This note corrects conditions in Proposition 3.4 and Theorem 5.2(ii) and comments on imprecisions in Propositions 4.2 and 4.4 in Fissler and Ziegel (2016)

Fissler, T

Ziegel, J

English

Spiral - Imperial College Digital Repository

arXiv:1901.08826v1  [math.ST]  25 Jan 2019Erratum: Higher Order Elicitability andOsband’s PrincipleTobias Fissler∗ Johanna F. Ziegel†January 28, 2019Abstract. This note corrects conditions in Proposition 3.4 and Theo-rem 5.2(ii) and comments on imprecisions in Propositions 4.2 and 4.4 inFissler and Ziegel (2016a).Keywords: Consistency; Decision theory; Elicitability; Expected Shortfall; Point fore-casts; Propriety; Scoring functions; Scoring rules; Spectral risk measures; Value at RiskAMS 2010 Subject Classification: 62C99; 91B061 Proposition 3.4As detailed Brehmer (2017) there are two technicalities that need to be resolved inFissler and Ziegel (2016a, Proposition 3.4): Firstly, due to the particular choice of theintegration path in the original version of Fissler and Ziegel (2016a, Proposition 3.4),the image of the integration path is not necessarily contained in int(A). Secondly, oneneeds to assume that the identification function V is locally bounded jointly in thetwo components. Proposition 1 gives a refined version of Fissler and Ziegel (2016a,Proposition 3.4).Proposition 1. Assume that int(A) ⊆ Rk is simply connected and let T : F → A bea surjective, elicitable and identifiable functional with a strict F-identification functionV : A × O → Rk and a strictly F-consistent scoring function S : A × O → R. Supposethat Assumption (V1), (V2), (S1) from Fissler and Ziegel (2016a) are satisfied. Let hbe the matrix-valued function appearing at Fissler and Ziegel (2016a, Equation (3.2)).∗Imperial College London, Department of Mathematics, Section Statistics, 180 Queen’s Gate, LondonSW7 2AZ, United Kingdom, e-mail: t.fissler@imperial.ac.uk†University of Bern, Department of Mathematics and Statistics, Institute of Mathemati-cal Statistics and Actuarial Science, Alpeneggstrasse 22, 3012 Bern, Switzerland, e-mail:johanna.ziegel@stat.unibe.ch1For any F ∈ F and any points x, z ∈ int(A) such that γ : [0, 1] → int(A) is anintegration path with γ(0) = x, γ(1) = z the score difference is necessarily of the formS̄(x, F )− S̄(z, F ) =∫γdS̄(·, F ) =∫ 10h(γ(λ))V̄(γ(λ), F)γ′(λ) dλ . (1)Moreover, if Assumptions (F1) and (VS1) from Fissler and Ziegel (2016a) are satisfiedand V is locally bounded, then there is a Lebesgue null set N ⊆ A × O such that for all(x, y) ∈ N c, (z, y) ∈ N c it necessarily holds thatS(x, y)− S(z, y) =∫γdS(·, y) =∫ 10h(γ(λ))V(γ(λ), y)γ′(λ) dλ , (2)where again γ : [0, 1] → int(A) is an integration path with γ(0) = x, γ(1) = z.Proof. Equation (1) follows from Fissler and Ziegel (2016a, Theorem 3.2) and Königsberger(2004, Satz 2, p. 183). The proof of (2) follows the lines of the original proof inFissler and Ziegel (2016b); cf. Brehmer (2017, Theorem 1.31) for details.2 Theorem 5.2(ii)The complication in the proof of Theorem 5.2(ii) in Fissler and Ziegel (2016a) can befound on p. 1702 directly under the equation defining the term R2. We write “Due tothe assumptions, the term Gr(y)+(pr/qr)Gk(w)y is increasing in y ∈ [tr, xr].” However,our assumptions do not ensure that the interval [tr, xr] is necessarily contained in theset A′r,w = {xr : ∃(z1, . . . , zk) ∈ A, xr = zr, xk = zk}. Hence, the condition at Equation(5.3) cannot readily be applied.In Subsection 2.1 we give a counterexample which demonstrates that the complicationdescribed above can indeed lead to a scoring function satisfying the conditions of The-orem 5.2(ii) in Fissler and Ziegel (2016a) which is not consistent. In Subsection 2.2 weintroduce a condition on the action domain which, in combination with the conditions inTheorem 5.2(ii), ensures (strict) consistency of the scoring function. We end with someremarks as to when this additional condition is satisfied.2.1 CounterexampleUsing the same notation as in Fissler and Ziegel (2016a), we confine ourselves to present-ing a counterexample for k = 2 and α ∈ (0, 1/2) (counterexamples for α ∈ [1/2, 1) canbe constructed in a similar manner). Consider the convex action domain A = {(x1, x2) ∈R2 : x1 ≥ 0, x2 ≤ |x1|}. Let G2 : R → R be a strictly convex twice differentiable functionand define G1 : [0,∞) → R viaG1(s) = G2(−s)/α, s ≥ 0 .2Observe that the condition at (5.3) in Fissler and Ziegel (2016a) holds since for (x1, x2) ∈AG′1(x1) + G′2(x2)/α =(G′2(x2)− G′2(−x1))/α{= 0, if x2 = −x1,> 0, if x2 > −x1,due to the fact that G2 is strictly convex. Consider the specific choice G2 = exp anda ≡ 0. This results in the scoreS(x1, x2, y) = (1{y ≤ x1} − α) exp(−x1)/α − 1{y ≤ x1} exp(−y)/α+ exp(x2)(x2 + (1{y ≤ x1} − α)x1/α − 1{y ≤ x1}y/α)− exp(x2) .To demonstrate that the score fails to be consistent for the pair T = (VaRα,ESα)with α = 0.05 consider the following distributions: First, F a point-distribution in 0.Then T (F ) = (0, 0) and we obtain S̄(T (F ), F ) = S(0, 0, 0) = −2 and, for example,S(2,−1.8, 0) ≈ −11.61. Second, F a normal distribution with mean µ = 0.2 and stan-dard deviation σ = 0.1. Then T (F ) ≈ (0.0355,−0.0063). A numerical integration yieldsthat S̄(T (F ), F ) ≈ −5.36 and, for example, S̄(2,−1.8, F ) ≈ −8.76.2.2 A sufficient condition for Theorem 5.2(ii)Proposition 2. Let the conditions of Theorem 5.2 in Fissler and Ziegel (2016a)(ii)prevail. Moreover, assume that for any x ∈ A and any t ∈ T (F), there exists a finitesequence (z(n))n=0,...,N ⊆ A such that1. z(0) = x;2. tr, z(N)r ∈ A′r,min{tk ,z(N)k}for all r ∈ {1, . . . , k − 1};3. for all n ∈ {1, . . . , N} there is an index r ∈ {1, . . . , k} such thata) tr ≤ z(n)r ≤ z(n−1)r or z(n−1)r ≤ z(n)r ≤ tr; andb) z(n)m = z(n−1)m for all m ∈ {1, . . . , k} \ {r};4. if xk < tk and z(n)k 6= z(n−1)k , then z(n−1)k < z(n)k ≤ −B(z(n), t) = −B(z(n−1), t),whereB(x, t) = −tk +k−1∑m=1pmqm(tm − xm)(qm − 1{tm < xm}). (3)Then the scoring function S defined at (5.2) is F-consistent for T . If additionally, thedistributions in F have unique qm-quantiles, m ∈ {1, . . . , k − 1}, Gk is strictly convexand the functions given at (5.3) are strictly increasing, then S is strictly F-consistentfor T .Proof. Let F ∈ F , t = T (F ) and x ∈ A. Let (z(n))n=0,...,N ⊆ A be a sequence thatsatisfies the above conditions. We will show that for all n ∈ {1, . . . , N}, we haveS̄(z(n−1), F ) − S̄(z(n), F ) ≥ 0 and S̄(z(N), F ) − S̄(t, F ) ≥ 0 with one inequality beingstrict under the conditions for strict F-consistency.3In the decomposition S̄(x, F ) − S̄(t, F ) = R1 + R2 on top of page 1702, we were notcompletely precise if F is not continuous. For a, b ∈ R, we defineI(a, b) ={(a, b], if a ≤ b(b, a], if a ≥ b,Ī(a, b) ={[a, b], if a ≤ b[b, a], if a ≥ b.For any z, z′ ∈ A and any w ∈ A′k it holds that S̄(z′, F )− S̄(z, F ) = R1 +R2 withR1 =k−1∑r=1(F (z′r)− qr)(Gr(z′r) +prqrGk(w)z′r)− (F (zr)− qr)(Gr(zr) +prqrGk(w)zr)− sgn(z′r − zr)∫I(z′r ,zr)(Gr(y) +prqrGk(w)y)dF (y).R2 = −Gk(z′k) + Gk(zk) +Gk(w)(z′k − zk) + (Gk(z′k)−Gk(w))(z′k + C(z′, F ))− (Gk(zk)−Gk(w))(zk + C(z, F ))withC(z, F ) =k−1∑m=1pmqm(∫(−∞,zm](zm − y)dF (y)− qmzm).If tr ≤ zr ≤ z′r, and y 7→ Gr(y)+prqrGk(w)y is increasing on [zr, z′r] then the rth summandof R1 is bounded below by(F (z′r)− qr)(Gr(z′r) +prqrGk(w)z′r)− (F (zr)− qr)(Gr(zr) +prqrGk(w)zr)− (F (z′r)− F (zr))(Gr(z′r) +prqrGk(w)z′r)= (F (zr)− qr)(Gr(z′r) +prqrGk(w)z′r −Gr(zr)−prqrGk(w)zr)≥ 0.If z′r ≤ zr ≤ tr, and y 7→ Gr(y)+prqrGk(w)y is increasing on [z′r, zr] then the rth summandof R1 can be rewritten as(F (z′r)− qr)(Gr(z′r) +prqrGk(w)z′r)− (F (zr)− qr)(Gr(zr) +prqrGk(w)zr)+∫[z′r,zr)(Gr(y) +prqrGk(w)y)dF (y)− (F (z′r)− F (z′r−))(Gr(z′r) +prqrGk(w)z′r)+ (F (zr)− F (zr−))(Gr(zr) +prqrGk(w)zr)≥ (F (z′r−)− qr)(Gr(z′r) +prqrGk(w)z′r)− (F (zr−)− qr)(Gr(zr) +prqrGk(w)zr)+ (F (zr−)− F (z′r−))(Gr(z′r) +prqrGk(w)z′r)= (F (zr−)− qr)(Gr(z′r) +prqrGk(w)z′r −Gr(zr)−prqrGk(w)zr)≥ 0.4In case that z′r < zr = tr or z′r > zr = tr, if y 7→ Gr(y) +prqrGk(w)y is strictly increasingon Ī(z′r, tr), and F has a unique qr-quantile, then even R1 > 0. Indeed, if F does notput any mass on the open interval int(I(z′r, tr)), then the last inequality is strict inboth cases. Otherwise, the first inequality is strict because y 7→ Gr(y) +prqrGk(w)y isstrictly increasing. In summary, setting w = min{z′k, zk}, exploiting condition 2 and theconvexity of A, this implies that both R1 ≥ 0 for z′ = z(n−1), z = z(n), n ∈ {1, . . . , N},and also for z′ = z(N), z = t. For the latter case, we even have R1 > 0 under theconditions for strict F-consistency and if (z(N)m )m=1,...,k−1 6= (tm)m=1,...,k−1.For the discussion of R2, one can use the identity −tk =∑k−1m=1(pm/qm)(∫(−∞,tm](tm−y)dF (y)− qmtm) to check that for any z ∈ A− tk ≤ C(z, F ) ≤ B(z, t), (4)where both inequalities are equalities for z = t. Let z′ = z(N), z = t and again w =min{z′k, zk}. Then due to (4) and the monotonicity of Gk the term R2 is bounded belowby−Gk(z′k) + Gk(tk) +Gk(w)(z′k − tk) + (Gk(z′k)−Gk(w))(z′k − tk) ≥ 0invoking the convexity of Gk. The inequality becomes strict if Gk is strictly convex andz′k 6= tk. Now, let z′ = z(n−1) and z = z(n) for n ∈ {1, . . . , N}. If tk ≤ zk ≤ z′k, then weobtainR2 ≥ −Gk(z′k) + Gk(zk) +Gk(z′k)z′k −Gk(zk)zk + (Gk(z′k)−Gk(zk))(−tk)= −Gk(z′k) + Gk(zk) +Gk(z′k)(z′k − tk)−Gk(zk)(zk − tk)≥ −Gk(z′k) + Gk(zk) +Gk(z′k)(z′k − tk)−Gk(z′k)(zk − tk) ≥ 0,where the penultimate inequality is due to the fact that Gk is increasing and the lastinequality follows due to the convexity of Gk. The last inequality is strict if Gk is strictlyconvex and if zk 6= z′k. If z′k ≤ zk ≤ tk and zk ≤ −B(z, t) = −B(z′, t), then, using (4)R2 = −Gk(z′k) + Gk(zk) +Gk(z′k)z′k −Gk(zk)zk + (Gk(z′k)−Gk(zk))C(z′, F )≥ −Gk(z′k) + Gk(zk) +Gk(z′k)z′k −Gk(zk)zk + (Gk(z′k)−Gk(zk))(−zk) ≥ 0by convexity of Gk. Again, the inequality is strict if Gk is strictly convex and if zk 6= z′k. Insummary, R2 ≥ 0 for z′ = z(n−1), z = z(n), n ∈ {1, . . . , N}, and also for z′ = z(N), z = t.For the latter case, we even have R2 > 0 if Gk is strictly convex and if tk 6= z(N)k .We would like to remark that the additional condition stated in Proposition 2 holds inmost practically relevant cases of action domains A. In fact, in the following situations,one can set N = 0 meaning that the original proof in Fissler and Ziegel (2016a) isapplicable: A = Rk; A = A0, where A0 is the maximal sensible action domain for thefunctional T defined in Theorem 5.2 in Fissler and Ziegel (2016a) for q1 < · · · < qk−1.11With ‘maximal sensible’ we mean that T (F) ⊆ A0 for any class of distributions F .5It is given byA0 :={x ∈ Rk : x1 ≤ · · · ≤ xk−1, xk ≤k−1∑m=1pmxm}.In particular, this construction retrieves the result of Corollary 5.5 in Fissler and Ziegel(2016a), considering the maximal action domain A0 = {x ∈ R2 : x1 ≥ x2} for thefunctional T = (VaRα,ESα). Also, A = A+0 := A0 ∩ Rk+ and A = A−0 := A0 ∩ Rk− satisfythe condition. The action domain considered in Theorem C.3 in Nolde and Ziegel (2017),corresponding to A = R × (−∞, 0) in our sign convention, also satisfies the conditiongiven in Proposition 2.For the action domain A = {x ∈ R2 : x2 ≤ x1 ≤ x2 + c} of Proposition 4.10 inFissler and Ziegel (2017) one generally needs N > 0. However, the existence and con-struction of the sequence (z(n))n=0,...,N is obvious.Acerbi and Szekely (2014) introduced a family of scoring functions SW , W ∈ R,with corresponding action domains AW = {x ∈ R2 : x2 > Wx1}; see page 1697 inFissler and Ziegel (2016a) for a discussion. The results are as follows: For W > 1the additional condition of Proposition 2 is satisfied (note that the potentially prob-lematic point (0, 0) is not in AW then.) For W = 1, the condition is empty sinceA1 ∩ T (F) ⊆ A1 ∩ A0 = ∅. For W ∈ (0, 1) the condition fails to be satisfied. For W = 0the condition holds. For W ∈ [(α − 1)/α, 0) the condition fails to be satisfied. ForW < (α− 1)/α the condition holds.3 Remarks on Propositions 4.2 and 4.4In Fissler and Ziegel (2016a) Proposition 4.2(i) we claimed that gm > 0 and in Proposi-tion 4.4(i) we claimed that the function h(x)r,l=1,...,k is positive definite for all x ∈ int(A).In the proof of Theorem 5.2(iii), we made similar claims about the function gk and thederivatives of the functions appearing at (5.3). It turns out that we were slightly im-precise with these claims. Indeed, these functions are all non-negative (positive semi-definite). Moreover, we can show that the functions gm in Proposition 4.2(i) (as wellas the functions appearing in the proof of Theorem 5.2(iii)) are strictly positive almosteverywhere.We will first demonstrate that the function h appearing in Proposition 4.4(i) is positivesemi-definite. Assume there is some t ∈ int(A) and some v ∈ Sk−1 such that v⊤h(t)v < 0.Recall that due to the assumptions and Theorem 3.2, the function h is continuous. Hence,there is an open neighbourhood t ∈ U ⊆ int(A) such that v⊤h(x)v < 0 for all x ∈ U .Invoking the surjectivity of T , let F ∈ F such that T (F ) = t. Then there is an ε > 0such that – using the notation on top of page 1701 in Fissler and Ziegel (2016a) –ddsS̄(t+ sv, F ) = q̄(F )sv⊤h(t+ sv)v{> 0, for s ∈ (−ε, 0),< 0, for s ∈ (0, ε).This is a contradiction to the F-consistency of S.6Now, we will show that the functions gm appearing in Proposition 4.2(i) are positivealmost everywhere, the arguments for the remaining functions being similar. Let m ∈{1, . . . , k}. The argument at the top of page 1701 in Fissler and Ziegel (2016a) showthat for all x ∈ int(A′m) there is an ε > 0 such that gm > 0 on (x−ε, x+ε)\{x}. Due toa compactness argument, gm is positive on any compact set almost everywhere. Sinceint(A′m) is σ-compact, gm > 0 almost everywhere. The continuity of gm also implies thatgm ≥ 0 everywhere.ReferencesC. Acerbi and B. Szekely. Backtesting expected shortfall. Risk Magazine, 27:76–81, 2014.J. R. Brehmer. Elicitability and its application in risk management. Master’s thesis, Universityof Mannheim, 2017. URL http://arxiv.org/abs/1707.09604.T. Fissler and J. F. Ziegel. Higher order elicitability and Osband’s principle. Annals of Statistics,44:1680–1707, 2016a.T. Fissler and J. F. Ziegel. Supplement to “Higher order elicitability and Osband’s principle”.Annals of Statistics, 44:1680–1707, 2016b.T. Fissler and J. F. Ziegel. Order-sensitivity and equivariance of scoring functions. Prepint, 2017.URL https://arxiv.org/abs/1711.09628v1.K. Königsberger. Analysis 2. Springer-Verlag, Berlin Heidelberg New York, 5th edition, 2004.N. Nolde and J. F. Ziegel. Elicitability and backtesting: Perspectives for banking regulation.Annals of Applied Statistics, 11:1833–1874, 2017. Paper with discussion and rejoinder.7

Erratum: Higher order elicitability and Osband's principle.

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Erratum: Higher order elicitability and Osband's principle.

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