Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach

Mai, Jan-Frederik; Schenk, Steffen; Scherer, Matthias

doi:10.1515/strm-2015-0009

Cited by 12 publications

(10 citation statements)

References 19 publications

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“…The question can be answered by the following example: Define an exchangeable min-id sequence via the extended chronometer (H t ) t∈R = − log (1 − Γ t ), where (Γ t ) t∈R denotes a Dirichlet process, which can be seen as a prior distribution on random distribution functions in a Bayesian framework [12]. The authors of [37] have shown that H is indeed infinitely divisible, which implies that the associated exchangeable sequence X is min-id. Moreover, [37] show that the upper and lower bivariate tail dependence coefficients ρ u 2 and ρ l 2 of X can take any value in (0, 1) Thus, exchangeable min-id sequences can exhibit arbitrary positive bivariate upper and lower tail dependence, which shows that the dependence structure of exchangeable min-id sequences is much richer than the dependence structure of exchangeable min-stable sequences.…”

Section: Main Results: Linking Exchangeable Min-id Sequences To Exten...mentioning

confidence: 99%

“…The authors of [37] have shown that H is indeed infinitely divisible, which implies that the associated exchangeable sequence X is min-id. Moreover, [37] show that the upper and lower bivariate tail dependence coefficients ρ u 2 and ρ l 2 of X can take any value in (0, 1) Thus, exchangeable min-id sequences can exhibit arbitrary positive bivariate upper and lower tail dependence, which shows that the dependence structure of exchangeable min-id sequences is much richer than the dependence structure of exchangeable min-stable sequences.…”

Section: Main Results: Linking Exchangeable Min-id Sequences To Exten...mentioning

confidence: 99%

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Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

Brück¹,

Mai²,

Scherer³

2020

Preprint

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We establish a correspondence between exchangeable sequences of random variables whose finite-dimensional distributions are min-(or max-) infinitely divisible and non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of a min-id sequence is shown to be the sum of a very simple "drift measure" and a mixture of product probability measures, which corresponds uniquely to the Lévy measure of a non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, minstable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.

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Section: Main Results: Linking Exchangeable Min-id Sequences To Exten...mentioning

confidence: 99%

Section: Main Results: Linking Exchangeable Min-id Sequences To Exten...mentioning

confidence: 99%

Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

Brück¹,

Mai²,

Scherer³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…in probability forecasting (Dawid, 1984)), but, to our knowledge, has not been utilised in the context of sensitivity analysis. Nonetheless, there are some conceptual parallels of our work with Mara and Tarantola (2012), who consider multivariate normal variables in a variance-based sensitivity framework, Mai et al (2015), who study model robustness via a transformation of the input vector, and Kraus and Czado (2017), who carry out bank stress testing using graphical dependence models.…”

Section: Relation To Existing Literaturementioning

confidence: 98%

Cascade Sensitivity Measures

2021

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In risk analysis, sensitivity measures quantify the extent to which the probability distribution of a model output is affected by changes (stresses) in individual random input factors.For input factors that are statistically dependent, we argue that a stress on one input should also precipitate stresses in other input factors. We introduce a novel sensitivity measure, termed cascade sensitivity, defined as a derivative of a risk measure applied on the output, in the direction of an input factor. The derivative is taken after suitably transforming the random vector of inputs, thus explicitly capturing the direct impact of the stressed input factor, as well as indirect effects via other inputs. Furthermore, alternative representations of the cascade sensitivity measure are derived, allowing us to address practical issues, such as incomplete specification of the model and high computational costs. The applicability of the methodology is illustrated through the analysis of a commercially used insurance risk model.

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“…This is [61,Theorem 3.5]. In order to prove necessity, the principle of inclusion and exclusion can be used to express the survival copula of Ĉ as an alternating sum of lower-dimensional margins of Ĉ.…”

Section: The Dirichlet Prior and Radial Symmetrymentioning

confidence: 99%

The infinite extendibility problem for exchangeable real-valued random vectors

Mai

2019

Preprint

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We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on R d , which is: Can a given random vector X = (X 1 , . . . , X d ) be represented in distribution as the first d members of an infinite exchangeable sequence of random variables? This is the case if and only if X has a stochastic representation that is "conditionally iid" according to the seminal De Finetti's Theorem. Of particular interest are cases in which the original motivation behind the model X is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when X takes values in {0, 1} d , has a spherical law, a law with 1 -norm symmetric survival function, or a law with ∞ -norm symmetric density. The solutions in all these cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on R, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when X has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a Lévy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems. In comparison to former literature on the topic, this survey purposely dispenses with generalizations to the related and larger concept of finite exchangeability or to more general state spaces than R. Instead, it aims to constitute an up-to-date comprehensive collection of known and compelling solutions of the real-valued extendibility problem, accessible for both applied and theoretical probabilists, presented in a lecture-like fashion.

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Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach

Abstract: It is standard in quantitative risk management to model a random vector

Cited by 12 publications

References 19 publications

Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

Cascade Sensitivity Measures

The infinite extendibility problem for exchangeable real-valued random vectors

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