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Asymmetric distributions are frequently seen in real-world datasets due to a number of factors, such as sample biases and nonlinear interactions between the variables observed. Thus, in order to better characterize real-world phenomena, studying asymmetric distribution is of great interest. In this work, we derive stress–strength reliability formulas of the type P (X < Y) when both X and Y follow p-max stable laws with three parameters, which are inherently asymmetric. The new relations are given in terms of extreme-value H-functions and have been obtained under fewer parameter restrictions when compared to similar results in the literature. We estimate the parameters of the p-max stable laws by a stochastic optimization method and the stress–strength probability by a maximum likelihood procedure. The performance of the analytical models is evaluated through simulations and real-life dataset modeling. [ABSTRACT FROM AUTHOR]

The modality is an important topic for modelling. Using parametric models is an efficient way when real data set shows trimodality. In this paper, we propose a new class of trimodal probability distributions, that is, probability distributions that have up to three modes. Trimodality itself is achieved by applying a proper transformation to density function of certain continuous probability distributions. At first, we obtain preliminary results for an arbitrary density function $ g(x) $ g (x) and, next, we focus on the Gaussian case, studying trimodal Gaussian model more deeply. The Gaussian distribution is applied to produce the trimodal form of Gaussian known as normal distribution. The tractability of analytical expression of normal distribution and properties of the trimodal normal distribution are important reasons why we choose normal distribution. Furthermore, the existing distributions should be improved to be capable of modelling efficiently when there exists a trimodal form in a data set. After new density function is proposed, estimating its parameters is important. Since Mathematica 12.0 software has optimization tools and important modelling techniques, computational steps are performed using this software. The bootstrapped form of real data sets are applied to show the modelling ability of the proposed distribution when real data sets show trimodality. [ABSTRACT FROM AUTHOR]

In reliability studies, we are interested in the behaviour of a system when it interacts with its surrounding environment. To assess the system's behaviour in a reliability sense, we can take the system's intrinsic quality as strength and the outcome of interactions as stress. Failure is observed whenever stress exceeds strength. Taking Y as a random variable representing the stress the system experiences and random variable X as its strength, the probability of not failing can be taken as a proxy for the reliability of the component and given as P (Y < X) = 1 − P (X < Y) . This way, in the present paper, it is considered that X and Y follow generalized extreme value distributions, which represent a family of continuous probability distributions that have been extensively applied in engineering and economic contexts. Our contribution deals with a more general scenario where stress and strength are not independent and copulas are used to model the dependence between the involved random variables. In such modelling framework, we explored the proper selection of copula models characterizing the dependence structure. The Gumbel–Hougaard, Frank, and Clayton copulas were used for modelling bivariate data sets. In each case, information criteria were considered to compare the modelling capabilities of each copula. Two economic applications, as well as an engineering one, on real data sets are discussed. Overall, an easy-to-use methodological framework is described, allowing practitioners to apply it to their own research projects. [ABSTRACT FROM AUTHOR]

Analyzing the statistical behavior of the assets' returns has shown to be an interesting approach to perform asset selection. In this work, we explore a stress-strength reliability approach to perform asset selection based on probabilities of the type P(X < Y) when both X and Y follow a generalized extreme value (GEV) distribution with three parameters. At first, we derive new analytical and closed form relations in terms of the extreme value H-function, which have been obtained under fewer parameter restrictions compared to similar results in the literature. To show the performance of our results, we include a Monte-Carlo simulation study and we investigate the application of the reliability measure P(X < Y) in selecting financial assets with returns characterized by the distributions X and Y. Therefore, rather than the conventional approach of comparing the expected values of X and Y based on modern portfolio theory, we delve into the metric P(X < Y) as an alternative parameter for assessing better returns. [ABSTRACT FROM AUTHOR]

Abstract Drift estimation for Generalized Ornstein-Uhlenbeck processes with Lévy noise and up to 3rd-order autoregressive behavior are studied. Under exponential trend and cosine oscillations, we propose two discrete-time versions of the MLE for the drift parameter. Both estimates are derived using jump filtering techniques. Simulation results showing their performance are included. [ABSTRACT FROM AUTHOR]

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