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QuestionSome ExamplesSome AnswersSome MoreReferencesSum of Two Standard Uniform Random VariablesRuodu Wanghttp://sas.uwaterloo.ca/ wangDepartment of Statistics and Actuarial ScienceUniversity of Waterloo, CanadaDependence Modeling in Finance, Insurance and Environmental ScienceMunich, GermanyMay 17, 2016Based on joint work with Bin Wang (Beijing)Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables1/25

QuestionSome ExamplesSome AnswersSome MoreReferencesA QuestionIn this talk we discuss this problem:X1 U[ 1, 1], X2 U[ 1, 1]what is a distribution (cdf) of X1 X2 ?A difficult problem with no applications (?)Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables2/25

QuestionSome ExamplesSome AnswersSome MoreReferencesGeneric FormulationIn an atomless probability space:F1 , . . . , Fn are n distributionsXi Fi , i 1, . . . , nSn X1 · · · XnDenote the set of possible aggregate distributionsDn Dn (F1 , · · · , Fn ) {cdf of Sn Xi Fi , i 1, · · · , n}.Primary question: Characterization of Dn .Dn is non-empty, convex, and closed w.r.t. weak convergenceRuodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables3/25

QuestionSome ExamplesSome AnswersSome MoreReferencesGeneric FormulationFor example:Xi : individual risks; Sn : risk aggregationfixed marginal; unknown copulaClassic setup in Quantitative Risk ManagementSecondary question: what is supF Dn ρ(F ) for some functionalρ (risk measure, utility, moments, .)?Risk aggregation with dependence uncertainty, an active fieldover the past few years:Embrechts et. al. (2014 Risks) and the references thereinBooks: Rüschendorf (2013), McNeil-Frey-Embrechts (2015)20 papers in the past 3 yearsRuodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables4/25

QuestionSome ExamplesSome AnswersSome MoreReferencesSome ObservationsAssume that F1 , . . . , Fn have finite means µ1 , . . . , µn , respectively.Necessary conditions:Sn cx F1 1 (U) · · · Fn 1 (U)In particular, E[Sn ] µ1 · · · µnRange(Sn ) Pni 1Range(Xi )Suppose E[T ] µ1 · · · µn . ThenFT Dn (F1 , . . . , Fn ) (F T , F1 , . . . , Fn ) is jointly mixableFor a theory of joint mixabilityW.-Peng-Yang (2013 FS), Wang-W. (2016 MOR)Surveys: Puccetti-W. (2015 STS), W. (2015 PS)Numerical method: Puccetti-W. (2015 JCAM)Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables5/25

QuestionSome ExamplesSome AnswersSome MoreReferencesSome ObservationsJoint mixability is an open research areaA general analytical characterization of Dn or joint mixabilityis far away from being clearWe tune down and look at standard uniform distributions andn 2Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables6/25

QuestionSome ExamplesSome AnswersSome MoreReferencesProgress of the Talk1Question2Some Examples3Some Answers4Some More5ReferencesRuodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables7/25

QuestionSome ExamplesSome AnswersSome MoreReferencesSimple ExamplesX1 U[ 1, 1], X2 U[ 1, 1], S2 X1 X2 .Obvious constraintsE[S2 ] 0range of S2 in [ 2, 2]Var(S2 ) 4/3Ruodu Wang(wang@uwaterloo.ca)S2 cx 2X1(sufficient?)Sum of two uniform random variables8/25

QuestionSome ExamplesSome AnswersSome MoreReferencesSimple ExamplesAre the following distributions possible for S2 ?Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables9/25

QuestionSome ExamplesSome AnswersSome MoreReferencesSimple Examples: More.Examples and counter-examples: Mao-W. (2015 JMVA) and Wang-W. (2016 MOR)Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables10/25

QuestionSome ExamplesSome AnswersSome MoreReferencesA Small Copula Game.P(S2 4/5) 1/2, P(S2 4/5) 1/210.6Y0.2-0.2-0.6-1-1-0.6-0.20.20.61XRuodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables11/25

QuestionSome ExamplesSome AnswersSome MoreReferencesProgress of the Talk1Question2Some Examples3Some Answers4Some More5ReferencesRuodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables12/25

QuestionSome ExamplesSome AnswersSome MoreReferencesExisting ResultsLet D2 D2 (U[ 1, 1], U[ 1, 1]). Below are implied by results inWang-W. (2016 MOR)Let F be any distribution with a monotone density function.then F D2 if and only if F is supported in [ 2, 2] and haszero mean.Let F be any distribution with a unimodal and symmetricdensity function. Then F D2 if and only if F is supported in[ 2, 2] and has zero mean.U[ a, a] D2 if and only if a [0, 2] (a special case of both).The case U[ 1, 1] D2 is given in Rüschendorf (1982 JAP).Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables13/25

QuestionSome ExamplesSome AnswersSome MoreReferencesUnimodal DensitiesA natural candidate to investigate is the class of distributions witha unimodal density.Theorem 1Let F be a distribution with a unimodal density on [ 2, 2] andzero mean. Then F D2 .Both the two previous results are special casesFor bimodal densities we do not have anything concreteRuodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables14/25

QuestionSome ExamplesSome AnswersSome MoreReferencesDensities Dominating a UniformA second candidate is a distribution which dominates a portion ofa uniform distribution.Theorem 2Let F be a distribution supported in [a b, a] with zero mean anddensity function f . If there exists h 0 such that f 3b4hon[ h/2, h/2], then F D2 .The density of F dominates 3b/4 times that of U[ h/2, h/2]Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables15/25

QuestionSome ExamplesSome AnswersSome MoreReferencesBi-atomic DistributionsContinuous distributions seem to be a dead end; what aboutdiscrete distributions? Let us start with the simplest cases.Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables16/25

QuestionSome ExamplesSome AnswersSome MoreReferencesBi-atomic DistributionsTheorem 3Let F be a bi-atomic distribution with zero mean supported on{a b, a}. Then F D2 if and only if 2/b N.2/b 12/b 5/4For given b a 0, there is only one distribution on{a b, a} with mean zero.Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables17/25

QuestionSome ExamplesSome AnswersSome MoreReferencesTri-atomic DistributionsFor a tri-atomic distribution F , write F (f1 , f2 , f3 ) where f1 , f2 , f3are the probability masses of FOn given three points, the set of tri-atomic distributions withmean zero has one degree of freedom.We study the case of F having an “equidistant support”{a 2b, a b, a}.For x 0, define a “measure of non-integrity” dxebxcdxc min 1, 1 [0, 1] .xxObviously dxc 0 x N.Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables18/25

QuestionSome ExamplesSome AnswersSome MoreReferencesTri-atomic DistributionsTheorem 4Suppose that F (f1 , f2 , f3 ) is a tri-atomic distribution with zeromean supported in {a 2b, a b, a}, 0 and a b. ThenF D2 if and only if it is the following three cases.(i) a b and f2 d b1 c.1b N.11b 2 N(ii) a b and(iii) a b,and f2 2a .cf. Theorem 3 (condition 2/b N)Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables19/25

QuestionSome ExamplesSome AnswersSome MoreReferencesTri-atomic DistributionsThe corresponding distributions in Theorem 4:(i) (f1 , f2 , f3 ) cx{(0, 1, 0), 21 (1 d b1 c, 2d b1 c, 1 d b1 c)}.(ii) (f1 , f2 , f3 ) cx{(0, ba , 1 ba ), 12 ( ba , 0, 2 ba )}.(iii) (f1 , f2 , f3 ) cx{(0, ba , 1 ba ), 12 ( ba 2a , a, 2 Ruodu Wang(wang@uwaterloo.ca)ab 2a )}.Sum of two uniform random variables20/25

QuestionSome ExamplesSome AnswersSome MoreReferencesProgress of the Talk1Question2Some Examples3Some Answers4Some More5ReferencesRuodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables21/25

QuestionSome ExamplesSome AnswersSome MoreReferencesSome More to ExpectIt is possible to further characterize n-atomic distributionswith an equidistant support (things get ugly though).We guess: for any distribution Fwith an equidistant support, orwith finite density and a bounded support,there exists a number M 0 such thatF D2 (U[ m, m], U[ m, m]) for all m N and m M.Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables22/25

QuestionSome ExamplesSome AnswersSome MoreReferencesSome More to ThinkTwo uniforms with different lengths?Three or more uniform distributions?Other types of distributions?Applications?We yet know very little about the problem of D2Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables23/25

QuestionSome ExamplesSome AnswersSome MoreReferencesReferences IEmbrechts, P., Puccetti, G., Rüschendorf, L., Wang, R. and Beleraj, A. (2014). An academic response toBasel 3.5. Risks, 2(1), 25–48.Mao, T. and Wang, R. (2015). On aggregation sets and lower-convex sets. Journal of Multivariate Analysis,136, 12–25.McNeil, A. J., Frey, R. and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniquesand Tools. Revised Edition. Princeton, NJ: Princeton University Press.Puccetti, G. and Wang, R. (2015). Detecting complete and joint mixability. Journal of Computational andApplied Mathematics, 280, 174–187.Puccetti, G. and Wang, R. (2015). Extremal dependence concepts. Statistical Science, 30(4), 485–517.Rüschendorf, L. (1982). Random variables with maximum sums. Advances in Applied Probability, 14(3),623–632.Rüschendorf, L. (2013). Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations andPortfolios. Springer, Heidelberg.Wang, B. and Wang, R. (2016). Joint mixability. Mathematics of Operations Research, forthcoming.Wang, R. (2015). Current open questions in complete mixability. Probability Surveys, 12, 13–32.Wang, R., Peng, L. and Yang, J. (2013). Bounds for the sum of dependent risks and worst Value-at-Riskwith monotone marginal densities. Finance and Stochastics 17 (2), 395–417.Ruodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables24/25

QuestionSome ExamplesSome AnswersSome MoreReferencesDanke SchönThank you for your kind attentionRuodu Wang(wang@uwaterloo.ca)Sum of two uniform random variables25/25

For a theory of joint mixability W.-Peng-Yang (2013 FS), Wang-W. (2016 MOR) Surveys: Puccetti-W. (2015 STS), W. (2015 PS) . Ruodu Wang (wang@uwaterloo.ca) Sum of two uniform random variables 24/25. Question Some Examples Some Answers Some More References Danke Sch on Thank you for your kind attention Ruodu Wang (wang@uwaterloo.ca) Sum of two .

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