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Generalized linear models (GLMs) form one of the most popular classes of models in statistics. The gamma variant is used, for instance, in actuarial science for the modelling of claim amounts in insurance. A flaw of GLMs is that they are not robust against outliers (i.e., against erroneous or extreme data points). A difference in trends in the bulk of the data and the outliers thus yields skewed inference and predictions. To address this problem, robust methods have been introduced. The most commonly applied robust method is frequentist and consists in an estimator which is derived from a modification of the derivative of the log-likelihood. The objective is to propose an alternative approach which is modelling-based and thus fundamentally different. Such an approach allows for an understanding and interpretation of the modelling, and it can be applied for both frequentist and Bayesian statistical analyses. The proposed approach possesses appealing theoretical and empirical properties. [ABSTRACT FROM AUTHOR]

This work examines a class of path-dependent stochastic systems which are hybrid with wideband noise, Poisson jumps and a singularly perturbed Markov chain. The addition of multi-scale Markov chain allows for modeling of discrete events with both fast and slow fluctuation. While this more realistic approach presents analytical challenges due to the non-Markovian formulation resulting from the wideband noise and the singularly perturbed Markov chain. By virtue of the weak convergence method and Itô functional formula, we prove that as ɛ → 0 , we obtain a Markovian switching jump diffusion. Finally, we offer several examples to illustrate our findings. • This is the first time to consider a class of stochastic systems with hybrid switching, wideband noise, Poisson jumps as well as time-delay perturbations. • Two-time-scale Markovian chain can describe the systems with both strong and weak interactions, which becomes really helpful when we dealt with a large-scale system. • Our problem is path-dependent and its phase space is infinite-dimensional, functional Itô formula expected to be applied to deal with such complex situation. • By virtue of the weak convergence method and Itô functional formula, we get the limit system for the singularly perturbed functional systems. • Integro-differential systems, Ginzburg–Landau equations and Replicator dynamics are presented to illustrate our findings. [ABSTRACT FROM AUTHOR]

In this paper, a special sequence of controlled branching processes is considered. Using a martingale problem approach, we prove that under some mild conditions the limit of such processes is a kind of continuous branching process with dependent immigration constructed in Li (2019). The conditions are given in terms of probability generating functions and can be realized. [ABSTRACT FROM AUTHOR]

We propose a new double inertial projection algorithm by combining the subgradient extragradient algorithm with the projection contraction algorithm. On one hand, our algorithm requires the mapping to be Lipschitz continuous, but without the Lipschitz constant. On the other hand, this algorithm only requires the mapping is quasimonotone.Under some mild conditions, we obtain a weak convergence result of the algorithm. In addition, we give a strong convergence result of the algorithm when the mapping is strongly pseudomonotone. Some numerical experiments are given to show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]