Proofs, Tables and Figures

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1 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem ec Proofs, Tales and Figures In this electronic companion to the paper, we provide proofs of the theorems and lemmas. This companion also contains several accompanying tales and figures to the paper. Tale EC. Range of critical fractile values where Assumption holds. Distriution When is Assumption satisfied? Notes Normalµ, σ Exponentialλ Lognormalµ, σ Paretox m, UniformA, B Gamma, β Beta, β Power Law Logisticµ, s GEVµ, σ, ξ Chik Chi-squaredk Laplaceµ, β Weiullλ,k +h +h 0 + erf σ +h 0 +h 0 +h γ, +h Γ B +β ;,β +h B,β 0 +h +h erf: error function Γ: gamma function, γ: incomplete gamma function B: eta function +h e ξ for ξ 0 P k, k for k ; P : regularized gamma function +h { Γ k γ k, k, if k +h 0, if k< +h +h { e k k, if k 0 if k< Tale EC. Average errors % with samples from an exponential distriution. a Sample average approximation Sample size Distriution fitting Sample size

2 ec e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem Figure EC. Upper ound for a log-concave distriution with +h quantile q. Tale EC.3 Average errors % with samples from a normal distriution. a Sample average approximation Sample size Distriution fitting Sample size

3 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem ec3 Tale EC.4 Average errors % with samples from a Pareto distriution. a Sample average approximation Sample size Distriution fitting Sample size Tale EC.5 Average errors % with samples from a Beta distriution. a Sample average approximation Sample size Distriution fitting Sample size

4 ec4 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem Tale EC.6 Average errors % with samples from a mixed normal distriution. a Sample average approximation Sample size Distriution fitting Sample size

5 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem ec5 EC.. Proof of Theorem As a preliminary for the proof, let us first state a version of Bernstein s inequality Bernstein 97: Theorem EC. Bernstein s inequality. Let X,X,...,X N that X c almost surely, and VarX =σ. Then, for any t>0, N Pr X i E[X ] t N i= Nt. σ +tc/3 For the proof of Theorem, we will require the following proposition. Proposition EC.. Suppose ˆQ N is the +h Then, for any γ > 0, Proof. Pr C ˆQ N γ and + C ˆQ N γ exp e i.i.d. random variales such quantile of a random sample from D with size N. 3Nγ 6h +8γ + h Let F e the complementary cdf of D, i.e., F q=prd q= F q+prd = q. For a random sample {D,...,D N } drawn from D, let ˆQ N e the +h. sample quantile. Define ˆF N q N ˆ F N q N N [ D i q], i= N [ D i q]. i= For simplicity, define γ +h and β. Define the events B [ +h +C ˆQ N < γ]=[f ˆQ N < β ] and L [ C ˆQ N > γ]=[ F ˆQ N < β ]. To prove Proposition EC., we need to find an upper ound for PrB and for PrL. Define the quantile q inf{q : F q β }. SinceF is nondecreasing, we have that B =[ˆQ N < q ]. Consider a monotonically decreasing, nonnegative sequence {τ k } k=, where τ k 0. Define the sequence of events {B k } k=, where B k [ ˆQ N q τ k ]=[ˆF N q τ k β]. Note that since ˆF N q τ k ˆF N q τ k+, then it follows that B k B k+. Thus, we have that B k lim k B k B, which implies PrB k Pr B. NotealsothatB B, thusprb Pr B. From the definition of q, oserve that for every k, there exists ε k > such that F q τ k = β ε k < β. Notethat F q τ k F q τ k < β β + ε k. EC.

6 ec6 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem Thus, we have that PrB k =PrˆF N q τ k β, =PrˆF N q τ k F q τ k ε k, Nε k/ F q τ k F q τ k + ε, EC. k 3 Nε k /, EC.3 β β+β + ε k 3 where EC. follows from Bernstein s inequality and EC.3 follows from inequality EC.. Now, since ε k >, forallk, we have that N/ PrB k β β +β, 3 N/ β β+ 4 minβ, β 3 N/ β β+ =exp 4 3, 3Nγ 6h +8γ + h δ. Thus, PrB Pr B δ. In fact, y going through a similar argument, we can show that PrL δ. Thus, y the union ound, we have that Pr C ˆQ N > γ or + C ˆQ N < γ =PrB L PrB+PrL δ, proving Proposition EC.. Q.E.D. We can now proceed with the proof of Theorem. Note that S LRS ϵ consists of all q for which Cq γ and + Cq γ, withγ = ϵ min, h. From Proposition EC., the SAA solution from 3 a random sample with size N lies in Sϵ LRS with proaility at least Nϵ min{, h} exp = exp 8h +8ϵ + hmin{, h} Nϵ exp 8 + 8ϵ EC.. Proof of Theorem 3 Nϵ min{, h} 8 max{, h} +8ϵ + h min{, h}. + h Since C is convex, S f ϵ [q, can e equivalently expressed as {q : C q C q and q q }. Note that, C q= + hf q F q = + h [ q q fq +Oq q ] = ϵh q fq +Oϵ, EC.4. which follows from Taylor series approximation and from the definition of q in 7.

7 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem ec7 To prove Theorem 3, note that the event that Q N S f ϵ [q,, where = C q, is equivalent to the intersection of events [ Q N q ] and [C Q N ]. We will prove an upper ound on the proaility of [ Q N <q ] and on the proaility of [C Q N > ]. It follows similar lines to the proof of Lemma 3.5 in Levi et al. 007, except we will use Bernstein s inequality instead of Hoeffding s inequality. Define β and γ. First, let us ound the proaility of B [ Q +h +h N <q ]. For a realvalued sequence {τ k } k= where τ k 0, define B k [ Q [ N q τ k ]= + + h ˆF N q τ k ] =[ˆF N q τ k β + γ]. Note that since ˆF N is monotonically increasing, it follows that B k B k+.thus,if B is the limiting event of the sequence of events {B k } k=, thenb k B, implying that PrB k Pr B. Note also that B B, thusprb Pr B. Therefore, to ound PrB, we only need to find a uniform upper ound for PrB k. Note that for any k, there exists ε k > 0 such that F q τ k =β ε k.thus, F q τ k F q τ k =β ε k β + ε k < β β + ε k. From Bernstein s inequality, we have that PrB k =Pr ˆFN q τ k β + γ =Pr ˆFN q τ k F q τ k γ + ε k Nγ + ε k F q τ k F q τ k + γ + 3 εk Nγ + ε k =exp β γ+ε k εk β γ + β ε k + 3 Nγ + ε k Nγ β β γ+β + β β γ+β + γ+ε k 3 γ 3 where the inequality follows when β γ 0. Hence, for all k, 3N PrB k. 4h +4 + h Since = C q, from EC.4 we have that PrB k 6Nϵh q fq +Oϵ 3/ 4h + Oϵ / Uϵ. EC.5 Now, let us ound the proaility of L [C Q [ ] N > ] = F Q N < h.defineq +h +h 0 { sup q : F } q h.thus,l =[ Q +h +h N >q 0 ]. Note that Q N =sup{q : h + h ˆ FN q }. For a real-valued sequence {τ k } k= where τ k 0, define L k [ Q [ N q 0 + τ k ]= h + h ˆ FN q 0 + τ k ] [ = ˆ F N q 0 + τ k h + h ] [ ] = ˆ FN q 0 + τ k β γ. + h

8 ec8 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem Since ˆ FN is nonincreasing, then it follows that L k L k+.thus,if L is the limiting event of the sequence {L k } k=, thenl k L, implying that PrL k Pr L. Note also that L L, implying that PrL Pr L. Therefore, to prove a ound on PrL, it is sufficient to prove a uniform upper ound on PrL k. Note that for some ϵ k > 0, we have that F q 0 + τ k = β γ ϵ k.thus,l k =[ˆ FN q 0 + τ k F q 0 + τ k γ + ϵ k ]. Finally, from Bernstein s inequality, we have that Nγ + ϵ k PrL k F q 0 + τ k F q 0 + τ k + γ + 3 ϵk Nγ + ϵ k =exp Therefore, we have that for all k, β γ ϵ k β +γ + ϵ k + γ+ϵ k 3 Nγ + ϵ k =exp β γ ϵ k β + γ + β γ ϵ k + γ+ϵ k 3 Nγ + ϵ k β γβ + γ + β γ+ γ+ϵ k 3 Nγ β γβ + γ + β γ+ γ 3 Nγ =exp β β γ + 4 β γ+. γ 3 PrL k 3N 4h +47h 5 6. Since = C q, we have from EC.4 that PrL k 6Nϵh q fq +Oϵ 3/ Uϵ. 4h + Oϵ / EC.6 Summarizing from EC.5 and EC.6, we have that PrB Pr B Uϵ and that PrL Pr L Uϵ. Thus, { Pr Q N <q or C Q } N >C q =PrB L PrB+PrL Uϵ exp 4 Nϵ q fq, as ϵ 0. EC.3. Proof of Lemma Denote y gx or + gx the left-side or right-side derivative of a function g at x. The failure rate and reverse hazard rate is given y rx= fx F x fx and rx=.sincef is a log-concave F x

9 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem ec9 distriution, it has an increasing failure rate. This implies that log rx = log fx log F x is increasing, and log rx 0 for all x. Thus, γ + γ 0 + h h = γ + fq F q log fq + fq F q = log rq 0. EC.7 A log-concave distriution also has a decreasing reversed hazard rate. This implies that log rx = log fx log F x is decreasing and + log rx 0 for all x. Thus, γ γ 0 + h = γ fq F q + log fq fq F q = + log rq 0. EC.8 Comining EC.7 and EC.8, we have that +h h γ γ 0 +h. EC.4. Proof of Lemma Note that since log f is concave, then log fx log γ 0 + γ x t, for all x such that fx > 0. Taking the exponent on oth sides proves our result. EC.5. Proof of Lemma 3 Note that d dx F x d dx F x y our assumption that f x f x. Moreover, since F t=f t, then F x F x for all x t and F x F x for all x t. Note that ED t D >t= PrD 0 >t+ s D >tds, = F F t 0 t + sds, F F t 0 t + sds, = ED t D >t With the same technique, we can also prove that Et D D t Et D D t. Comining these results proves the lemma. EC.6. Proof of Lemma 4 We first introduce the following notation: G β + log + β + β log β min{β, β}, Uβ β β log β, β Lβ β β log β. β We need to prove that each of the three functions are nonnegative.. Let us prove the result for G. First, we prove the result for the case when β. Note that + β G = log β. β

10 ec0 e-companion to Levi, Perakis and Uichanco: Data-driven Newsvendor Prolem The derivative is nonnegative if and only if G + βe β βe β 0. Note that for 0, G = β e β + βe β β βe β, β e β + β βe β, β e β e β 0 Note that G 0 = 0, thus, G 0 for all 0. Now define G + βe β βe β. Note that G G if 0. We have G = β e β e β 0, for 0. Note that G 0 = 0, thus, G G 0 for all 0. Thus, G is nondecreasing in 0, and non-increasing in 0. Since at = 0, this function is zero, then G 0 for all. Now we can also prove the result for β, if we define the function β = β Q.E.D. and G=G.. Let us prove the result for U. The result is true if and only if log β β. Note that log β is a convex function of β, thus the linear approximation at β = i.e., the function β ounds it from elow. Q.E.D. 3. Let us prove the result for L. Defining β = β, note that Lβ=U β 0, which follows from. EC.7. Proof of Theorem 4 Recall that if q Sϵ f [q,, then Cq + ϵcq. Also, Sϵ f [q, can e equivalently expressed as {q : C q C q and q q }. Let Q N e defined in 8, ut with = ϵh min{,h} + +h Oϵ. Since, C q= + hf q F q = + h [ q q fq +Oq q ] = ϵh q fq +Oϵ, then it follows from Proposition that C q when the demand distriution is log-concave. This implies that [ ] [ Q N q C Q ] [ ] [ N Q N q C Q ] N C q. Thus, we only need to derive a lower ound on the proaility of the left-hand side event to prove Theorem 4. Modifying the proof of Theorem 3 y letting = ϵh min{,h} + Oϵ, we can prove +h that Pr Q N <q or C Q N > U ϵ exp h} Nϵmin{,, as ϵ h