with the hope that the combined estimator will reduce variance at minimal cost in bias. 3.5. Recapitulation We see that as the batch size m → , the batched area, batched CvM, and NBM estimators are all asymptotically unbiased for 2 . In addition, the variances of these estimators are all inversely proportional to the sample-sizeto-batch-size ratio b (at least for sufficiently large batch size). Further, it has been shown in, e.g., Alexopoulos et al. (2000) and Chien et al. (1997), that the above variance results hold as b also becomes large along with the batch size m; this and the bias results from Equations (14), (16), and (17) allow us to conclude that the estimators are consistent in mean square as m and b → . Of course, for fixed m and b and, hence, for fixed sample size n = mb, some estimators will tend to perform better than others. Certainly, we want to use estimators with low bias and variance; but for fixed n, decreasing one usually comes at the expense of increasing the other—the wellknown trade-off that we have already mentioned. One could argue that NBM, as the benchmark method, has moderate bias and variance. The good news is that STS batched area and batched CvM estimators with certain weighting functions can outperform NBM in terms of large-sample bias, and in the case of CvM, in terms of variance as well. The better news is that, in the sequel, we will show that the use of overlapping batches with respect to any particular estimator preserves its expected value, while reducing its variance.
4. Estimators from Overlapping Batches In this section, we consider the use of estimators based on overlapping batches, as in Meketon and Schmeiser (1984). Now we do the following: (i) divide the run into a number of overlapping batches; (ii) compute an STS estimator
from each batch; and (iii) take the average of the estimators. The presentation roughly follows that of the previous section. In §4.1, we establish notation to be used in the discussion of overlapping estimators. In §§4.2–4.4, we deal with the STS overlapping area, STS overlapping CvM, and OBM estimators, respectively. As in §3, we give results on the expected values and variances of the various estimators. Alexopoulos et al. (2007a) provide complete proofs of all the results presented in this section, while the appendix at the end of this paper contains abridged proofs of selected key results that highlight the general methods of analysis used. 4.1. Overlapping Batching Basics Here we form n − m + 1 overlapping batches, each of size m, from the sample Yj j = 1 n. In particular, the observations Yi+k k = 0 m − 1 constitute overlapping batch i for i = 1 n − m + 1. We will continue to let b ≡ n/m as before; but obviously, when speaking in the context of overlapping batches, b can no longer be interpreted as “the number of batches.” To parallel the discussion in §3.1, we define the standardized time series from overlapping batch i as TiOm t ≡
mt Y¯iOm − Y¯iO mt √ m for t ∈ 0 1 and i = 1 n − m + 1
where O ≡ Y¯ij
j−1 1 Y j k=0 i+k
for i = 1n−m+1 and j = 1m
If Assumption FCLT holds, then in terms of (13) we have
O T sm m · −→ s · m→
for fixed s ∈ 0 b − 1
4.2. Overlapping Area Estimator We define the area estimator computed exclusively from overlapping batch i by AOi f m ≡
2 m k 1 k TiOm f m k=1 m m for i = 1 n − m + 1
The overlapping area estimator for 2 is O f b m ≡
n−m+1 O 1 A f m n − m + 1 i=1 i
The next theorem gives the limiting distribution of O f b m . Its proof is in the appendix.
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Theorem 3. If Assumption FCLT and (4) hold, then
O f b m −→ O f b m→
2 1 b−1 1 ≡ f u s u du ds b−1 0 0
(18)
0
The first-order unbiased weighting functions f2 · and fcos j · from Example 1 all satisfy the condition F 1 = 0, making the calculation of pkl 0 y particularly easy.
Moreover, Theorem 1 immediately gives E O f b m = 2 +
If we happen to be dealing with weighting functions fk · and fl · such that Fk 1 = Fl 1 = 0, then most of the terms in (20) disappear. In this case, an application of integration by parts yields the equivalent expression 1−y Fl u Fk y + u du for y ∈ 0 1 (21) pkl 0 y =
F 1 − F 1 2 + F2 1 + o 1/m 2m
(19)
Therefore, the expected value of the overlapping area estimator matches that of the batched area estimator, which is no surprise because both estimators are simply linear combinations of the Ai f m i = 1 2 or of the AOi f m i = 1 2 , each having the same expected value. The next theorem, whose proof is immediate from the CMT, shows that the asymptotic covariance of overlapping area estimators formed from two weighting functions, say fk · and fl · , is as it should be. Theorem 4. Let fk · and fl · denote two weighting functions that satisfy (4) and that are used to compute the overlapping area estimators O fk b m and O fl b m , respectively; and assume that for each fixed b, the family of random variables O fk b m 2 O fl b m 2 m = 1 2 is uniformly integrable. If b is fixed and m → , then we have Cov O fk b m O fl b m −→ Cov O fk b O fl b
Lemma 2. If (4) holds for the weighting functions fk · and fl · , and if we take 2 2 qkl y ≡ pkl
0 y + plk
0 y for y ∈ 0 1
then for fixed b 2, we have Cov O fk b O fl b 2 4 1 =
b − 1 − y qkl y dy
b − 1 2 0
(22)
To calculate Cov O fk b O fl b , we apply (20) (or (21) if appropriate) to the definition of qkl y and plug into (22). Of course, as a by-product, we take f · = fk · = fl · when we want to calculate Var O f b as a special case of Cov O fk b O fl b . Contrary to our findings in §§2.2 and 3.2, where we did not use overlapping batches, some examples show that the variance of the overlapping area estimator does depend on the choice of weighting function. Example 4. For the overlapping constant-weighted area estimator, we have from Equations (20) and (22) that for b 2, Var O f0 b m −→ Var O f0 b
m→
m→
But how do we calculate this asymptotic covariance? We start with a couple of necessary lemmas, whose proofs are in the appendix. Lemma 1. If (4) holds for the weighting functions fk · and fl · , and if we take pkl s r ≡
1 0
1 0
fk u fl v Cov s u r v du dv for s r ∈ 0 1
then pkl 0 y = Fk 1 Fl 1 − y − Fl 1 − y − Fl 1 y 1−y + Fk y Fl 1 − fl u Fk y + u du 0
=
24b − 31 4 24 4 ∼ 2 35 b − 1 35b
as b →
This compares impressively to the batched constantweighted area estimator’s asymptotic (m → ) variance, Var f0 b = 2 4 /b (Equation (15)). In other words, Var O f0 b /Var f0 b ∼ 12/35 as b → , a substantial savings. Example 5. For the overlapping area estimator with firstorder unbiased quadratic weighting function f2 · from Example 1, we have from Equations (21) and (22) that for b 2, Var O f2 b =
3514b − 4359 4 4290 b − 1 2
for y ∈ 0 1 (20)
In this case, Var O f2 b /Var f2 b ∼ 0410 as b → .
where Fk · , Fl · , Fk · , and Fl · are analogous to the functions defined in (8).
Example 6. For the overlapping area estimators from the family of orthonormal first-order unbiased weighting
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functions fcos j · j = 1 2 , we have from Example 1 and Equations (21) and (22) that for b 2, Var O fcosj b = ∼
16( 2 j 2 + 30 b − 20( 2 j 2 + 33 4 24( 2 j 2 b − 1 2 8( 2 j 2 + 15 4 12( 2 j 2 b
as b →
for i = j?
The answer is: no, but almost. To see this, we can carry out the explicit calculations. Because the weights under consideration are first-order unbiased, we use Equation (21) to obtain 1−y pij 0 y = Fcos j u Fcos i y + u du 0
=
−j sin 2(iy + i sin 2(jy ( i2 − j 2
(24)
i2 + j 2 2b − 3 2 i2 + j 2 4 4 ∼ 2 2 2 2 2 2 2 ( i − j b − 1 ( i − j 2 2 b as b → if i = j (25)
We now have the machinery to look at new estimators formed by averaging the overlapping versions of the original orthonormal area estimators, all of which are first-order unbiased. For example, we can average the first k such estimators, k 1 ¯Ok fcos b m ≡ O fcos j b m −→ ¯Ok fcos b m→ k j=1
≡
k 1 O fcos j b k j=1
k2 Var ¯Ok fcos b m −→ k2 Var ¯Ok fcos b
j=1
Var O fcos j b + 2
k−1
k
j=1 l=j+1
=
14400( 2 + 47726 b − 18000( 2 + 66689 4 64800( 2 b − 1 2
0297 4 b and, similarly, ∼
as b →
0227 4 as b → Var ¯O4 fcos b ∼ b These results represent the lowest variances that we have achieved so far. Finally, for completeness, Theorem 5 establishes that the overlapping area estimator O f b m converges in mean square (quadratic mean) to 2 as both the sample-size-tobatch-size ratio b and the batch size m grow large. Theorem 5. If (4) holds, then O f b converges in mean square to 2 as b → ,
Covjl
O f b −→ 2
(27)
b→
Now suppose that Assumption FCLT also holds. If the sequence of cumulative distribution functions (c.d.f.’s) O f b m z ≡ PrO f b m z −→ O f b z ≡ m→
O
Pr f b z uniformly on z b z ∈ − 2 and b = B B + 1 for some sufficiently large positive integer B, and if the family of random variables O f b m 2 b m = 1 2 is uniformly integrable, then O f b m converges in mean square to 2 as b m → , qm
O f b m −→ 2
(28)
b m→
4.3. Overlapping CvM Estimator
for i = 1 n − m + 1 and the overlapping CvM estimator for 2 by O g b m ≡
m→
k
Var ¯O3 fcos b
as b →
We define the overlapping CvM estimator computed from overlapping batch i by 2 m k 1 k O O g Ti m Ci g m ≡ m k=1 m m
Because ¯Ok fcos b m is the average of k first-order unbiased estimators, it too is first-order unbiased. Its variance is
=
0429 4 b
∼
qm
for i = j
from which we have the bonus equality pij 0 y = pji 0 y . Then, for b 2, (22) and (24) give 4 4 1 Covij =
b − 1 − y pij2 0 y dy
b − 1 2 0 =
384( 2 +1090 b − 480( 2 +1455 4 1152( 2 b −1 2
Var ¯O2 fcos b =
(23)
Thus, Var O fcos j b /Var fcos j b ∼ 0397 as b → for j = 1 and decreases to 1/3 as j → . A defining property of the orthonormal estimators A fcos j n j = 1 2 , as discussed in Example 1, is that their corresponding limiting random variables A fcos j j = 1 2 are i.i.d. 2 12 . This observation leads one to wonder whether the independence property carries over to the analogous overlapping area estimators. In other words, are O fcosi b and O fcos j b , say, independent when i = j? Or equivalently, because the underlying random summands are normal, do we have Covij ≡ Cov O fcos i b O fcos j b = 0
Equations (23), (25), and (26), along with some algebra, give us, for b 2,
(26)
n−m+1 O 1 C g m n − m + 1 i=1 i
The following theorem establishes the limiting distribution of O g b m for fixed b as m → .
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Theorem 6. If Assumption FCLT and (9) hold, then
Example 7. For the overlapping constant-weighted CvM estimator, we have that for b 2,
O g b m −→ O g b
Var O g0 b m −→ Var O g0 b
m→
m→
1 b−1 1 ≡ g u 2 2 s u du ds b−1 0 0
(29)
Moreover, Theorem 2 implies that G − 1 + o 1/m m
E O g b m = 2 +
Therefore, the expected value of the overlapping CvM estimator is the same as that of the batched CvM estimator. We turn our attention to the variance of the overlapping CvM estimator. The next theorem, whose proof results from a direct application of the CMT, shows that the asymptotic variance of the estimator is reasonable. Theorem 7. Suppose that (9) holds. For each fixed b, assume that the family of random variables O g b m 2 m = 1 2 is uniformly integrable. If b is fixed and m → , then we have Var O g b m −→ Var O g b m→
Example 8. For the overlapping CvM estimator with quadratic weight g2, t from Example 2, we have that for b 2, Var O g2, b =
then for b 2, (30)
Before proceeding, we need to come up with a more tractable expression for q y . Similar to the development in §4.2, Lemma 4 and Equation (35) (in the appendix) imply that 1−y y q y = y 2 g v v2 g u u2 du dv + 1 − y 2 +
1−y
0
+ y2
1−y
1 1−y
g v 1
0
g v 1 − v 2
y+v
y
1−y 0
y 0
g u u2 du dv
g u u−y +vy −uv −uvy 2 dudv
g v 1 − v 2
+ 1 + y 2
g v v2
1 y
g u 1 − u 2 du dv 1
y+v
10768b − 13605 4 0777 4 ∼ 13860 b − 1 2 b
as b →
This compares to the batched quadratic CvM estimator’s asymptotic variance, Var g2, b = 121 4 / 70b . In this case, Var O g2, b /Var g2, b ∼ 0450 as b → .
for y ∈ 0 1
0
as b →
This compares favorably to the batched constantweighted CvM estimator’s asymptotic (m → ) variance, Var g0 b = 4 4 / 5b . In other words, the ratio Var O g0 b /Var g0 b ∼ 11/21 as b → .
g4, t ≡
0
4 4 1 Var O g b =
b − 1 − y q y dy
b − 1 2 0
88b − 115 4 44 4 ∼ 2 210 b − 1 105b
Example 9. Finally, consider the quartic weighting function
Lemma 3. If (9) holds and we take 1 1 q y ≡ g u g v Cov2 0 u y v du dv 0
=
g u 1 − u 2 du dv for y ∈ 0 1
Plugging this result into Equation (30) and carrying out the tedious algebra allows us to calculate the desired variance for a particular weighting function.
−1310 19270t 25230t 2 + − 21 21 7 3 4 16120t 8060t + − 3 3
In Goldsman et al. (1999), this weighting function is shown to yield the first-order unbiased CvM estimator for 2 having the minimum variance over the space of quartic weights. Indeed, it turns out that Var C g4, = 1042 4 . From Equation (30), we find that the corresponding overlapping quartic CvM estimator has Var O g4, b ∼
0477 4 b
as b →
Thus, Var O g4, b /Var g4, b ∼ 0458 as b → . Similar to Theorem 5, Theorem 8 (proven in the appendix) establishes that the overlapping CvM estimator O g b m converges in mean square to 2 as the samplesize-to-batch-size ratio b and the batch size m tend to infinity. Theorem 8. If (9) holds, then O g b converges in mean square to 2 as b → , qm
O g b −→ 2 b→
(31)
Now suppose that Assumption FCLT also holds. If the sequence of c.d.f.’s O g b m z ≡ Pr O g b m z −→ O g b z ≡ Pr O g b z uniformly on m→
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Table 1.
Approximate asymptotic bias and variance for different estimators.
Nonoverlapping estimators
m/ Bias
f b m f0 b m f2 b m fcos j b m
Equation (14) 3 o 1 o 1
g b m g0 b m g2, b m g4, b m b m
G−1 5 o 1 o 1 1
b/ 4 Var 2 2 2 2
Equation (10) 08 1729 1042 2
z b z ∈ − 2 and b = B B + 1 for some sufficiently large positive integer B, and if the family O g b m 2 b m = 1 2 is uniformly integrable, then O g b m converges in mean square to 2 as b m → , qm
O g b m −→ 2
(32)
b m→
4.4. OBM Estimator Using the notation from §4.1, we define the ith overlapping batch mean as Y¯iOm ≡ m−1 k=0 Yi+k /m for i = 1 n−m+1. The OBM estimator for 2 , originally studied by Meketon and Schmeiser (1984) (with a slightly different scaling constant), is b m ≡
n−m+1 nm
Y¯iOm − Yn 2
n − m + 1 n − m i=1
(33)
Under the mild moment condition (6) of Theorem 1, E b m = 2 +
b 2 + 1 + o 1/m mb b − 1
(34)
which is a close match with the corresponding NBM result given by Equation (17). For a simple derivation of (34), see the appendix (cf. Goldsman and Meketon 1986 and Song and Schmeiser 1995). As for the OBM estimator’s variance, Damerdji (1995, Appendix B) finds that lim Var b m
m→
=
4b 3 − 11b 2 + 4b + 6 4 4 4 ∼ 3 b − 1 4 3b
as b →
5. Summary and Conclusions In this paper, we derived key asymptotic properties of overlapping versions of the STS area and Cramér–von Mises variance estimators. Paralleling the discussion in §3.5, we see that as the batch size m → , the overlapping area,
Overlapping estimators
m/ Bias
b/ 4 Var
O f b m O f0 b m O f2 b m O fcos j b m O fcos 1 b m ¯O2 fcos b m ¯O3 fcos b m ¯O4 fcos b m O g b m O g0 b m O g2, b m O g4, b m b m
Equation (19) 3 o 1 o 1 o 1 o 1 o 1 o 1 G−1 5 o 1 o 1 1
Equation (22) 0686 0819
8( 2 j 2 + 15 / 12( 2 j 2 0793 0429 0297 0227 Equation (30) 0419 0777 0477 1333
overlapping CvM, and OBM estimators are all asymptotically unbiased for 2 . Most importantly, the overlapping area and overlapping CvM estimators are consistent in mean square as b m → . Table 1 gives an approximate synopsis of all of our asymptotic results. The bias properties of the overlapping estimators more or less match those of their nonoverlapping counterparts. In addition, the overlapping area and overlapping CvM estimators using certain “unbiased” weighting functions have better small-sample bias properties than does OBM. The variances of our overlapping estimators are all inversely proportional to the ratio b = n/m (for sufficiently large b and m). In fact, we see that the overlapping STS estimators also outperform their nonoverlapping counterparts, as well as NBM and OBM, in terms of variance; and thus many of the overlapping estimators do the same with respect to MSE. In Alexopoulos et al. (2007b), we summarize the results of a comprehensive experimental performance evaluation of the small-sample properties of the proposed overlapping variance estimators, and we establish the computational complexity of efficient numerical schemes for implementing these variance estimators.
6. Electronic Companion An electronic companion to this paper is available as part of the online version that can be found at http:/ / or. journal. informs.org/.
Appendix In this appendix, we prove Lemmas 1–3, Equation (34), and Theorems 3 and 8. The proofs of Theorems 5 and 6 are detailed in Alexopoulos et al. (2007a). We begin with a useful result. Lemma 4 (Patel and Read 1996). If N1 N2 are bivariate normal with marginal means equal to zero, then Cov N12 N22 = 2Cov2 N1 N2 .
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Proof of Lemma 1. To begin, note that for 0 u v y 1,
=
Cov 0 u y v = Cov u −u 1 y +v − y
=
−v y +1 − y = min uy +v − 1−v min uy −umin 1y +v +u 1−v y −uvy if 0 u y 1 0 v 1−y 1 −u 1−v 1−y if 0 u y 1 0 1−y v 1 u−y +vy −uv −uvy = if 0 y u y +v 1 0 v 1−y 1 − 1−u 1−v y if 0 y u 1 0 1−y v 1
1−u v 1+y if 0 y +v u 1 0 v 1−y 1 This implies that 1−y y pkl 0y = −y fl v v fk u ududv 0
− 1−y +
1 1−y
1−y
fl v
0
−y
0
1 1−y
fl v 1−v
+ 1+y
y 0
fk u ududv
y+v
fk u u−y +vy −uv −uvy dudv
y
fl v 1−v
1−y 0
1 y
fl v v
= = =
1 0
0
1 0
1
0
fk u 1−u dudv. (35)
0 1 y
1 0 1 0
qkl y dy dr +
b−1 r
1
r−1
qkl r − s ds dr
(by independent increments)
b−1 1
1
0
qkl y dr dy + b − 2
b − 1 − y qkl y dy
qkl y dy dr
1 − y qkl y dy + b − 2
=
1
0
0
qkl y dy
1
qkl y dy
1 b−1 b−1 1 1 g u g v 2 0 0 0 0
· Cov2 s u r v du dv ds dr b−1 r 1 1 g u g v = 0
y+v
r
qkl r − s ds dr +
Proof of Lemma 3. Similar to the proof of Lemma 2, we have
b − 1 2 Var O g b 4 4 1 b−1 b−1 1 1 = g u g v 4 0 0 0 0 · Cov 2 s u 2 r v du dv ds dr
fk u 1−u dudv 1
r
=
0
b−1 r
0
0
0
0
· Cov2 s u r v du dv ds dr
1 0
(by Lemma 4)
1 0
g u g v
Now note that x integration by parts for a weighting function f x gives 0 sf s ds = xF x − F x ; and repeated use of this fact on (35) eventually gives the conclusion of the lemma.
· Cov2 0 u r−s v du dv ds dr (by stationarity) b−1 r q r − s ds dr =
Proof of Lemma 2.
b − 1 2 Cov O fk b O fl b 2 4 1 b−1 b−1 = Cov2 fk u s u du
=
0
= = = = =
0
0
0
0
b−1 r b−1 b−1
0 r 0 r 0
b−1 r
0
0
1 0
r 0
0
0
1
=
fl v r v dv ds dr
2 pkl
s r ds dr +
b−1 b−1
0
r
0
(by Lemma 4) 2 pkl
s r ds dr
0
1 0
0 1
0
r
2 2 pkl
0 r − s + plk
0 r − s ds dr
qkl r − s ds dr +
1
b−1 r 0
qkl r − s ds dr
1
b−1 r 0
q r − s ds dr
b − 1 − y q y dy
(as in the proof of Lemma 2) Proof of Equation (34). Define Zk ≡ ki=1 Yi and Vk ≡ Var Zk , k = 1 2 . Goldsman and Meketon (1986) show that Cov Zi Zj = 21 Vi + Vj − Vi−j , i j, and Vk = k 2 + − 2
2 2 pkl
s r + pkl
r s ds dr 2 2 pkl
s r + plk
s r ds dr
q r − s ds dr +
i=k
k − i Ri = k 2 + + o 1 for k 0
(36)
Then, by the preceding expression for Cov Zi Zj , we have nm
n−m+1 i=1
=
n−m+1 i=1
Cov Y¯iOm Yn
Cov Zi+m−1 − Zi−1 Zn =
n i=n−m+1
Vi −
m−1 i=0
Vi (37)
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Thus, by Equations (33) and (37), we have b 1 E bm = Vm + V n−m b n−m n n m−1 2 − V− V
n−m+1 n−m i=n−m+1 i i=0 i
Definition A-4. Let
The result follows after repeated application of Equation (36).
Proposition A-1. If the weighting function f · satisfies (4) and if >m · and > · are defined by (41) and (40), respectively, with the event > as in (43), then
Asymptotic Distribution of Overlapping Area Estimator: Proof of Theorem 3
Pr ∈ D 0 b − > = 1
Definition A-1. Let D 0 b denote the space of functions on 0 b that are right-continuous and have left-hand limits. Let 9b denote the class of strictly increasing, continuous mappings of 0 b onto itself such that for every : ∈ 9b , we have : 0 = 0 and : b = b. If X Z ∈ D 0 b , then the Skorohod metric ;b X Z defining the “distance” between X and Z in D 0 b is the infimum of those positive < for which there exists a : ∈ 9b such that sup : t − t
Z ∈ defined by
2 1 b−1 1 > Z ≡ f u =Z s u du ds (40) b−1 0 0 For m = 1 2 , we define the approximate overlapping area map >m Z ∈ D 0 b → >m Z ∈ by
b−1 m+1 m 1 1 >m Z ≡ f u k m
b − 1 m + 1 i=1 m k=1 2 · =Z s i m u k m (41) where for m = 1 2 , we take s i m ≡ i − 1 /m for i = 1 b −1 m+1 and u k m ≡ k/m for k = 1 m. In view of (1), (41), and the definitions of s i m and u k m , we have =Xm s i m u k m = TiOm k/m for m = 1 2 i = 1 b − 1 m + 1 and k = 1 m. It follows from the development in §4.2 that 2
b−1 m+1 m 1 k 1 k O >m Xm = Tim f
b −1 m+1 i=1 m k=1 m m = O f bm
> ≡ x ∈ D 0b for some sequencexm ⊂ D 0b with lim ;b xm x = 0the sequence>m xm m→ does not converge to > x
(42)
(43)
(44)
where Pr· denotes the Wiener measure. Proof of Proposition A-1. To prove (44), we exploit the almost-sure continuity of sample paths of ; see p. 64 of Billingsley (1968). Thus, without loss of generality we may restrict attention to an event b ⊂ D 0 b such that b ⊂ ∈ D 0 b is continuous on 0 b
and
Pr ∈ b = 1 To prove (44), we show that for every ∈ b and every sequence xm converging to in D 0 b , the corresponding sequence >m xm converges to > in so that b ⊂ D 0 b − > . Choose ∈ b and ? > 0 arbitrarily. If xm ⊂ D 0 b converges to , then we prove that there exists M sufficiently large so that >m xm − > < ? for every m M. By the triangle inequality, >m xm − > >m xm − >m + >m − > for m = 1 2
(45)
Consider first the term >m xm − >m in (45). Choose A ∈ 0 1 arbitrarily; later we determine how to set A in relation to the given quantity ? > 0. The almost-sure continuity of · and Theorem 4.47 of Apostol (1974) imply that t is uniformly continuous on 0 b ; thus there exists B A > 0 such that for all t t ∈ 0 b with t − t < B A we have t − t < A (46) Because xm converges to in D 0 b , there exists M1 A such that ;b xm < minA B A for m M1 A . It follows from (38) that for every m M1 A , there exists :m ∈ 9b such that sup :m t − t A
t∈ 0 b
and
sup xm t − :m t A
t∈ 0 b
(47)
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By (39) together with repeated applications of (47), (46), and the triangle inequality, we see that for every s u ∈ 0 b − 1 × 0 1 and for m M1 A , =xm s u − = s u uxm s + 1 − s + 1 + uxm s − s + xm s + u − s + u + xm s − s
>m − > < ?/2
xm s +1 − :m s +1 + :m s +1 − s +1 + 2xm s − :m s + 2 :m s − s + xm s + u − :m s + u + :m s + u − s + u 8A
(48)
= s u − = s u
u − u s + 1 + s
+ s + u − s + u
(49)
and it follows immediately from (49), the uniform continuity (46) of on 0 b , and the boundedness of on 0 b that = s u is continuous at every s u ∈ 0 b − 1 × 0 1 . Thus, f u = s u is also a continuous function of s u ∈ 0 b − 1 × 0 1 , and we see that f u = s u <
H ∗ ≡ sup f u < u∈ 0 1
for m = 1 2
and (50)
by Theorem 4.27 of Apostol (1974). For each m M1 A , we use (48) and (50) to obtain an upper bound on >m xm − >m that is a continuous function of A: >m xm − >m
b−1 m+1 m 2 1 k f
b − 1 m + 1 i=1 m k=1 m
k k · =xm s i m − = s i m m m m 1 k k k · = + = f x s i m s i m m m k=1 m m m 16 2 H ∗ A G∗ + 4H ∗ A (51) Now it is clear from (51) that we can take A ? sufficiently small so that 16 2 H ∗ A ? G∗ + 4H ∗ A ? < ?/2
>m xm − > ?/2 + ?/2 = ? for m M
CMT (Theorem 5.5 of Billingsley 1968).
+ s + 1 − s + 1 + 2 s − s
sup
Finally, taking M = maxM1 A ? M2 and combining (45), (51), (52), and (53), we obtain
m→
+ s + u − s + u
s u ∈ 0 b−1 × 0 1
(53)
Proof of Theorem 3. Combining the FCLT, (42), (43), and (44), we see that the desired asymptotic result >m Xm −→ > follows directly from the generalized
= s u − = s u + = s u − = s u
G∗ ≡
for m M2
The conclusion of Proposition A-1 follows immediately.
Note that for s u s u ∈ 0 b − 1 × 0 1 ,
Next, we consider the term >m − > in (45). In view of the continuity of = s u at every s u , we see that 2l=1 f ul = s ul is continuous for all u1 u2 s ∈ 0 1 2 × 0 b − 1 and hence is Riemann integrable on 0 1 2 × 0 b − 1 ; see Theorem 14.5 of Apostol (1974). Thus, we can find M2 sufficiently large so that
(52)
Mean-Square Convergence of Overlapping Variance Estimators: Proof of Theorem 8 From definition (13) of s · , the observation that s · is a function of t t ∈ s s + 1 alone, and the independent increments property of · , we see that s1 s2 ∈ 0 b − 1 and s1 − s2 > 1 imply s1 · s2 · are independent. From this property, Tonelli’s Theorem (to justify the integral and expectation interchange), and Lemma 4, we have Var O g b 4 b−1 b−1 1 1 = g u1 g u2
b − 1 2 0 0 0 0
=
2 4
b − 1 2
· Cov 2 s1 u1 2 s2 u2 du1 du2 ds1 ds2 1 1 g u1 g u2 s1 s2 ∈ 0 b − 1 s1 − s2 1
0
0
·Cov2 s1 u1 s2 u2 du1 du2 ds1 ds2 (54) Further, by the Cauchy-Schwarz inequality, Cov s1 u1 s2 u2 E 2 s1 u1 E 2 s2 u2 1 sup Var 0 t = 4 0t1 so that M ≡ supg u1 g u2 Cov2 s1 u1 s2 u2
s1 s2 u1 u2 ∈ 0 b − 1 2 × 0 1 2 <
(55)
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Thus, by (54) and (55), Var O gb
=
4
2 M
b −1 2
1
s1 s2 ∈ 0 b −1 s1 −s2 1
0
1 0
du1 du2 ds1 ds2
2 4 M
2b −3 −→ 0 b→
b −1 2
(56)
This proves (31). Because O g b m z −→ O g b z m→
2
uniformly on z b z ∈ − and b = B B + 1 for B sufficiently large, it follows that for every real z = 2 , we have the double limit lim O g b m z
b m→
= lim O g b z = b→
0
if z < 2
1
if z > 2
(57)
by (31) because mean-square convergence of the sequence O g b b = 1 2 to 2 implies convergence in distribution of that sequence to 2 (Loève 1977). From (57), we see that the double sequence O g b m b m = 1 2 also converges in distribution to 2 so that O g b m −→ 2 because convergence in distribution b m→
to a constant is equivalent to convergence in probability to that constant (Loève 1977). Finally, (32) follows from the latter result, the uniform integrability of O g b m 2 b m = 1 2 , and the Lp convergence theorem (Loève 1977).
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