robust risk aggregation and merging p-values · background useful results p-values merging via...

Background Useful results P-values Merging via averaging Simulation Conclusion

Robust Risk Aggregation and Merging P-values

Ruodu Wang

http://sas.uwaterloo.ca/~wang

Department of Statistics and Actuarial Science

University of Waterloo

4th European Actuarial Journal Conference

KU Leuven, Belgium September 11, 2018

Ruodu Wang ([email protected]) Risk aggregation and merging p-values 1/59

http://sas.uwaterloo.ca/~wang

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Agenda

1 Robust risk aggregation problems

2 Some results in robust VaR aggregation

3 P-values and hypothesis testing

4 Merging p-values via averaging

5 Simulation study

6 Concluding remarks and open questions


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Fundamental problem in Finance/Insurance

Basic setup.

I A vector of risk factors: X = (X1, . . . ,Xd)

I A financial position Ψ(X)

I A risk measure ρ

Calculate ρ(Ψ(X))

Most relevant choices:

I ρ = VaRp or ρ = ESp (TVaRp)

I Ψ(X) =∑d

i=1 Xi

Challenge: We need a joint model for the random vector X


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Frechet problem

Model assumption: Xi ∼ Fi , Fi known, i = 1, . . . , d .

Let Sd = Sd(F1, . . . ,Fd) =

d∑

i=1

Xi : Xi ∼ Fi , i = 1, . . . , d

.

I Every element in Sd is a possible risk position

I Determination (distributions-wise) of Sd : very challenging.

I Example:

• F = U[0, 1]

• Characterization of S2(F ,F ) is an open question

• S3(F ,F ,F ) characterized analytically recently

S ∈ S3(F ,F ,F ) ⇔ S ≤cx U[0, 3]


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VaR and ES

Two regulatory risk measures in Basel III & IV and Solvency II

Value-at Risk (VaR)

For p ∈ (0, 1], X ∈ L0,

VaRp(X ) = F−1X (p) = infx ∈ R : FX (x) ≥ p.

Expected Shortfall (ES, or TVaR, CVaR, CTE, AVaR, ...)

For p ∈ (0, 1), X ∈ L1,

ESp(X ) =1

1− p

∫ 1

pVaRq(X )dq =

(F cont.)E [X |X > VaRp(X )] .


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Worst- and best-values of VaR and ES

The Frechet problems

I For p ∈ (0, 1),

VaRp(Sd) = supVaRp(S) : S ∈ Sd(F1, . . . ,Fd),

VaRp(Sd) = infVaRp(S) : S ∈ Sd(F1, . . . ,Fd).

I Same notation for ESp

I ES is subadditive: ESp(Sd) =∑d

i=1 ESp(Xi ).

I VaRp(Sd), VaRp(Sd), and ESp(Sd): generally open questions


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Basel III & IV ES calculation

In the Basel FRTB internal model approach, for market risk:

Capital Charge = λESp

(d∑

i=1

Xi

)︸︷︷︸

internal model

+(1− λ)d∑

i=1

ESp(Xi )︸︷︷︸ESp(Sd )

,

where

I Xi is the total random loss from a risk class, i = 1, . . . , d

• commodity, equity, credit spread, interest rate, exchange

I T = 10-day, p = 0.975, λ = 0.5

I ESp is calculated under a stressed scenario

Dependence uncertainty!


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Solvency II SCR calculation

Copied from Solvency II, 2009


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Robust risk aggregation

Calculation of VaRp(Sd), VaRp(Sd), and ESp(Sd)

I Dependence modeling, (co/counter)-monotonicity, ...

• Dhaene, Denuit, Goovaert, Vanduffel, ...

I Risk measures

• Artzner-Delbaen-Eber-Heath, Wang, ...

I Mass transportation

• Ruschendorf, Embrechts, Puccetti, ...

I Joint mixability

I Computation, neural network methods, ...

I Uncertainty, ambiguity, decision making, ...


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5 Simulation study



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Some results in robust VaR aggregation

Some results

I Embrechts-Puccetti’06, F&S

I W.-Peng-Yang’13, F&S

I Bernard-Jiang-W.’14, IME

I Wang-W.’15, JMVA

I Embrechts-Wang-W.’15 F&S

I Jakobsons-Han-W.’16, SAJ

I Bignozzi-Mao-Wang-W.’16, Extremes

I Wang-W.’16, MOR

This talk - work with Vladimir Vovk (Royal Holloway, London)

I Vovk-W.’18


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Simple VaR-ES relation

I For p ∈ (0, 1), define

ES←p (·) =1

p

∫ p

0VaRq(·)dq.

I By definition

ES←p ≤ VaRp ≤ ESp.

I By subaddivity of ESp and superadditivity of ES←p ,

d∑i=1

ES←p (Xi ) ≤ VaRp

(d∑

i=1

Xi

)≤

d∑i=1

ESp(Xi ).

I Asymptotically sharp as d →∞ (Embrechts-Wang-W’15).


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Dual bound of Embrechts-Puccetti’06

Assume a homogeneous model F1 = · · · = Fd = F .

Derived from Theorem 4.2 of Embrechts-Puccetti’06

For a continuous cdf F and X1, . . . ,Xd ∼ F ,

P(X1 + · · ·+ Xd ≥ ds) ≤ infr∈[0,s)

∫ ds−(d−1)rr (1− F (x))dx

s − r.

I Based on Ruschendorf’82

I probability bounds ⇔ VaR bounds


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VaRp formula of W.-Peng-Yang’13

Derived from Theorem 3.4 of W.-Peng-Yang’13

Suppose that the density of F is decreasing on its support. For

p ∈ (0, 1) and X ∼ F ,

VaRp(Sd) = dE[X |F−1(p + (d − 1)c) ≤ X ≤ F−1(1− c)],

where c is the smallest number in [0, 1d (1− p)] such that

∫ 1−c

p+(d−1)cF−1(t)dt ≥ 1− p − dc

d((d − 1)F−1(p + (d − 1)c) + F−1(1− c)).

I only decreasing density in the tail part is required

I dual form in Puccetti-Ruschendorf’13


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VaRp formula of W.-Peng-Yang’13

Derived from Theorem 3.4 of W.-Peng-Yang’13

Suppose that the density of F is decreasing on its support. For

p ∈ (0, 1) and X ∼ F ,

VaRp(Sd) = max(d − 1)F−1(0) + F−1(p), dES←p (X ).

I An asymmetry between upper and lower bounds ...


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Asymptotic equivalence of Wang-W.’15

Corollary 3.7 of Wang-W.’15

For p ∈ (0, 1) and X ∼ F ,

limd→∞

1

dVaRp(Sd) = ESp(X ).

I VaRp(Sd)/ESp(Sd)→ 1.

I First results obtained by Puccetti-Ruschendorf’14

I Inhomogeneous portfolios: Embrechts-Wang-W.’15

I Other risk measures, rank dependent utilities:

W.-Bignozzi-Tsanakas’15, Cai-Liu-W.’18, W.-Xu-Zhou’18


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Aggregation ratio of Bignozzi-Mao-Wang-W.’16

Derived from Proposition 3.5 of Bignozzi-Mao-Wang-W.’16

For a Pareto distribution F with index β ∈ (0, 1),

limd→∞

VaRp(Sd)

d1/βF−1(p)=

(1

1− β

) 1β

.

I If β ≥ 1, asymptotic equivalence implies

limd→∞

VaRp(Sd)

dF−1(p)=

ESp(X )

VaRp(X )≈ β

β − 1.

I For MRV models (e.g. Embrechts-Lambrigger-Wuthrich’09),

limp↑1

VaRp(X1 + · · ·+ Xd)

d1/βF−1(p)≤ 1.


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VaRp formula of Jakobsons-Han-W.’16

Assume inhomogeneous models

Derived from Corollary 4.7 of Jakobsons-Han-W.’16

Suppose that the density of each F1, . . . ,Fd is decreasing on its

support. For p ∈ (0, 1),

VaRp(Sd) = max

maxi=1,...,d

F−1i (p) +

∑j 6=i

F−1j (0)

, µp

,

where µp =∑d

i=1 ES←p (Xi ), Xi ∼ Fi , i = 1, . . . , d .

I Formula for VaRp(Sd) involves solving an implicit ODE


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Summary on VaR aggregation

d = 2

I solved analytically (Makarov’81, Ruschendorf’82)

I based on counter-monotonicity

d ≥ 3

I solved analytically for decreasing densities

• also solved for increasing densities by putting a negative sign

• relies on joint mixability for monotone densities (Wang-W’16)

I generalization to other distributions is limited

I homogeneous relaxation in Bernard-Jiang-W.’14


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Summary on VaR aggregation

Remarks.

I Efficient numerical algorithm: the Rearrangement Algorithm• Puccetti-Ruschendorf’12, Embrechts-Puccetti-Ruschendorf’13,

Hofert-Memartoluie-Saunders-Wirjanto’17,

Bernard-Bondarenko-Vanduffel’18, ...

I Risk aggregation with partial dependence information• Bernard-Vanduffel’15, Puccetti-Ruschendorf-Manko’16,

Lux-Papapantoleon’17, Bernard-Ruschendorf-Vanduffel’17,

Bernard-Ruschendorf-Vanduffel-W.’17, ...

I Risk aggregation with marginal and dependence uncertainty• Li-Shao-W.-Yang’18, Shao’18, ...

I Connection to distributionally robust optimization


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Robust VaR aggregation

Some disadvantages:

I often too high risk values for practical use

I analytical results require monotone density assumptions

I assumed no dependence information

I assumed full marginal information

Are there real-world problems with these features?


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5 Simulation study



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P-values

STAT 101

A (true) p-value p for testing a hypothesis H0:

I p is uniform on [0, 1] under H0 ⇔ PH0(p ≤ ε) = ε for ε ∈ [0, 1]

I Specify a significant level α, typically α = 0.05, 0.01, 0.005 ...

I Rejects H0 if (realized) p ≤ α• cannot reject H0 if p > α

I Probability of type I error = PH0(reject H0) = α


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Merging p-values

Suppose we are testing the same hypothesis using K ≥ 2 different

statistical tests and obtain p-values p1, . . . , pK . How can we

combine them into a single p-value?

Examples.

I backtesting credit risk ratings: typically 17 binomial tests

I backtesting market risk models: several quantile level tests

I meta-analysis


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Meta-analysis

A typical example from meta-analysis

The sex differences dataset, from p.35 of Hedges-Olkin’85


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The Bonferroni method

A question of a long history

I Fisher’48: assumes p1, . . . , pK are independent

Without any assumptions on the p-values p1, . . . , pK ...

I The Bonferroni method (Dunn’58):

F (p1, . . . , pK ) = K min(p1, . . . , pK ).

I Ruger’78:

F (p1, . . . , pK ) =K

kp(k).

In particular, 2 times the median or the maximum.

I Hommel’83:

F (p1, . . . , pK ) =

(1 +

1

2+ · · ·+ 1

K

)min

k=1,...,K

K

kp(k).


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I overly conservative ... if tests are similar?

I dictated by a single experiment?

I what if some p-values are more important (e.g. bigger

experiments)?

Particular interest: the case of heavily dependent tests.

I is it more natural to look at some average of p-values?


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Merging functions

In any atomless probability space (Ω,A,P) ...

Definition 1

(i) A (conservative) p-value is a random variable P that satisfies

P(P ≤ ε) ≤ ε, ε ∈ (0, 1).

(ii) A merging function is an increasing Borel function

F : [0, 1]K → [0,∞) such that F (P1, . . . ,PK ) is a p-value for

all p-values P1, . . . ,PK .

I Controlled type I error

I Merging functions may be applied iteratively in multiple layers


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Merging functions

For an increasing Borel function F : [0, 1]K → [0,∞), equivalent

are:

I F is a merging function;

I F (U1, . . . ,UK ) is a p-value for all U1, . . . ,UK ∈ U ;

I for all ε ∈ (0, 1), P(F ≤ ε) ≤ ε, where

P(F ≤ ε) = sup P(F (U1, . . . ,UK ) ≤ ε) | U1, . . . ,UK ∈ U .

U : the set of all uniform [0, 1] random variables


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Precise merging functions

Definition 2

A merging function F is precise if, for all ε ∈ (0, 1), P(F ≤ ε) = ε.

I A precise merging function cannot be improved

Examples.

I The Bonferroni method F (p1, . . . , pK ) = K min(p1, . . . , pK )

I F (p1, . . . , pK ) = max(p1, . . . , pK )

I F (p1, . . . , pK ) = p1 (trivial)


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Precise merging functions

The Bonferroni method F (p1, . . . , pK ) = K min(p1, . . . , pK )

P(K min(p1, . . . , pK ) ≤ ε) = P

(K⋃i=1

Kpi ≤ ε

)

≤K∑i=1

P (Kpi ≤ ε)

=K∑i=1

ε

K= ε.

The inequality is an equality if Kpi ≤ ε, i = 1, . . . ,K are

mutually exclusive.


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Asymptotically precise merging functions

Definition 3

A family of merging functions FK on [0, 1]K , K = 2, 3, . . ., is called

asymptotically precise if, for any a ∈ (0, 1), aFK is not a merging

function for a large enough K .

I This family of merging functions cannot be improved by a

constant multiplier


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5 Simulation study



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Merging p-values via averaging

A general notion of averaging

I Axiomatized by Kolmogorov’30,

Mφ,K (p1, . . . , pK ) = φ−1

(φ(p1) + · · ·+ φ(pK )

K

),

where φ : [0, 1]→ [−∞,∞] is continuous and strictly

monotonic.

I Most common forms, for r ∈ R \ 0,

Mr ,K (p1, . . . , pK ) =

(pr1 + · · ·+ prK

K

)1/r

.


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Special cases:

I Arithmetic: M1,K (p1, . . . , pK ) = 1K

∑Kk=1 pk

I Harmonic: M−1,K (p1, . . . , pK ) =(

1K

∑Kk=1

1pk

)−1

I Quadratic: M2,K (p1, . . . , pK ) =√

1K

∑Kk=1 p

2k

Limiting cases:

I Geometric: M0,K (p1, . . . , pK ) =(∏K

k=1 pk

)1/K

I Maximum: M∞,K (p1, . . . , pK ) = max(p1, . . . , pK )

I Minimum: M−∞,K (p1, . . . , pK ) = min(p1, . . . , pK )

The cases r ∈ −1, 0, 1 are known as Platonic means.


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The arithmetic average M1,K (p1, . . . , pK ) = 1K

∑Kk=1 pk is not a

merging function (Ruschendorf’82, Meng’93):

P(M1,K ≤ ε) = min(2ε, 1).

I 2M1,K is a precise merging function

Task. Find ar ,K > 0 such that ar ,KMr ,K is a merging function

I Mr ,K increases in r

• The constants ar ,K should decrease in r .


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Translation to a risk aggregation problem

For α ∈ (0, 1] and a random variable X , define

qα(X ) = infx ∈ R : P(X ≤ x) ≥ α = VaRα(X ).

and for a function F : [0, 1]K → [0,∞), define

qα(F ) = inf qα(F (U1, . . . ,UK )) | U1, . . . ,UK ∈ U .


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Lemma 1

For a > 0, r ∈ [−∞,∞], and F = aMr ,K , equivalent are:

(i) F is a merging function, i.e. P(F ≤ ε) ≤ ε for all ε ∈ (0, 1);

(ii) qε(F ) ≥ ε for all ε ∈ (0, 1);

(iii) P(F ≤ ε) ≤ ε for some ε ∈ (0, 1);

(iv) qε(F ) ≥ ε for some ε ∈ (0, 1).

The same conclusion holds if all ≤ and ≥ are replaced by =.

I In statistical practice one only needs to have P(F ≤ ε) ≤ ε for

a specific ε, e.g. 0.05, 0.01, ...


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It boils down to calculate qε(Mr ,K ), or equivalently:

(i) for r > 0, aggregation of Beta risks

(qε(Mr ,K ))r = infU1,...,UK∈U

qε

(1

K(U r

1 + · · ·+ U rK )

)(ii) for r = 0, aggregation of exponential risks

ln(qε(Mr ,K )) = infU1,...,UK∈U

qε

(1

K(lnU1 + · · ·+ lnUK )

)(iii) for r < 0, aggregation of Pareto risks

(qε(Mr ,K ))r = supU1,...,UK∈U

q1−ε

(1

K(U r

1 + · · ·+ U rK )

)Ruodu Wang ([email protected]) Risk aggregation and merging p-values 39/59

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Breakdown of U r (or lnU) for r ∈ R

super-heavy

Pareto

integrable

Pareto

negative

exponential

increasing

Beta density

decreasing

Beta density

Pareto(1) uniform

r = −1

harmonic

r = 0

geometric

r = 1

arithmetic

r


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The integrable case: r > −1

Proposition 1

For r ∈ (−1,∞], (r + 1)1/rMr ,K , K ∈ 2, 3, . . ., is a family of

merging functions and it is asymptotically precise.

Proof.

I r > 0, qε(∑K

k=1 Urk) ≥

∑Kk=1 ES

←ε (U r

k) = K 1r+1ε

r

I r = 0, qε(∑K

k=1 lnUk) ≥∑K

k=1 ES←ε (lnUk) = K (ln ε+ 1)

I r < 0, q1−ε(∑K

k=1 Urk) ≤

∑Kk=1 ES1−ε(U

rk) = K 1

r+1εr

I In all cases, qε((r + 1)1/rMr ,K ) ≥ ε

I Use the VaR/ES asymptotic equivalence of Wang-W.’15.

(1 + r)1/r |r=0 = e.


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Main results summary

Constant multiplier ar ,K in front of Mr ,K

rr+1

K1+1/r (1 + r)1/r

e

(1 + r)1/r K1/r

K 1e lnK 2

r = −1

harmonic

r = 0

geometric

r = 1

arithmetic

r = −∞minimum

r =∞maximum

r = 1K−1 r = K − 1

blue: precise; green: asymptotically precise; red: rough


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Main results summary

Methodology breakdown details

super-heavyaggregation

asymptoticequivalence

VaR formula

joint mixability VaR formula

trivial trivialVaR formula uniform

r = −1

harmonic

r = 0

geometric

r = 1

arithmetic

r = −∞minimum

r =∞maximum

r = 1K−1 r = K − 1

purple: Rushcendorf’82; blue: W.-Peng-Yang’13; brown: Wang-W’11

green: Wang-W.’15; red: Bignozzi-Mao-Wang-W.’16


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Weighted averaging

Consider weighted averaging functions

Mφ,w(p1, . . . , pK ) = φ−1 (w1φ(p1) + · · ·+ wKφ(pK )) ,

and in particular,

Mr ,w(p1, . . . , pK ) = (w1pr1 + · · ·+ wKp

rK )1/r ,

where w = (w1, . . . ,wK ) ∈ ∆K .

I Intuitively, the weights reflect the prior importance of the

p-values.

∆K =

(w1, . . . ,wK ) ∈ [0, 1]K∣∣w1 + · · ·+ wK = 1

is the standard K -simplex.


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Weighted averaging

Proposition 2

For w = (w1, . . . ,wK ) ∈ ∆K , w = max(w) and r ∈ (−1,∞),

(i) (r + 1)1/rMr ,w is a merging function;

(ii) (r + 1)1/rMr ,w is precise ⇔ w ≤ 1/2 and r ∈ [ w1−w ,

1−ww ];

(iii) if r ∈ [1,∞), min(r + 1, 1w )1/rMr ,w is a precise merging

function.

Proof.

I (r + 1)1/rMr ,w is precise ⇔ VaR = ES ⇔ joint mixability

I For r ∈ (0, 1), use the characterization of joint mixability for

monotone densities of Wang-W.’16.

I For r ≥ 1, use the VaRp formula of Jakobsons-Han-W.’16.


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Weighted averaging

The special case r = 1 (weighted arithmetic mean):

I w ≤ 1/2

• no single experiment outweighs the total of all the others

• the optimal multiplier is 2 (same as uniform weights)

I w > 1/2

• a single experiment outweighs the total of all the others

• assuming w1 = max(w),

1

wM1,w(p1, . . . , pK ) = p1 +

K∑k=2

wk

wpk

• weighted adjustments are added to the p-value obtained from

the most important experiment


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Weighted averaging

Conjecture

For a > 0 and any r and K , if aMr ,K is a merging function, then

aMr ,w is also a merging function for all w ∈ ∆K .

I A deeper conjecture: Sd(F1, . . . ,Fd) ⊂ Sd(F , . . . , F ) where

F−1 = 1d

∑di=1 F

−1i .

I Bernard-Jiang-W.’14: Sd(F1, . . . ,Fd) ⊂ Sd(F , . . . , F ) where

F = 1d

∑di=1 Fi .


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5 Simulation study



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Simulation

A simulation study for the performance of ar ,KMr ,K

I 6 merging functions: r ∈ −∞,−1, 0, 1, 2,∞• Bonferroni (minimum), harmonic, geometric, arithmetic,

quadratic, maximum

I a−1,K = 1.83 lnK for K = 20 and a−1,K = 1.7 lnK for K ≥ 50

I a0,K = e

I others are precise


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Simulation

Simulation setup

I K tests for whether a given set of data has zero mean

• n iid normal observations with mean d/√n and known

variance 1

• two-sided z-tests

I o = %-overlap of observations between any two tests

I c = correlation of non-overlapping observations between tests

I α = 0.05, significance level

I the procedure is repeated N times


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Simulation

Power, K=50, n=50, o=0, c=0, d=2, N=1000

0.0

0.2

0.4

0.6

0.8

1.0

Bonferroni harmonic geometric arithmetic quadratic maximum

0.0045 0.0102 0.0842 0.2917 0.4469 0.8699(0.0086) (0.0136) (0.0294) (0.0608) (0.0797) (0.1166)

Power, K=50, n=50, o=0.7, c=0.7, d=2, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


1.3213 0.6079 0.3049 0.2676 0.2687 0.3516(2.9538) (1.0054) (0.4524) (0.3594) (0.3311) (0.3286)

Power, K=50, n=50, o=0, c=0, d=3, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0 1e−04 0.0052 0.0666 0.1553 0.4622(0.0001) (0.0003) (0.0026) (0.0266) (0.0681) (0.2280)

Power, K=50, n=50, o=0.7, c=0.7, d=3, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0.2748 0.1498 0.0818 0.0777 0.0834 0.1316(1.2512) (0.4745) (0.2215) (0.1826) (0.1744) (0.2087)


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Simulation

Power, K=50, n=50, o=0, c=0, d=4, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0 0 1e−04 0.0097 0.0331 0.1175(0.0000) (0.0000) (0.0001) (0.0074) (0.0325) (0.1293)

Power, K=50, n=50, o=0.7, c=0.7, d=4, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0.0181 0.0117 0.0077 0.0088 0.0109 0.0221(0.1596) (0.0690) (0.0368) (0.0352) (0.0381) (0.0647)

Power, K=50, n=50, o=0.5, c=0.5, d=3, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0.0315 0.0379 0.0402 0.0632 0.0905 0.2017(0.1304) (0.1343) (0.1010) (0.1156) (0.1359) (0.2464)

Power, K=50, n=50, o=0.9, c=0.9, d=3, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0.9101 0.2093 0.0885 0.0673 0.0602 0.056(2.8428) (0.5801) (0.2412) (0.1805) (0.1589) (0.1302)


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Simulation

Power, K=50, n=50, o=0.95, c=0, d=3, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0.6755 0.1921 0.0845 0.0668 0.0619 0.0693(2.4028) (0.5412) (0.2284) (0.1735) (0.1549) (0.1423)

Power, K=50, n=50, o=0, c=0.95, d=3, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0.309 0.1431 0.0712 0.0627 0.0638 0.0929(1.1628) (0.4210) (0.1906) (0.1537) (0.1446) (0.1709)

Power, K=20, n=50, o=0.7, c=0.7, d=3, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0.1404 0.1091 0.0704 0.0669 0.0718 0.097(0.5987) (0.3567) (0.2007) (0.1657) (0.1590) (0.1760)

Power, K=500, n=50, o=0.7, c=0.7, d=3, N=1000

0.0

0.2

0.4

0.6

0.8

1.0


0.8772 0.1713 0.0622 0.0613 0.0682 0.17(4.2020) (0.5478) (0.1688) (0.1444) (0.1420) (0.2347)


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Simulation

Observations.

I The Bonferroni method

• performs very well in cases of no or moderate correlation

• often has a large standard deviation

• is able to utilize information from independent tests

I The arithmetic method

• performs very well in cases of heavy overlap or correlation

• has a relatively stable average p-value

I The geometric method performs quite nicely in most cases

I The harmonic method is between Bonferroni and geometric

I The maximum and quadratic methods are not remarkable


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5 Simulation study



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Concluding remarks

Further directions

I Power analysis for classic statistical models

I Adaptive selection of the merging function

I Relation between dependence and the choice of merging

functions


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Open questions on risk aggregation

Mathematical questions on robust risk aggregation:

I Characterization of Sd and joint mixability

I Analytical formulas for VaRp, VaRp and ESp

I Aggregation of random vectors

I Partial information on dependence

I RDU and CPT risk aggregation

I Other aggregation functionals


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Open questions on risk aggregation

A few concrete questions:

I For a given F , determine whether F ∈ S(U[0, 1],U[0, 1])?

I For given correlation matrix Σ and F1, . . . ,Fd , is the set

VΣ = X : Corr(X) = Σ, Xi ∼ Fi , i = 1, . . . , d

empty?

I If VΣ 6= ∅, what are the values of

supVaRp(S) : X ∈ VΣ and supESp(S) : X ∈ VΣ?

Here S = X1 + · · ·+ Xd .


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Thank you

Thank you for your kind attention

The manuscript is available at

I SSRN: https://ssrn.com/abstract=3166304

I arXiv: https://arxiv.org/abs/1212.4966


https://ssrn.com/abstract=3166304

https://arxiv.org/abs/1212.4966

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Appendix

References I

Bernard, C., Jiang, X. and Wang, R. (2014). Risk aggregation with dependence

uncertainty. Insurance: Mathematics and Economics, 54, 93–108.

Bignozzi, V., Mao, T., Wang, B. and Wang, R. (2016). Diversification limit of

quantiles under dependence uncertainty. Extremes, 19(2), 143–170.

Embrechts, P. and Puccetti, G. (2006). Bounds for functions of dependent risks.

Finance and Stochastics, 10, 341–352.

Jakobsons, E., Han, X. and Wang, R. (2016). General convex order on risk

aggregation. Scandinavian Actuarial Journal, 2016(8), 713–740.

Wang, B. and Wang, R. (2015). Extreme negative dependence and risk

aggregation. Journal of Multivariate Analysis, 136, 12–25.

Wang, B. and Wang, R. (2016). Joint mixability. Mathematics of Operations

Research, 41(3), 808–826.

Wang, R., Peng, L. and Yang, J. (2013). Bounds for the sum of dependent risks

and worst Value-at-Risk with monotone marginal densities. Finance and

Stochastics, 17(2), 395–417.


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Appendix

References II

Dunn, O. J. (1958). Estimation of the means for dependent variables. Annals of

Mathematical Statistics. 29(4), 1095–1111.

Fisher, R. A. (1948). Combining independent tests of significance. American

Statistician, 2, 30.

Hedges, L. V. and Olkin, I. (1985). Statistical Methods for Meta-Analysis.

Orlando, FL: Academic Press.

Holm, S. (1979). A simple sequentially rejective multiple test procedure.

Scandinavian Journal of Statistics, 6, 65–70.

Mattner, L. (2012). Combining individually valid and arbitrarily dependent

P-variables. In Abstract Book of the Tenth German Probability and Statistics

Days, p. 104. Institut fur Mathematik, Johannes Gutenberg-Universitat Mainz.

Meng, X.-L. (1993). Posterior predictive p-values. Annals of Statistics, 22,

1142–1160.

Ruschendorf, L. (1982). Random variables with maximum sums. Advances in

Applied Probability, 14(3), 623–632.


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Appendix

Analysis for the sex differences data

For the sex differences dataset, the combined p-values are

(compared with 0.05 significance level; weighted by sample size)

I Bonferroni: 0.029 (significant)

I harmonic: 0.045 (weighted 0.041) (significant)

I geometric: 0.157 (weighted 0.198) (not significant)

I arithmetic: 0.613 (weighted 0.793) (not significant)


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Appendix

Analysis for the passive smoking data

For the passive smoking dataset (Hartung-Knapp-Sinha’08,

Table 3.1, p.31, K = 19), the combined p-values are (compared

with 0.05 significance level)

I Bonferroni: 0.051 (not significant)

I harmonic: 0.126 (not significant)

I geometric: 0.254 (not significant)

I arithmetic: 0.449 (not significant)


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Appendix

The non-integrable Pareto case: r < −1

No VaR/ES asymptotic equivalence for r ≤ −1.

Proposition 3

For r ∈ (−∞,−1), rr+1K

1+1/rMr ,K , K ∈ 2, 3, . . ., is a family of

merging functions and it is asymptotically precise.

Proof.

I To show rr+1K

1+1/rMr ,K is a merging function, directly apply

the dual bound of Embrechts-Puccetti’06.

I To show the asymptotic precision, use the aggregation ratio of

Bignozzi-Mao-Wang-W.’16 for super-heavy Pareto risks.

Letting r → −∞ one recovers the Bonferroni method: KM−∞,K .

back


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Appendix

Precise results for the Beta case: r ≥ 1/(K − 1)

Proposition 4

For K ∈ 2, 3, . . . and r ∈ (−1,∞),

(i) (r + 1)1/rMr ,K is a precise merging function ⇔r ∈ [ 1

K−1 ,K − 1].

(ii) If r ≥ K − 1, K 1/rMr ,K is a precise merging function.

Proof.

I r ≥ 1, U r has a decreasing density

I r ∈ [ 1K−1 , 1], U r has an increasing density

I The VaRp and VaRp formulas of W.-Peng-Yang’13 give the

precise value of qε(Mr ,K )


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Appendix

Precise results for the Beta case: r ≥ 1/(K − 1)

Examples.

I min(r + 1,K )1/rMr ,K is precise for r ≥ 1/(K − 1).

I The arithmetic average times 2 is precise for K ≥ 2

I The quadratic average times√

3 is precise for K ≥ 3

I Letting r →∞, the maximum M∞,K is precise


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Appendix

Geometric averaging

Proposition 5

For each K ∈ 2, 3, . . ., aKM0,K is a precise merging function,

where

aK =1

cKexp (−(K − 1)(1− KcK ))

and cK is the unique solution to the equation

ln(1/c − (K − 1)) = K − K 2c

over c ∈ (0, 1/K ). Moreover, aK ≤ e and aK → e as K →∞.

Proof.

I Obtained from the VaRp formula of W.-Peng-Yang’13.


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Appendix

Geometric averaging

Table: Numeric values of aK/e for the geometric mean

K aK/e K aK/e K aK/e

2 0.7357589 5 0.9925858 10 0.9999545

3 0.9286392 6 0.9974005 15 0.9999997

4 0.9779033 7 0.9990669 20 1.0000000

I In practice, use aK ≈ e for K ≥ 5

I eM0,K is always a merging function (noted by Mattner’12)


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Appendix

Harmonic averaging

Proposition 6

For K > 2, (e lnK )M−1,K is a merging function.

Proof.

I For a given K > 2, e lnK = minr<−1r

r+1K1+1/r

I (e lnK )Mr ,K is a merging function for some r < −1

I M−1,K ≥ Mr ,K for r < −1


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Appendix

Harmonic averaging

Proposition 7

Set aK = (yK+K)2

(yK+1)K , K > 2, where yK is the unique solution to the

equation

y2 = K ((y + 1) ln(y + 1)− y), y ∈ (0,∞).

Then aKM−1,K is a precise merging function. Moreover,

aK/ lnK → 1 as K →∞.

Proof.

I Again obtained from the VaRp formula of W.-Peng-Yang’13.


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Appendix

Harmonic averaging

Table: Numeric values of aK/ lnK for the harmonic mean

K aK/ lnK K aK/ lnK K aK/ lnK

3 2.499192 10 1.980287 100 1.619631

4 2.321831 20 1.828861 200 1.561359

5 2.214749 50 1.693497 400 1.514096

I The rate of convergence aK/ lnK → 1 is very slow

I Suggestions:

• for K ≥ 3, use (2.5 lnK )M−1,K

• for K ≥ 10, use (2 lnK )M−1,K

• for K ≥ 50, use (1.7 lnK )M−1,K back


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