On New Types of Multivariate Trigonometric Copulas

Copulas are useful functions for modeling multivariate distributions through their univariate marginal distributions and dependence structures. They have a wide range of applications in all fields of science that deal with multivariate data. While there is a plethora of copulas, those based on trigonometric functions, especially in dimensions greater than two, have received much less attention. They are, however, of interest because of the properties of oscillation and periodicity of the trigonometric functions, which can appear in certain models of correlation of natural phenomena. In order to fill this gap, this paper introduces and investigates two new types of “multivariate trigonometric copulas”. Their main theoretical properties are studied, and some perspectives for applications are sketched for future work. In particular, we show that the proposed copulas are symmetric, not associative, with no orthant dependence, and with copula densities that have wide oscillations, which remains an uncommon property in the field. The expressions of their multivariate Spearman’s rho are also determined. Furthermore, the first type of the proposed copulas has the interesting feature of having a multivariate Spearman’s rho equal to 0 for all of the dimensions. Some graphic evidence supports the findings. Some mathematical formulas involving the product of n trigonometric functions may be of independent interest.

A Generative Model for Correlated Graph Signals

A graph signal is a random vector with a partially known statistical description. The observations are usually sufficient to determine marginal distributions of graph node variables and their pairwise correlations representing the graph edges. However, the curse of dimensionality often prevents estimating a full joint distribution of all variables from the available observations. This paper introduces a computationally effective generative model to sample from arbitrary but known marginal distributions with defined pairwise correlations. Numerical experiments show that the proposed generative model is generally accurate for correlation coefficients with magnitudes up to about 0.3, whilst larger correlations can be obtained at the cost of distribution approximation accuracy. The generative models of graph signals can also be used to sample multivariate distributions for which closed-form mathematical expressions are not known or are too complex.

Simulation of Automotive Warranty Data

<p>This thesis will investigate the prediction of the number of claims in a two dimensional automotive warranty claim model for the case of minimal repair.The method involved fitting marginal distributions for age of claim and mileage of claim seperately. Next, various copulas were fitted to establish the correlation between age and mileage, and assessed for fit. The Gumbel copula is chosen as optimal. From this Gumbel copula, a simulation of warranty claims is undertaken. The method produced a good fit for claim age but performed less well for claim mileage, due to the asymmetry of the correlation between mileage and age. Further research directions to improve the accuracy and usefulness of this model are suggested.</p>

Simulation of Automotive Warranty Data

<p>This thesis will investigate the prediction of the number of claims in a two dimensional automotive warranty claim model for the case of minimal repair.The method involved fitting marginal distributions for age of claim and mileage of claim seperately. Next, various copulas were fitted to establish the correlation between age and mileage, and assessed for fit. The Gumbel copula is chosen as optimal. From this Gumbel copula, a simulation of warranty claims is undertaken. The method produced a good fit for claim age but performed less well for claim mileage, due to the asymmetry of the correlation between mileage and age. Further research directions to improve the accuracy and usefulness of this model are suggested.</p>

Sufficiency of Markov Policies for Continuous-Time Jump Markov Decision Processes

One of the basic facts known for discrete-time Markov decision processes is that, if the probability distribution of an initial state is fixed, then for every policy it is easy to construct a (randomized) Markov policy with the same marginal distributions of state-action pairs as for the original policy. This equality of marginal distributions implies that the values of major objective criteria, including expected discounted total costs and average rewards per unit time, are equal for these two policies. This paper investigates the validity of the similar fact for continuous-time jump Markov decision processes (CTJMDPs). It is shown in this paper that the equality of marginal distributions takes place for a CTJMDP if the corresponding Markov policy defines a nonexplosive jump Markov process. If this Markov process is explosive, then at each time instance, the marginal probability, that a state-action pair belongs to a measurable set of state-action pairs, is not greater for the described Markov policy than the same probability for the original policy. These results are applied in this paper to CTJMDPs with expected discounted total costs and average costs per unit time. It is shown for these criteria that, if the initial state distribution is fixed, then for every policy, there exists a Markov policy with the same or better value of the objective function.

Nonparametric Estimation of Multivariate Extreme Value Copulas with Known and Unknown Marginal Distributions

The purpose of this paper is estimating the dependence function of multivariate extreme values copulas. Different nonparametric estimators are developed in the literature assuming that marginal distributions are known. However, this assumption is unrealistic in practice. To overcome the drawbacks of these estimators, we substituted the extreme value marginal distribution by the empirical distribution function. Monte Carlo experiments are carried out to compare the performance of the Pickands, Deheuvels, Hall-Tajvidi, Zhang and Gudendorf-Segers estimators. Empirical results showed that the empirical distribution function improved the estimators’ performance for different sample sizes.

On subcopula estimation for discrete models

PurposeTo discuss subcopula estimation for discrete models.Design/methodology/approachThe convergence of estimators is considered under the weak convergence of distribution functions and its equivalent properties known in prior works.FindingsThe domain of the true subcopula associated with discrete random variables is found to be discrete on the interior of the unit hypercube. The construction of an estimator in which their domains have the same form as that of the true subcopula is provided, in case, the marginal distributions are binomial.Originality/valueTo the best of our knowledge, this is the first time such an estimator is defined and proved to be converged to the true subcopula.

Evaluation of multi-parameter dependencies in weather and climate simulations

&lt;p&gt;Weather and climate simulations based on numerical models provide 4-dimensional reconstructions of multiple meteorological parameters describing the atmopsheric state. Yet, the vast majority of evaluation studies focus on the evaluation of single parameters in time and space without looking at the statistical multivariate dependence between the parameters. This, however, is necessary especially with respect to specific events where two or more parameters are involved, i.e., so called compound events. Previous studies have investigated the representation of natural hazards such as wildfires, heat stress, droughts by evaluating corresponding indices based on two or more parameters. Thereby the evaluation process stays in a single parameter framework with well established verification methods at hand.&amp;#160;&lt;/p&gt;&lt;p&gt;In this work, we present a more sophisticated and generalized approach to investigate physical dependencies between parameters by employing copula theory. With this method, we aim at evaluating the multivariate statistical dependence between two parameters (i.e. the copula) separately from their marginal distributions. This separation enables a more detailed investigation of compound indices (CI) based on the involved parameters. The differences in CI derived from model simulations and observations can now be related to deficiencies of the numerical model due to (i) a misrepresentation of the marginal distributions of the contributing variables, (ii) a misrepresentation of the statistical dependence between the parameters (the copula), (iii) or both. While the method is applicable to all combinations of two parameters, we will present the results of a specific joint copula-based evaluation of temperature and humidity which are the basis for natural hazards mentioned above.&lt;/p&gt;