A new generalized newsvendor model with random demand
aa r X i v : . [ s t a t . A P ] J u l A new generalized newsvendor model with randomdemand
Soham Ghosh, Mamta Sahare and Sujay Mukhoti ∗ Operations Management and Quantitative Techniques AreaIndian Institute of Management, IndoreRau-Pithampur Road, Rau, Indore, Madhya Pradesh, India- 453556
Keywords:
Inventory Management, Newsvendor Problem, Power LossFunction, Random Demand, Optimal Order Quantity Estimation, BrokenSample.
Abstract
Newsvendor problem is an extensively researched topic in inventorymanagement. In this class of inventory problems, shortage and excesscosts are considered to be proportional to the quantity lost. But, forcritical goods or commodities, inventory decision is a typical exam-ple where, excess or shortage may lead to greater losses than merelythe total cost. Such a problem has not been discussed much in theliterature. Moreover, majority of the existing literature assumes thedemand distribution to be completely known. In this paper, we pro-pose a generalization of the newsvendor problem for critical goods orcommodities with higher shortage or excess losses but of same degree.We also assume that, the parameters of the demand distribution areunknown. We also discuss different estimators of the optimal orderquantity based on a random sample of demand. In particular, we pro-vide different estimators based on (i) full sample and (ii) broken sampledata (i.e with single order statistic). We also report comparison of theestimators using simulated bias and mean square error (MSE).
Newsvendor problem is one of the most extensively discussed problems inthe inventory management literature due to its applicability in differentfields [Silver et al., 1998]. This inventory problem relies on offsetting theshortage and leftover cost in order to obtain optimal order quantity. Inthe standard newsvendor problem, each of the shortage and leftover costsare assumed to be proportional to the loss amount, i.e linear in the order ∗ [email protected] In a single period classical newvendor problem, the vendor has to orderthe inventory before observing the demand so that the excess and shortagecosts are balanced out. Let us denote the demand by a positive randomvariable X with cumulative distribution function (CDF) F θ ( x ) , θ ∈ Θ andprobability density function f θ ( x ). We further assume that the first momentof X exists finitely. Suppose Q ∈ R + is the inventory level at the beginningof period. Then, shortage loss is defined as ( X − Q ) + and the excess loss isdefined as ( Q − X ) + , where d + = max ( d, C s and C e denote the perunit shortage and excess costs (constant) respectively and the correspondingcosts are defined as C s ( X − Q ) + and C e ( Q − X ) + . Thus the total cost canbe written as a piece-wise linear function in the following manner: χ = (cid:26) C s ( X − Q ) if X > QC e ( Q − X ) if X ≤ Q (1)The optimal order quantity ( Q ∗ ) is obtained by minimizing the expectedtotal cost, E [ χ ]. The analytical solution in this problem is given by Q ∗ = F − θ (cid:16) C s C e + C s (cid:17) , that is the γ th quantile of the demand distribution ( γ = C s C s + C e ).We propose the following extension the classical newsvendor problemreplacing the piece-wise linear loss functions by piece-wise power losses ofsame degree. Thus, the generalized cost function is given by χ m = (cid:26) C s ( X − Q ) m if X > QC e ( Q − X ) m if X ≤ Q (2)The powers of losses on both sides of Q on the support of X being same wewould refer to this problem as symmetric generalized (SyGen) newsvendorproblem. 4he expected total cost in SyGen newsvendort problem is given by, E [ χ m ] = Z Q C e ( Q − x ) m f θ ( x ) dx + Z ∞ Q C s ( x − Q ) m f θ ( x ) dx In the subsequent sections, we show that the expected cost admits minimumfor uniform and exponential distributions.In order to do so, the first order condition (FOC) is given by ∂E [ χ m ] ∂Q = Z Q mC e ( Q − x ) m − f θ ( x ) dx − Z ∞ Q mC s ( x − Q ) m − f θ ( x ) dx = 0 ⇒ C e Z Q ( Q − x ) m − f θ ( x ) dx = C s Z ∞ Q ( x − Q ) m − f θ ( x ) dx (3)It is difficult to provide further insight without having more informationon f θ ( x ). In the following section, we assume two choices for the demanddistribution - (i) Uniform and (ii)
Exponential . In this section, we consider the problem of determining optimal order quan-tity in SyGen newsvendor set up where the demand is assumed to be a
U nif orm random variable over the support (0 , b ) . The pdf of demanddistribution is given by, f θ ( x ) = (cid:26) b if < x < b otherwise The minimum demand here is assumed to be zero without loss of gener-ality, because any non-zero lower limit of the support of demand could beconsidered as a pre-order and hence can be pre-booked. Using Leibnitzrule and routine algebra, it can be shown that the optimal order quan-tity is given by Q ∗ = b α m , where α m = (cid:16) C e C s (cid:17) m . Corresponding opti-mal cost is C s × b m m +1 × C e (cid:16) C /me + C /ms (cid:17) m . Notice that (cid:16) C /me + C /ms (cid:17) m = C s (cid:16) α /m (cid:17) m ≥ C s , and hence we get an upper bound of the optimalcost as C e b m m +1 . Notice, large α m would imply small order quantity. Inother words, if the cost of excess inventory is much larger than the shortagecost, then the newsvendor would order less quantity to avoid high penalty.Similarly, small α m would result in ordering closer to the maximum possi-ble demand ( b ) to avoid high shortage penalty. However, if the degree ofloss ( m ) is very high, then the optimum choice would be to order half themaximum possible demand, i.e. Q ∗ → b , as m → ∞ .5 .2 Optimal Order Quantity for SyGen Newsvendor withExponential Demand Next we consider the demand to be exponentially distributed with mean λ .The pdf of exponential distribution is given by, f λ ( x ) = 1 λ e − xλ ; x > , λ > E [ χ m ] = Z Q C e ( Q − x ) m f θ ( x ) dx + Z ∞ Q C s ( x − Q ) m f θ ( x ) dx (4)In the following theorem we derive the first order condition for optimalinventory in SyGen newsvendor problem with exponential demand. Theorem 2.1.
Let the demand in a SyGen newsvendor problem be an expo-nential random variable X with mean λ > . Then the first order conditionfor minimizing the expected cost is given by ψ ( Q/λ ) = e − Qλ (cid:20) C s C e − ( − m (cid:21) (5) where ψ (cid:18) Qλ (cid:19) == m − X j =0 ( − j (cid:18) Qλ (cid:19) m − j − m − j − .Proof. Let us define, I m = R Q ( Q − x ) m − λ e − xλ dx and J m = R ∞ Q ( x − Q ) m − λ e − xλ dx .Hence, the FOC in eq.(3) becomes C e I m = C s J m (6)Assuming Q − x = u , we get I m = e − Qλ λ Z Q u m − e uλ du = e − Qλ λ × h λe uλ u m − i Q − λ ( m − I m − , (integrating by parts)= Q m − − λ ( m − Q m − + λ ( m − m − I m − . . . . . . . . . = m − X j =0 Q m − − j ( − λ ) j ( m − m − j − e − Qλ λ m − ( − m ( m − x − Q = v in J m , we get J m = e − Qλ λ Z ∞ v m − e − vλ dv = e − Qλ Γ( m ) λ m − C e I m = C s J m ⇒ m − X j =0 ( − j (cid:18) Qλ (cid:19) m − j − m − j − e − Qλ (cid:20) C s C e − ( − m (cid:21) = γ m e − Qλ As a consequence of the above FOC condition for exponential demand,we need to inspect the existence of non-negative zeroes of eq.(5). We firstprovide lower and upper bound of ψ ( Q/λ ) in the following theorem.
Theorem 2.2.
For γ m > and m ≥ , the following inequality holds: − (1 + γ m ) (cid:18) u − u + 1 (cid:19) + S m − < ψ ( u ) < (1 + γ m )( u −
1) + S m − where, ψ ( u ) = S m − − e − u γ m and S m − k = m − k X j =0 ( − j u m − j − ( m − j − .Proof. Letting Qλ = u , we obtain from eq. (5), ψ ( u ) = m − X j =0 ( − j u m − j − ( m − j − − e − u γ m = 0 , where, γ m = (cid:20) C s C e − ( − m (cid:21) Since, e − u > − u, , for m ≥
3, we get ψ ( u ) < m − X j =0 ( − j u m − j − ( m − j − − (1 − u ) γ m = (( − m − − γ m ) + (( − m − + γ m ) u + m − X j =0 ( − j u m − j − ( m − j − − m − − γ m ) + (( − m − + γ m ) u + m − X j =0 ( − j u m − j − ( m − j − − m − + γ m )( u −
1) + m − X j =0 ( − j u m − j − ( m − j − ≤ (1 + γ m )( u −
1) + S m − , [ since, ( − m − ≤
1] (7)On the other hand,using e − u < − u + u , it can be shown from eq. (5)7hat for m ≥ ψ ( u ) ≥ m − X j =0 ( − j u m − j − ( m − j − − (cid:18) − u + u (cid:19) γ m = (( − m − − γ m ) + (( − m − + γ m ) u + (( − m − − γ m ) u m − X j =0 ( − j u m − j − ( m − j − − m − − γ m ) − (( − m − − γ m ) u + (( − m − − γ m ) u m − X j =0 ( − j u m − j − ( m − j − − m − − γ m )(1 − u + u m − X j =0 ( − j u m − j − ( m − j − ≥ (( − − γ m )(1 − u + u S m − , [ since, ( − m − ≥ − − (1 + γ m )(1 − u + u S m − (8)The following few remarks could be made immediately from the abovetheorem. Remark.
For m = 2 k , γ m < C s < C e , in which case the directionof the above inequality will be altered. Further, for m = 2 k + 1, γ m is alwayspositive as C s , C e > Remark. If m = 2 k + 1, γ m >
0. In that case, we get from the lowerboundary function in eq.(7), g L ( u ) = u k (2 k )! − u k − (2 k − ... + u − u − (1 + γ m ) u u (1 + γ m ) − (1 + γ m )Therefore, by Descarte’s rule of sign, the maximum number of positive realroots is 2 k − g U ( u ) = u k (2 k )! − u k − (2 k − ... + u − u
3! + u u (1 + γ m ) − (1 + γ m )Hence, by similar argument as in g L ( u ), at least one positive root of g U ( u )will exist. Thus, both the boundary functions will have a positive rootleading to the existence of positive root of ψ ( u ).8 emark. If m ( ≥
4) is even then, following the similar line of argument as inthe previous remark, it can be shown that both g L ( u ) and g U ( u ) will have2 k − Remark.
In the particular case of m = 3, it the boundary functions reducesto g L ( U ) = (2 + C s C e )( u − u + 1)& g U ( u ) = ( u − γ m + 1) + u − (1 + γ m ) + p γ m + 4 γ m + 3.Since eq. (5) is a transcendental equation in Q, numerical methodswould be required to find zeros, which in turn would provide the optimalorder quantity. However, the solution would be dependent on the shortageand excess cost ratio. In what follows, we describe the nature of solutionsin different scenarios for α = C s C e .In case of equal shortage and excess costs ( C s = C e ), α = 1 and cor-responding optimal order quantities ( Q ∗ ) are given in the table below fordifferent m : Table 1: Optimal Order Quantity for C s = C e .m Q ∗ λ . λ . λ
10 3 . λ
20 6 . λ On the other hand, if the shortage cost is much lower than the excesscost ( α → Q ∗ ) is very small (tends to0). In case of 0 < α <
1, the optimum order quantity can be computedfrom the following equation: m − X j =0 ( − j (cid:18) Qλ (cid:19) m − j − m − j − γ m e − Qλ The two figures in appendix B, fig.1 provide the optimal order quantityas a multiple of the average demand ( λ ) obtained for α ∈ (0 ,
1) and m =2 , , , , , , , , , α s reduce with increasing m . Here we may interpret m asthe degree of seriousness of the losses. Hence, the observation made above9ay also be restated in the following manner. As the degree of serious-ness of the loss increases, the newsvendor becomes indifferent to both thelosses. That is, beyond a risk level the newsvendor will not react much tothe increase in any type of loss and order at a steady level, i.e become riskneutral.In the following section we consider the parameters of the demand dis-tributions discussed above, to be unknown and study the estimation of theoptimal order quantity. In this section we consider the problem of estimating the optimal orderquantity in SyGen newsvendor setup, based on a random sample of fixed sizeon demand, when the parameters of the two demand distributions discussedabove are unknown. U (0 , b ) Demand
As in the previous section, we first consider the demand distribution tobe unif orm (0 , b ), where b is unknown. We elaborate on the estimation ofoptimal order quantity in SyGen news vendor problem with available iiddemand observations X , X . . . X n . Method of moment type estimator ofthe optimal order quantity could be constructed by plugging in the same forthe unknown parameter b . Thus,ˆ Q = 2¯ x α m (9)which is an unbiased estimator as well. The variance of ˆ Q is given by V ( ˆ Q ) = b n (1+ α m ) . In fact the uniformly minimum variance unbiased esti-mator (UMVUE) of the optimal order quantity can be obtained using theorder statistics X (1) < X (2) . . . < X ( n ) . The UMVUE is given as follows:ˆ Q = ( n + 1) X ( n ) n (1 + α m ) (10)The variance of the UMVUE is given by V ( ˆ Q ) = b (1+ α m ) n ( n +2) . Since X ( n ) is the maximum likelihood estimator (MLE) of b , the MLE of optimal orderquantity can also be obtained asˆ Q = X ( n ) α m (11)Note that ˆ Q is biased with Bias ( ˆ Q ) = − Q ∗ n +1 and mean square error( M SE ) = Q ∗ n +1)( n +2) . Comparing ˆ Q , ˆ Q and ˆ Q in terms of their10ariances (MSE for ˆ Q ), it can be easily seen that the UMVUE providesthe best estimator among the three. In particular, V ( ˆ Q ) < M SE ( ˆ Q )
0. Let X , X , . . . , X n be a random sample ondemand. Thus the problem becomes estimation of optimal order quantityin this SyGen setup. In order to provide a good estimator of the optimal order quantity basedon the full sample, we replace the parametric functions of λ involved inthe FOC eq. (5) by their suitable estimators. In particular, we focus onreplacing (i) λ by its MLE, (ii) e − Qλ = ¯ F ( Q ) by corresponding UMVUEs.Performances of the estimated optimal order quantities, Q ∗ , is measured bythe corresponding bias and MSE. FIRST ESTIMATING EQUATION
Replacing λ in eq. (5) by its MLE ¯ X , the first estimating equation isobtained as follows: m − X j =0 ( − j (cid:18) Q ¯ X (cid:19) m − j − m − j − e − Q ¯ X γ m (12)Let the solution of this estimating equation be denoted by Q ∗ . Though Q ∗ is a plug-in estimator, being a function of MLE, it still would be expectedto perform well in terms of bias and MSE. SECOND ESTIMATING EQUATION
The UMVUE of e − Qλ = ¯ F ( Q ) based on the SRS X , X , ..., X n drawnfrom exp ( λ ) population is given by T SRS = (cid:18) − QW (cid:19) n − where, W = P ni =1 X i = n ¯ X and ( d ) + = max ( d, e − Qλ by T SRS in eq(5). Also, the UMVUE of λ , appearing in the coefficients of Q m − j − , ∀ ≤ j ≤ m −
1, in eq(5), is given by ¯ X . Replacing the above11wo estimators in place of their corresponding estimands in FOC, the secondestimating equation is obtained as m − X j =0 ( − j (cid:18) n − m − j − (cid:19) (cid:18) QW (cid:19) m − j − = γ m (cid:18) − QW (cid:19) n − (13)Let the solution of the above estimating equation be denoted by ˆ Q ∗ . Thisis also a plug-in estimator and we compare it with ˆ Q ∗ in terms of bias andMSE, as provided in section 4. In case the full data is not available, it is more likely that the seller wouldbe able to recall the worst day or the best day in terms of demand. Theworst day demand would be represented by the smallest order statistic X (1) ,and the best day would be counted as the largest order statistic X ( n ) . Ingeneral, if the observation on the i th smallest demand out of a sample of size n is available, viz. X ( i ) , then we may consider the following two possibleways of estimating the optimal order quantity. One is to use estimatingequation obtained by replacing λ with its unbiased estimator based on X ( i ) in both sides of the eq. (5) [see Sengupta and Mukhuti, 2006, and referencestherein]. Another estimating equation can be obtained by replacing λ by itsunbiased estimator using X ( i ) in the left hand side expression of eq. (5) and X ( i ) based unbiased estimator of ¯ F ( Q ) in the right hand side of the sameequation.Sinha et al. [2006] provided an unbiased estimator of ¯ F ( Q ) based on X ( i +1) , which is as follows: h i +1 ( Z i +1 ) = ∞ X j =0 ∞ X j =0 . . . ∞ X j i =0 d j j ...j i I ( Z i +1 > α j α j . . . α j i i Q ) (14)where Z i = ( n − i + 1) X ( i ) , α k = n − i + kn − i , k = 1 , . . . i , d j j ...j i = ( − P j i (cid:0) ni (cid:1) × i Y k =1 " (cid:0) ik (cid:1) α k j k , P extending over all even suffixes of j . Further, an unbiasedestimator of λ based on X ( i ) is given by ˆ λ = X ( i ) a i , where a i = i X j =1 n − j + 1 .The first approach to estimate Q would be through the estimating equationobtained by replacing λ with ˆ λ in eq. 5. Let the corresponding solution bedenoted by ˆ Q ∗ (1) . The second approach for estimating Q ∗ is to replace ¯ F ( Q )by h i ( Z i ) and λ by ˆ λ in eq. (5). We denote the corresponding estimator byˆ Q ∗ (2) . 12n the case of only the best day demand data being available, the largestorder statistic X ( n ) is observed. However, due to high complexity of com-putation we don’t investigate this case in details. In this section we present simulation studies for exponential demand in or-der to estimate the optimum order quantities from the estimating equa-tions discussed above. Let us consider standard exponential demand dis-tribution (exp(1)) and observe performances of the estimated optimal or-der quantities corresponding to different estimating equations over differ-ent values of m and α m . In this simulation we draw samples of size n ,(= 10 , , , , , , n , the esti-mated optimal quantities are determined from each of the estimating equa-tions described in the previous section. This procedure is repeated 1000times. We compute the bias in the estimated optimal order quantities bythe average of ( ˆ Q ∗ − Q ∗ ) over these 1000 repetitions, where Q ∗ is true opti-mal order quantity and ˆ Q ∗ is its estimate. The tables below report bias andMSE of different estimators of the optimal order quantity.Full sample bias and MSE of ˆ Q ∗ are reported in tables 2a-2c, respectivelyand the same for ˆ Q ∗ are given in tables 3a-3c. It can be observed from thefigures that the bias and MSEs of these two estimators of Q ∗ are comparable.Also, none of the two estimators uniformly outperforms the other in termsof absolute bias or MSE across the given degrees of importance of loss, m ,in either small or large sample cases.Comparing the bias and MSE of the estimators of Q ∗ based on 2 nd order statistic ( viz. ˆ Q ∗ (1) and ˆ Q ∗ (2) ) given in tables Tab.4a-4c and Tab.5a-5crespectively, similar observations as in the full sample case, can be made.However, in this case, the margin in bias and/or MSE for certain ( α , m, n )are much larger. For example, MSE of ˆ Q ∗ (1) is quite smaller than that of ˆ Q ∗ (2) for m = 50 over all the considered sample sizes, when α = 2, whereas similarmargins could be observed favoring ˆ Q ∗ (2) in case of α = 1 and m = 10, forall sample sizes except 100 and 1000. Thus, neither of the two estimatorsoutperform the other. Contributions of our study in this paper are 2-fold. First we have proposeda generalization of the standard news vendor problem assuming random de-mand and higher degree of shortage and excess loss. In particular, we havedeveloped a symmetric generalized news vendor cost structure using powerlosses of same degree for both of shortage and excess inventory. We have pre-sented the method to determine the optimal order quantity. In particular,13e have presented the analytical expression of the optimal order quantityfor uniform demand. In case of exponential demand, determination of op-timal order quantity requires finding zeroes of a transcendental equation.We have proven the existence of real roots of the equation, ensuring thatthere exists a realistic solution to the proposed general model. Our secondcontribution is to provide different estimators of the optimal order quantity.We have provided estimators of the optimal order quantity using (i) fullsample and (ii) broken sample data on demand. Whereas, analytical formof the estimator and its properties are easy to verify for uniform demanddistribution, it is difficult for exponential distribution. We have presenteda simulation study to compare different estimators. In this paper, we havepresented estimators based on full sample as well as broken sample like sin-gle order statistic. Finally, we have provided a simulation study to gaugethe performance of the proposed estimators in terms of bias and MSE.A natural extension of this work could be to consider asymmetric short-age and excess losses. However, asymmetric power type shortage and excesslosses would result in different dimensions of shortage and excess costs mak-ing them incomparable. Unpublished manuscript by Baraiya and Mukhoti[2019] proposes an inventory model for such a case.
Acknowledgement rd author remains deeply indebted to Late Prof. S Sengupta and Prof.Bikas K Sinha for their intriguing discussions and guidance on broken sam-ple estimation. Research of 1 st author is funded by Indian Institute of Man-agement Indore. The authors acknowledge gratefully the valuable commentsby the participants of the SMDTDS-2020 conference organized by IAPQRat Kolkata. References
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Management Science , 12(3):206–222, 1965.16
Tables
Table 2a: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ for C s = 2 C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.0052 -0.0070 -0.0010 0.0011 -0.0007 0.0004 -0.00060.0597 0.0113 0.0060 0.0012 0.0005 0.0001 0.00013 0.6206 0.5937 0.6070 0.6116 0.6076 0.6099 0.60790.6775 0.4074 0.3976 0.3800 0.3718 0.3726 0.36984 13.1606 12.9312 13.0446 13.0836 13.0495 13.0695 13.0523194.3676 171.1996 172.2751 171.6106 170.4826 170.8554 170.38375 -0.0927 -0.1180 -0.1055 -0.1012 -0.1050 -0.1028 -0.10470.2664 0.0624 0.0369 0.0155 0.0134 0.0111 0.011210 3.0587 2.9607 3.0092 3.0258 3.0112 3.0198 3.012413.2180 9.4926 9.4406 9.2340 9.1028 9.1270 9.078620 -4.4660 -4.4904 -4.4784 -4.4742 -4.4778 -4.4757 -4.477620.1848 20.2090 20.0797 20.0235 20.0533 20.0325 20.048750 -10.9240 -10.9797 -10.9522 -10.9427 -10.9510 -10.9461 -10.9503120.5828 120.7890 120.0743 119.7677 119.9350 119.8197 119.910017able 2b: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ for C s = C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.0978 0.0805 0.0891 0.0920 0.0895 0.0910 0.08970.1299 0.0291 0.0200 0.0109 0.0091 0.0085 0.00823 1.8820 1.8318 1.8566 1.8651 1.8577 1.8621 1.85834.5529 3.5459 3.5480 3.4993 3.4603 3.4694 3.45434 -0.5893 -0.6052 -0.5973 -0.5946 -0.5970 -0.5956 -0.59680.4485 0.3853 0.3669 0.3556 0.3573 0.3549 0.35635 0.0128 -0.0172 -0.0024 0.0027 -0.0017 0.0009 -0.00140.3614 0.0683 0.0361 0.0074 0.0033 0.0007 0.000410 11.3833 11.1515 11.2661 11.3055 11.2711 11.2913 11.2739151.2083 128.4259 129.0847 128.2545 127.2342 127.5375 127.122320 -4.3518 -4.3806 -4.3664 -4.3615 -4.3657 -4.3632 -4.365419.2711 19.2521 19.0983 19.0292 19.0627 19.0385 19.057050 -8.2308 -8.3314 -8.2817 -8.2646 -8.2795 -8.2708 -8.278371.8199 70.1792 68.9928 68.3861 68.5878 68.4136 68.5345Table 2c: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ for C s = 0 . C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.1296 0.1074 0.1184 0.1222 0.1189 0.1208 0.11910.2148 0.0488 0.0338 0.0189 0.0159 0.0150 0.01443 4.5171 4.4217 4.4689 4.4851 4.4709 4.4793 4.472124.0654 20.2408 20.3366 20.1907 20.0227 20.0712 20.00344 -0.5046 -0.5252 -0.5150 -0.5115 -0.5146 -0.5128 -0.51430.4257 0.3080 0.2823 0.2651 0.2664 0.2633 0.26475 1.2617 1.2088 1.2349 1.2439 1.2361 1.2407 1.23672.7166 1.6729 1.6374 1.5702 1.5381 1.5416 1.530610 -2.2265 -2.2468 -2.2367 -2.2333 -2.2363 -2.2345 -2.23615.1226 5.0790 5.0195 4.9909 5.0025 4.9934 5.000120 -4.2357 -4.2689 -4.2525 -4.2468 -4.2517 -4.2488 -4.251318.3848 18.3069 18.1277 18.0444 18.0814 18.0536 18.074350 0.1000 -0.1344 -0.0185 0.0213 -0.0135 0.0069 -0.010722.1078 4.1763 2.2061 0.4496 0.2016 0.0449 0.022318able 3a: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ for C e = 2 C s m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 -0.0040 -0.0062 0.0024 0.0000 -0.0003 -0.0001 -0.00020.0566 0.0110 0.0060 0.0012 0.0006 0.0001 0.00013 0.6517 0.6058 0.6197 0.6102 0.6091 0.6090 0.60870.7187 0.4211 0.4138 0.3785 0.3740 0.3715 0.37094 13.4388 13.1458 13.2482 13.1041 13.0796 13.0654 13.0613201.9324 176.7958 177.6949 172.1608 171.2964 170.7480 170.61975 -0.0795 -0.1103 -0.0954 -0.1028 -0.1037 -0.1036 -0.10390.2606 0.0597 0.0352 0.0160 0.0134 0.0113 0.011110 3.1096 2.9906 3.0482 3.0195 3.0161 3.0164 3.015413.4791 9.6562 9.6832 9.1984 9.1368 9.1061 9.096920 -4.4347 -4.4792 -4.4667 -4.4754 -4.4765 -4.4765 -4.476819.9084 20.1073 19.9757 20.0342 20.0411 20.0399 20.041950 -10.9395 -10.9700 -10.9336 -10.9470 -10.9486 -10.9481 -10.9486120.8739 120.5714 119.6690 119.8620 119.8843 119.8641 119.8738Table 3b: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ for C s = 2 C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.1491 0.0942 0.1002 0.0918 0.0907 0.0904 0.09020.1502 0.0314 0.0223 0.0109 0.0095 0.0084 0.00833 1.9619 1.8564 1.8812 1.8628 1.8606 1.8604 1.85994.8798 3.6337 3.6416 3.4912 3.4723 3.4630 3.46024 -0.5811 -0.6003 -0.5910 -0.5956 -0.5962 -0.5962 -0.59630.4374 0.3791 0.3595 0.3569 0.3565 0.3556 0.35575 0.0309 -0.0070 0.0101 0.0009 -0.0002 -0.0002 -0.00040.3582 0.0668 0.0368 0.0076 0.0038 0.0007 0.000410 11.5039 11.2222 11.3586 11.2907 11.2825 11.2832 11.2810153.6697 129.9282 131.2102 127.9323 127.5194 127.3534 127.283720 -4.3348 -4.3713 -4.3546 -4.3633 -4.3643 -4.3642 -4.364519.1194 19.1696 18.9964 19.0449 19.0506 19.0472 19.049450 -8.3287 -8.3343 -8.2585 -8.2744 -8.2763 -8.2746 -8.275473.1996 70.2044 68.6129 68.5503 68.5387 68.4776 68.486919able 3c: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ for C s = 2 C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.1288 0.1119 0.1261 0.1205 0.1198 0.1200 0.11980.2084 0.0489 0.0359 0.0187 0.0164 0.0148 0.01463 4.7220 4.5005 4.5335 4.4846 4.4784 4.4765 4.475326.0939 20.9409 20.9274 20.1881 20.0944 20.0462 20.03214 -0.4644 -0.5131 -0.5038 -0.5122 -0.5132 -0.5134 -0.51370.3920 0.2951 0.2713 0.2660 0.2652 0.2639 0.26405 1.2892 1.2249 1.2560 1.2405 1.2387 1.2388 1.23832.7712 1.7080 1.6916 1.5625 1.5460 1.5369 1.534710 -2.2054 -2.2386 -2.2276 -2.2344 -2.2352 -2.2352 -2.23545.0296 5.0418 4.9791 4.9959 4.9978 4.9965 4.997320 -4.2275 -4.2598 -4.2397 -4.2490 -4.2502 -4.2500 -4.250318.3061 18.2278 18.0200 18.0635 18.0684 18.0636 18.065850 0.0614 -0.1994 -0.0146 -0.0146 -0.0126 -0.0034 -0.004621.3335 4.0419 2.2135 0.4609 0.2292 0.0443 0.0239Table 4a: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ (1) for C s = 2 C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.0311 0.0057 0.0044 0.0118 0.0130 0.0042 0.02570.3344 0.2931 0.2813 0.3175 0.2928 0.3210 0.36573 0.6780 0.6219 0.6189 0.6353 0.6379 0.6185 0.66602.0924 1.8219 1.7604 1.9575 1.8396 1.9542 2.23094 13.6486 13.1712 13.1460 13.2854 13.3079 13.1422 13.5463304.4965 277.3833 272.5336 289.0052 280.8245 286.5041 312.91125 -0.0388 -0.0915 -0.0943 -0.0789 -0.0765 -0.0947 -0.05011.4415 1.2740 1.2235 1.3767 1.2693 1.3950 1.578910 3.2672 3.0633 3.0525 3.1120 3.1216 3.0508 3.223532.2458 28.3435 27.5137 30.2142 28.6722 30.0715 34.005320 -4.4141 -4.4649 -4.4676 -4.4528 -4.4504 -4.4680 -4.425020.8213 21.1102 21.0868 21.0992 20.9786 21.2496 21.044050 -10.8054 -10.9214 -10.9275 -10.8936 -10.8882 -10.9285 -10.8303123.7353 125.4102 125.2968 125.3128 124.6758 126.1481 124.933820able 4b: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ (1) for C s = C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.1346 0.0986 0.0967 0.1073 0.1089 0.0965 0.12690.6903 0.6005 0.5763 0.6512 0.6016 0.6563 0.75193 1.9886 1.8843 1.8788 1.9092 1.9141 1.8779 1.96639.6012 8.5136 8.2929 9.0191 8.6186 8.9619 10.04764 -0.5556 -0.5886 -0.5903 -0.5807 -0.5791 -0.5906 -0.56260.8739 0.8432 0.8253 0.8751 0.8314 0.8929 0.93545 0.0765 0.0142 0.0109 0.0291 0.0320 0.0104 0.06322.0236 1.7737 1.7021 1.9211 1.7715 1.9423 2.212810 11.8767 11.3942 11.3687 11.5096 11.5323 11.3648 11.7733261.8440 235.9926 231.1351 247.4247 238.9786 245.4252 270.839020 -4.2906 -4.3505 -4.3536 -4.3362 -4.3333 -4.3541 -4.303420.2683 20.5605 20.5222 20.5714 20.4089 20.7477 20.554650 -8.0167 -8.2261 -8.2372 -8.1760 -8.1662 -8.2389 -8.061687.0193 87.6663 87.0425 88.5002 86.6497 89.7789 89.8955Table 4c: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ (1) for C s = 0 . C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.1768 0.1306 0.1282 0.1417 0.1439 0.1278 0.16691.1370 0.9889 0.9491 1.0724 0.9909 1.0807 1.23833 4.7201 4.5216 4.5111 4.5691 4.5784 4.5095 4.677642.7250 38.4152 37.5964 40.3345 38.9017 40.0158 44.26174 -0.4607 -0.5036 -0.5059 -0.4933 -0.4913 -0.5062 -0.46991.1676 1.0934 1.0618 1.1526 1.0797 1.1759 1.26675 1.3742 1.2641 1.2583 1.2905 1.2956 1.2574 1.35068.1699 7.1191 6.8820 7.6434 7.1903 7.6275 8.700510 -2.1833 -2.2255 -2.2278 -2.2154 -2.2135 -2.2281 -2.19245.6909 5.7651 5.7423 5.7875 5.7101 5.8538 5.817920 -4.1650 -4.2341 -4.2378 -4.2176 -4.2143 -4.2383 -4.179819.8266 20.1071 20.0502 20.1477 19.9362 20.3501 20.185150 0.5986 0.1109 0.0852 0.2276 0.2505 0.0812 0.4941123.7725 108.4855 104.1096 117.5048 108.3515 118.8006 135.347021able 5a: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ (2) for C s = 2 C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 -0.0625 -0.1219 -0.0799 -0.1007 -0.1041 -0.1025 -0.10560.2761 0.2108 0.2245 0.2384 0.2546 0.2371 0.25763 0.5977 0.4855 0.5937 0.5472 0.5395 0.5451 0.53761.9156 1.4015 1.6598 1.6739 1.7599 1.6658 1.77794 13.2132 12.6096 13.6049 13.2154 13.1489 13.1931 13.1255291.3362 251.0958 288.8477 284.1045 289.9123 282.9913 290.76495 -0.1042 -0.1701 -0.0614 -0.1039 -0.1112 -0.1064 -0.11381.4026 1.1268 1.2407 1.3157 1.4074 1.3100 1.425510 3.0143 2.7592 3.1798 3.0152 2.9871 3.0058 2.977229.9356 24.0600 28.6405 28.6393 29.8210 28.4890 30.024120 -4.5136 -4.5327 -4.4134 -4.4535 -4.4608 -4.4556 -4.462721.6035 21.5750 20.6595 21.0824 21.2326 21.0951 21.267850 -11.1292 -11.2668 -11.0399 -11.1287 -11.1439 -11.1338 -11.1492129.9304 131.7310 127.2745 129.5401 130.2713 129.6265 130.4667Table 5b: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ (2) for C s = C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.1857 0.2157 0.3067 0.2795 0.2740 0.2783 0.27250.8032 0.7401 0.8810 0.9173 0.9725 0.9135 0.98373 2.0321 1.8945 2.1214 2.0326 2.0174 2.0275 2.012110.2007 8.3781 9.8960 9.8237 10.1555 9.7758 10.21034 -0.5965 -0.6378 -0.5697 -0.5963 -0.6009 -0.5979 -0.60250.9022 0.8377 0.8101 0.8679 0.9087 0.8672 0.91755 -0.2011 -0.2469 -0.1175 -0.1674 -0.1741 -0.1685 -0.17661.5988 1.3269 1.4608 1.5476 1.6572 1.5434 1.679010 11.2783 10.6747 11.6700 11.2805 11.2140 11.2581 11.1905243.9466 206.0419 239.9422 236.7064 242.7717 235.6797 243.715020 -4.4437 -4.4980 -4.3694 -4.4175 -4.4258 -4.4206 -4.428721.3903 21.5558 20.5983 21.1017 21.2846 21.1207 21.331350 -8.4431 -8.6981 -8.2775 -8.4421 -8.4703 -8.4516 -8.480292.1351 92.1044 87.0470 90.8175 92.6434 90.8839 93.073522able 5c: Bias (in 1 st row) and MSE (in 2 nd row) of ˆ Q ∗ (2) for C s = 0 . C e m n =10 n =50 n =100 n =500 n =1000 n =5000 n =100002 0.0783 -0.0096 0.0751 0.0353 0.0286 0.0322 0.02611.0075 0.7557 0.8501 0.8859 0.9455 0.8799 0.95663 4.6372 4.3821 4.8027 4.6381 4.6100 4.6286 4.600142.3529 35.6494 41.5951 41.0597 42.1501 40.8788 42.32114 -0.4795 -0.4729 -0.3671 -0.4008 -0.4066 -0.4019 -0.40841.2028 1.0670 1.0989 1.1834 1.2596 1.1810 1.27575 1.2377 1.1001 1.3270 1.2382 1.2230 1.2331 1.21777.6032 5.9993 7.1566 7.2254 7.5813 7.1856 7.644610 -2.1798 -2.2379 -2.1481 -2.1815 -2.1869 -2.1827 -2.18895.7243 5.7721 5.4737 5.6682 5.7558 5.6704 5.776820 -4.4662 -4.6036 -4.4881 -4.5374 -4.5444 -4.5388 -4.547121.8757 22.6154 21.7307 22.2611 22.4430 22.2681 22.490250 -0.1642 -0.7678 0.2275 -0.1620 -0.2285 -0.1843 -0.2519116.7738 92.6822 103.8056 109.4833 117.0710 108.9685 118.550423 I Figures a Q* a m = m = m = m = m = a Q* a m = m = m = m = m = F i g u r e : O p t i m a l O r d e r Q u a n t i t y ( Q ∗ ) f o r d i ff e r e n t α ∈ ( , ) a nd m ..