Exact simulation of continuous max-id processes with applications to exchangeable max-id sequences

Brück, Florian

doi:10.1016/j.jmva.2022.105117

Cited by 1 publication

(3 citation statements)

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Section: Discussionmentioning

confidence: 99%

“…To circumvent this problem, Brück (2022) provides a simulation algorithm for the class of real-valued continuous max-id-processes, which ensures that a user-specified number of locations of the max-id-process are simulated exactly. Since every min-id sequence X may be transformed into a continuous max-id-process 1∕X with index set ℕ , the results of Brück (2022) may be applied to simulate an exchangeable minid sequence X . The key ingredient of the proposed simulation algorithm in Brück (2022) is the exponent measure of X (or 1∕X ), which can be easily constructed according to (4) or deduced from the Lévy measure of the associated nnnd id-process by Theorem 3.5.…”

Section: Discussionmentioning

confidence: 99%

“…Since every min-id sequence X may be transformed into a continuous max-id-process 1∕X with index set ℕ , the results of Brück (2022) may be applied to simulate an exchangeable minid sequence X . The key ingredient of the proposed simulation algorithm in Brück (2022) is the exponent measure of X (or 1∕X ), which can be easily constructed according to (4) or deduced from the Lévy measure of the associated nnnd id-process by Theorem 3.5. Thus, the results of this paper allow to construct exchangeable minid sequences in terms of their associated exponent or Lévy measures, while Brück (2022) provides a method for the exact simulation of their d-dimensional margins.…”

Section: Discussionmentioning

confidence: 99%

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

Brück

Mai²,

Scherer

2022

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We establish a one-to-one correspondence between (i) exchangeable sequences of random variables whose finite-dimensional distributions are minimum (or maximum) infinitely divisible and (ii) non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of an exchangeable minimum infinitely divisible sequence is shown to be the sum of a very simple “drift measure” and a mixture of product probability measures, which uniquely corresponds to the Lévy measure of a non-negative and non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. In probabilistic terms, the aforementioned infinitely divisible process is equal to the conditional cumulative hazard process associated with the exchangeable sequence of random variables with minimum (or maximum) infinitely divisible marginals. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many interesting classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable 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: Discussionmentioning

confidence: 99%