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Univariate continuous distributions are one of the fundamental components on which statistical modelling, ancient and modern, frequentist and Bayesian, multi-dimensional and complex, is based. In this article, I review and compare some of the main general techniques for providing families of typically unimodal distributions on ℝ with one or two, or possibly even three, shape parameters, controlling skewness and/or tailweight, in addition to their all-important location and scale parameters. One important and useful family is comprised of the 'skew-symmetric' distributions brought to prominence by Azzalini. As these are covered in considerable detail elsewhere in the literature, I focus more on their complements and competitors. Principal among these are distributions formed by transforming random variables, by what I call 'transformation of scale'—including two-piece distributions—and by probability integral transformation of nonuniform random variables. I also treat briefly the issues of multi-variate extension, of distributions on subsets of ℝ and of distributions on the circle. The review and comparison is not comprehensive, necessarily being selective and therefore somewhat personal.

Originating from a system theory and an input/output point of view, I introduce a new class of generalized distributions. A parametric nonlinear transformation converts a random variable X into a so-called Lambert W random variable Y, which allows a very flexible approach to model skewed data. Its shape depends on the shape of X and a skewness parameter γ. In particular, for symmetric X and nonzero γ the output Y is skewed. Its distribution and density function are particular variants of their input counterparts. Maximum likelihood and method of moments estimators are presented, and simulations show that in the symmetric case additional estimation of γ does not affect the quality of other parameter estimates. Applications in finance and biomedicine show the relevance of this class of distributions, which is particularly useful for slightly skewed data. A practical by-result of the Lambert W framework: data can be \"unskewed.\" The R package LambertW developed by the author is publicly available (CRAN).

In this study, a Level III reliability design of an armor block of rubble mound breakwater was developed using the optimized probabilistic wave height model for the Korean marine environment and Van der Meer equation. To demonstrate what distinguishes this study from the others, numerical simulation was first carried out, assuming that wave slope follows Gaussian distribution recommended by PIANC. Numerical results showed that Gaussian wave slope distribution overpredicted the failure probability of armor block, longer and shorter waves, and on the contrary, underpredicted waves of the medium period. After noting the limitations of Gaussian distribution, some efforts were made to develop an alternative for Gaussian distribution. As a result, non-Gaussian wave slope distribution was analytically derived from the joint distribution of wave amplitude and period by Longuet–Higgins using the random variables transformation technique. Numerical results showed that non-Gaussian distribution could effectively address the limitations of Gaussian distribution due to its capability to account for the nonlinear resonant wave–wave interaction and its effects on the wave slope distribution that significantly influences the armor block’s stability. Therefore, the non-Gaussian wave slope distribution presented in this study could play an indispensable role in addressing controversial issues such as whether or not enormous armor blocks like a Tetrapod of 100 t frequently mentioned in developing countermeasures against rough seas due to climate change is too conservatively designed.

In this paper, some exact solutions of the stochastic generalized nonlinear shallow water wave equation are investigated. This equation is important in fluid mechanics' fields since it can model the propagation of disturbances in water and other incompressible fluids. Opposite to what is usually considered in the literature, the two dispersion coefficients of the nonlinear terms are considered dependent random quantities as a more realistic case. The modified extended-tanh function (METF) method is combined with the random variable transformation (RVT) technique to get full probabilistic solutions of the problem via computing the probability density functions (PDFs) of the solution processes. Based on the probability density function, any statistical moment of the solution can be evaluated. Through two different applications for the input random variables (dispersion coefficients), my findings are applied efficiently. Finally, numerical results are presented graphically along the spatial dimension at a certain wave speed and time. The obtained results ratify that the proposed technique is efficient and powerful for obtaining analytical probabilistic solutions for the problem.

We present a new analytical method to find the asymptotic stable equilibria states based on the Markov chain technique. We reveal this method on the Susceptible-Infectious-Recovered (SIR)-type epidemiological model that we developed for viral diseases with long-term immunity memory. This is a large-scale model containing 15 nonlinear ordinary differential equations (ODEs), and classical methods have failed to analytically obtain its equilibria. The proposed method is used to conduct a comprehensive analysis by a stochastic representation of the dynamics of the model, followed by finding all asymptotic stable equilibrium states of the model for any values of parameters and initial conditions thanks to the symmetry of the population size over time.

The classical kinetic equation has been broadly used to describe reaction and deactivation processes in chemistry. The mathematical formulation of this deterministic nonlinear differential equation depends on reaction and deactivation rate constants. In practice, these rates must be calculated via laboratory experiments, hence involving measurement errors. Therefore, it is more realistic to treat these rates as random variables rather than deterministic constants. This leads to the randomization of the kinetic equation, and hence its solution becomes a stochastic process. In this paper we address the probabilistic analysis of a randomized kinetic model to describe reaction and deactivation by catalase of hydrogen peroxide decomposition at a given initial concentration. In the first part of the paper, we determine closed-form expressions for the probability density functions of important quantities of the aforementioned chemical process (the fractional conversion of hydrogen peroxide, the time until a fixed quantity of this fractional conversion is reached and the activity of the catalase). These expressions are obtained by taking extensive advantage of the so called Random Variable Transformation technique. In the second part, we apply the theoretical results obtained in the first part together with the principle of maximum entropy to model the hydrogen peroxide decomposition and aspergillus niger catalase deactivation using real data excerpted from the recent literature. Our results show full agreement with previous reported analysis but having as additional benefit that they provide a more complete description of both model inputs and outputs since we take into account the intrinsic uncertainties involved in modelling process.

Random initial value problems to non-homogeneous first-order linear differential equations with complex coefficients are probabilistically solved by computing the first probability density of the solution. For the sake of generality, coefficients and initial condition are assumed to be absolutely continuous complex random variables with an arbitrary joint probability density function. The probability of stability, as well as the density of the equilibrium point, are explicitly determined. The Random Variable Transformation technique is extensively utilized to conduct the overall analysis. Several examples are included to illustrate all the theoretical findings.

In this paper a complete probabilistic description for the solution of random homogeneous linear second-order differential equations via the computation of its two first probability density functions is given. As a consequence, all unidimensional and two-dimensional statistical moments can be straightforwardly determined, in particular, mean, variance and covariance functions, as well as the first-order conditional law. With the aim of providing more generality, in a first step, all involved input parameters are assumed to be statistically dependent random variables having an arbitrary joint probability density function. Second, the particular case that just initial conditions are random variables is also analysed. Both problems have common and distinctive feature which are highlighted in our analysis. The study is based on random variable transformation method. As a consequence of our study, the well-known deterministic results are nicely generalized. Several illustrative examples are included.

In this paper, we deal with computational uncertainty quantification for stochastic models with one random input parameter. The goal of the paper is twofold: First, to approximate the set of probability density functions of the solution stochastic process, and second, to show the capability of our theoretical findings to deal with some important epidemiological models. The approximations are constructed in terms of a polynomial evaluated at the random input parameter, by means of generalized polynomial chaos expansions and the stochastic Galerkin projection technique. The probability density function of the aforementioned univariate polynomial is computed via the random variable transformation method, by taking into account the domains where the polynomial is strictly monotone. The algebraic/exponential convergence of the Galerkin projections gives rapid convergence of these density functions. The examples are based on fundamental epidemiological models formulated via linear and nonlinear differential and difference equations, where one of the input parameters is assumed to be a random variable.

We provide a full stochastic description, via the first probability density function, of the solution of linear-quadratic logistic-type differential equation whose parameters involve both continuous and discrete random variables with arbitrary distributions. For the sake of generality, the initial condition is assumed to be a random variable too. We use the Dirac delta function to unify the treatment of hybrid (discrete-continuous) uncertainty. Under general hypotheses, we also compute the density of time until a certain value (usually representing the population) of the linear-quadratic logistic model is reached. The theoretical results are illustrated by means of several examples, including an application to modelling the number of users of Spotify using real data. We apply the Principle Maximum Entropy to assign plausible distributions to model parameters.