Univariate Probability Distributions Research Articles

Fixed point characterizations of continuous univariate probability distributions and their applications

By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein’s method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavors, we focus on explicit representations given through a formula for the density- or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known. To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.

Fréchet analysis of variance for random objects

Summary Fréchet mean and variance provide a way of obtaining a mean and variance for metric space-valued random variables, and can be used for statistical analysis of data objects that lie in abstract spaces devoid of algebraic structure and operations. Examples of such data objects include covariance matrices, graph Laplacians of networks and univariate probability distribution functions. We derive a central limit theorem for the Fréchet variance under mild regularity conditions, using empirical process theory, and also provide a consistent estimator of the asymptotic variance. These results lead to a test for comparing $k$ populations of metric space-valued data objects in terms of Fréchet means and variances. We examine the finite-sample performance of this novel inference procedure through simulation studies on several special cases that include probability distributions and graph Laplacians, leading to a test for comparing populations of networks. The proposed approach has good finite-sample performance in simulations for different kinds of random objects. We illustrate the proposed methods by analysing data on mortality profiles of various countries and resting-state functional magnetic resonance imaging data.

Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation

Conditional value-at-risk (CVaR) and value-at-risk, also called the superquantile and quantile, are frequently used to characterize the tails of probability distributions and are popular measures of risk in applications where the distribution represents the magnitude of a potential loss. buffered probability of exceedance (bPOE) is a recently introduced characterization of the tail which is the inverse of CVaR, much like the CDF is the inverse of the quantile. These quantities can prove very useful as the basis for a variety of risk-averse parametric engineering approaches. Their use, however, is often made difficult by the lack of well-known closed-form equations for calculating these quantities for commonly used probability distributions. In this paper, we derive formulas for the superquantile and bPOE for a variety of common univariate probability distributions. Besides providing a useful collection within a single reference, we use these formulas to incorporate the superquantile and bPOE into parametric procedures. In particular, we consider two: portfolio optimization and density estimation. First, when portfolio returns are assumed to follow particular distribution families, we show that finding the optimal portfolio via minimization of bPOE has advantages over superquantile minimization. We show that, given a fixed threshold, a single portfolio is the minimal bPOE portfolio for an entire class of distributions simultaneously. Second, we apply our formulas to parametric density estimation and propose the method of superquantiles (MOS), a simple variation of the method of moments where moments are replaced by superquantiles at different confidence levels. With the freedom to select various combinations of confidence levels, MOS allows the user to focus the fitting procedure on different portions of the distribution, such as the tail when fitting heavy-tailed asymmetric data.

Spatial met-ocean data analysis for the North Sea using copulas: application in lumping of offshore wind turbine fatigue load cases

Knowledge about stochastic environmental loads and the response of the substructure are essential in fatigue damage calculations for offshore wind turbines. Site-specific and high-quality met-ocean data that contains joint measurements of wind and waves however, are not always available. An approach based on copulas was used to generate lumped joint met-ocean data in a location where only marginals (i.e., individual, univariate probability distributions) are known. Empirical copulas were calculated for different stations in the North Sea for pairs of wind speed and wave height variables. These copulas are then combined with marginals from another station to generate lumped data. The generated data is compared with the known lumped joint data, and the difference of two methods is measured. The lumped values are used to estimate fatigue damage based on simple formulas. The results show that the joint behaviour of wind speed and wave height follows a similar pattern around the North Sea. In addition, the mean difference of generated lumped wind speeds based on this technique from lumped wind speeds based on real measurements is less than 5%. The mean difference of the fatigue damage caused by the generated lumped wind speeds relative to lumped wind speeds based on real measurements is less than 5% at two locations, and less than 15% in another location. This suggests that the approach is promising for estimating joint met-ocean data at new locations in the North Sea where only marginals are available, which is of major interest for wind farm planning and design.

A generalization of Basu-Dhar’s bivariate geometric distribution to the trivariate case

In this paper, it is introduced a new parametric distribution to be used in multivariate lifetime data analysis as an alternative for the use of some existing multivariate parametric models as the popular multivariate normal distribution which is the most widely used model assumed in the analysis of continuous multivariate data. Although the normal multivariate distribution has univariate marginal normal probability distributions and simple interpretations for all their parameters, it may not be well fitted by many data sets, especially in survival data applications, usually considering logarithm transformed data. In many cases the use of parametric multivariate discrete models could be more appropriate for the data analysis. In this paper, it is introduced a generalization of the bivariate Basu-Dhar geometric distribution to a trivariate case applied to count data. Some properties of this trivariate geometric distribution, including its marginal probability distributions, order statistics distributions, the probability generating function and some simulation studies are presented. It is also presented some discussion on an extension of the trivariate case for the multivariate case. Classical and Bayesian inferences are presented assuming censored or uncensored observations. To illustrate the proposed methodology, two applications with real lifetime data are considered as examples.

Plotting Likelihood-Ratio-Based Confidence Regions for Two-Parameter Univariate Probability Models

Plotting two-parameter confidence regions is nontrivial. Numerical methods often rely on a computationally expensive grid-like exploration of the parameter space. A recent advance reduces the two-dimensional problem to many one-dimensional problems employing a trigonometric transformation that assigns an angle from the maximum likelihood estimator, and an unknown radial distance to its confidence region boundary. This paradigm shift can improve computational runtime by orders of magnitude, but it is not robust. Specifically, parameters differing greatly in magnitude and/or challenging nonconvex confidence region shapes make the plot susceptible to inefficiencies and/or inaccuracies. This article improves the technique by (i) keeping confidence region boundary searches in the parameter space, (ii) selectively targeting confidence region boundary points in lieu of uniformly spaced angles from the maximum likelihood estimator and (iii) enabling access to regions otherwise unreachable due to multiple roots for select angles. Two heuristics are given for selection: an elliptic-inspired angle selection heuristic and an intelligent smoothing search heuristic. Finally, a jump-center heuristic permits plotting otherwise inaccessible multiroot regions. This article develops these heuristics for two-parameter likelihood-ratio-based confidence regions associated with univariate probability distributions, and introduces the R conf package, which automates the process and is publicly available via CRAN.

Asymmetric generalizations of symmetric univariate probability distributions obtained through quantile splicing

Balakrishnan et al. proposed a two-piece skew logistic distribution by making use of the cumulative distribution function (CDF) of half distributions as the building block, to give rise to an asymmetric family of two-piece distributions, through the inclusion of a single shape parameter. This paper proposes the construction of asymmetric families of two-piece distributions by making use of quantile functions of symmetric distributions as building blocks. This proposition will enable the derivation of a general formula for the L-moments of two-piece distributions. Examples will be presented, where the logistic, normal, Student’s t(2) and hyperbolic secant distributions are considered.

A Copula-Based Bayesian Network for Modeling Compound Flood Hazard from Riverine and Coastal Interactions at the Catchment Scale: An Application to the Houston Ship Channel, Texas

Traditional flood hazard analyses often rely on univariate probability distributions; however, in many coastal catchments, flooding is the result of complex hydrodynamic interactions between multiple drivers. For example, synoptic meteorological conditions can produce considerable rainfall-runoff, while also generating wind-driven elevated sea-levels. When these drivers interact in space and time, they can exacerbate flood impacts, a phenomenon known as compound flooding. In this paper, we build a Bayesian Network based on Gaussian copulas to generate the equivalent of 500 years of daily stochastic boundary conditions for a coastal watershed in Southeast Texas. In doing so, we overcome many of the limitations of conventional univariate approaches and are able to probabilistically represent compound floods caused by riverine and coastal interactions. We model the resulting water levels using a one-dimensional (1D) steady-state hydraulic model and find that flood stages in the catchment are strongly affected by backwater effects from tributary inflows and downstream water levels. By comparing our results against a bathtub modeling approach, we show that simplifying the multivariate dependence between flood drivers can lead to an underestimation of flood impacts, highlighting that accounting for multivariate dependence is critical for the accurate representation of flood risk in coastal catchments prone to compound events.

A Novel Semiparametric Hidden Markov Model for Process Failure Mode Identification

The emitting distributions of a hidden Markov model (HMM) are normally constructed using the cross moments of the process variables. Similar to the mean of a univariate probability distribution, the cross moment is the most fundamental statistic of a multivariate probability distribution, which is not capable of capturing the high-order statistical features of process data. To alleviate this limitation, the high-order equivalence of the cross moment demonstrated in this paper, as the complete dependence structure, is used to construct the emitting distribution for HMM. The complete dependence structure among the process variables is modeled in a Gaussian copula. A semiparametric data transformation is also proposed to ensure the necessary conditions for using a Gaussian copula are met. The final emitting distribution is constructed as a finite mixture of the copula models. The proposed HMM is tested on two industrial studies for performance validation. Note to Practitioners —Gaussian mixture model HMM (GMM-HMM) is an efficient and easy-to-implement tool for mode identification of dynamic industrial processes. These virtues of GMM-HMM mainly come from the use of Gaussian emitting distribution, which allows closed-form derivatives to be computed for the expectation and maximization (EM) estimation of mode parameters. The main novelty of the proposed copula mixture model HMM (SPCMM-HMM) is twofold; its emitting distribution also belongs to the exponential family, thus enabling efficient estimation of parameters through an exact EM procedure; meanwhile, it is also capable of characterizing complex dependence structures among process variables. However, the proposed SPCMM-HMM has more model parameters compared with the GMM-HMM. It excels in situations where the operation of the process system being monitored is highly integrated (with complex process variable interactions) and abundance of training data samples is available. In this paper, the interactions between process variables in both case studies are complex due to the implementation of multiple closed control loops, and a large amount of training data samples can be easily generated from simulation. The performance of the SPCMM-HMM is shown to be consistently better than that of the GMM-HMM under such settings, which are fairly common in modern industrial processes. Nonetheless, if the process system being considered is only designed for a simple operation or there is a scarcity of training data samples, the GMM-HMM is still the preferred method as it is less prone to overfitting.

CVaR distance between univariate probability distributions and approximation problems

The paper defines new distances between univariate probability distributions, based on the concept of the CVaR norm. We consider the problem of approximation of a discrete distribution by some other discrete distribution. The approximating distribution has a smaller number of atoms than the original one. Such problems, for instance, must be solved for generation of scenarios in stochastic programming. The quality of the approximation is evaluated with new distances suggested in this paper. We use CVaR constraints to assure that the approximating distribution has tail characteristics similar to the target distribution. The numerical algorithm is based on two main steps: (i) optimal placement of positions of atoms of the approximating distribution with fixed probabilities; (ii) optimization of probabilities with fixed positions of atoms. These two steps are iterated to find both optimal atom positions and probabilities. Numerical experiments show high efficiency of the proposed algorithms, solved with convex and linear programming.

Offline-Enhanced Reduced Basis Method Through Adaptive Construction of the Surrogate Training Set

The reduced basis method (RBM) is a popular certified model reduction approach for solving parametrized partial differential equations. One critical stage of the offline portion of the algorithm is a greedy algorithm, requiring maximization of an error estimate over parameter space. In practice this maximization is usually performed by replacing the parameter domain continuum with a discrete “training” set. When the dimension of parameter space is large, it is necessary to significantly increase the size of this training set in order to effectively search parameter space. Large training sets diminish the attractiveness of RBM algorithms since this proportionally increases the cost of the offline phase. In this work we propose novel strategies for offline RBM algorithms that mitigate the computational difficulty of maximizing error estimates over a training set. The main idea is to identify a subset of the training set, a “surrogate training set” (STS), on which to perform greedy algorithms. The STS we construct is much smaller in size than the full training set, yet our examples suggest that it is accurate enough to induce the solution manifold of interest at the current offline RBM iteration. We propose two algorithms to construct the STS: our first algorithm, the successive maximization method, is inspired by inverse transform sampling for non-standard univariate probability distributions. The second constructs an STS by identifying pivots in the Cholesky decomposition of an approximate error correlation matrix. We demonstrate the algorithm through numerical experiments, showing that it is capable of accelerating offline RBM procedures without degrading accuracy, assuming that the solution manifold has rapidly decaying Kolmogorov width.

Log-linear models to evaluate direction of effect in binary variables

In this article, we present a new approach to determining direction of effects in binary variables. A variable is considered explanatory if it explains the probability distribution of another variable. In cross-classifications of binary variables, the univariate probability distribution of variables can be considered explained if omitting the univariate effects of this variable does not lead to an ill-fitting model. Directional (non-hierarchical) log-linear models are introduced that allow statements concerning the direction of association in binary data. Cases in which variables of latent linear regression processes are partially observed as binary variables reveal a close conceptual link between the proposed log-linear approach and existing direction of effect methodology for metric variables. A Monte Carlo study is presented that shows that the proposed approach has good power and enables researchers to distinguish the correct model from the incorrect, reverse model. The approach can be extended to multiple explanatory and multiple outcome variables. Empirical data examples from research on aggression development in adolescence illustrate the proposed direction dependence approach.

Uncertainty of Hydrological Drought Characteristics with Copula Functions and Probability Distributions: A Case Study of Weihe River, China

This study investigates the sensitivity and uncertainty of hydrological droughts frequencies and severity in the Weihe Basin, China during 1960–2012, by using six commonly used univariate probability distributions and three Archimedean copulas to fit the marginal and joint distributions of drought characteristics. The Anderson-Darling method is used for testing the goodness-of-fit of the univariate model, and the Akaike information criterion (AIC) is applied to select the best distribution and copula functions. The results demonstrate that there is a very strong correlation between drought duration and drought severity in three stations. The drought return period varies depending on the selected marginal distributions and copula functions and, with an increase of the return period, the differences become larger. In addition, the estimated return periods (both co-occurrence and joint) from the best-fitted copulas are the closet to those from empirical distribution. Therefore, it is critical to select the appropriate marginal distribution and copula function to model the hydrological drought frequency and severity. The results of this study can not only help drought investigation to select a suitable probability distribution and copulas function, but are also useful for regional water resource management. However, a few limitations remain in this study, such as the assumption of stationary of runoff series.

The Metalog Distributions

The metalog distributions constitute a new system of continuous univariate probability distributions designed for flexibility, simplicity, and ease/speed of use in practice. The system is comprised of unbounded, semibounded, and bounded distributions, each of which offers nearly unlimited shape flexibility compared to previous systems of distributions. Explicit shape-flexibility comparisons are provided. Unlike other distributions that require nonlinear optimization for parameter estimation, the metalog quantile functions and probability density functions have simple closed-form expressions that are quantile parameterized linearly by cumulative-distribution-function data. Applications in fish biology and hydrology show how metalogs may aid data and distribution research by imposing fewer shape constraints than other commonly used distributions. Applications in decision analysis show how the metalog system can be specified with three assessed quantiles, how it facilities Monte Carlo simulation, and how applying it aided an actual decision that would have been made wrongly based on commonly used discrete methods. This work is licensed under a Creative Commons Attribution 4.0 International License. You are free to copy, distribute, transmit and adapt this work, but you must attribute this work as “Decision Analysis. Copyright © 2016 The Author(s). https://doi.org/10.1287/deca.2016.0338 , used under a Creative Commons Attribution License: https://creativecommons.org/licenses/by/4.0/ .”

Inventory management in food companies with statistically dependent demand

Purpose The objective of this paper is to propose a methodology based on random demand inventory models and dependence structures for a set of raw materials, referred to as “components”, used by food services that produce food rations referred to as “menus”. Design/methodology/approach The contribution margins of food services that produce menus are optimised using random dependent demand inventory models. The statistical dependence between the demand for components and/or menus is incorporated into the model through the multivariate Gaussian (or normal) distribution. The contribution margins are optimised by using probabilistic inventory models for each component and stochastic programming with a differential evolution algorithm. Findings When compared to the non-optimised system previously used by the company, the (average) expected contribution margin increases by 18.32 per cent when using a continuous review inventory model for groceries and uniperiodic models for perishable components (optimised system). Research limitations/implications The multivariate modeling can be improved by using (a) other non-Gaussian (marginal) univariate probability distributions, by means of the copula method that considers more complex statistical dependence structures; (b) time-dependence, through autoregressive time-series structures and moving average; (c) random modelling of lead-time; and (d) demands for components with values equal to zero using zero-inflated or adjusted probability distribution. Practical implications Professional management of the supply chain allows the users to register data concerning component identification, demand, and stock levels to subsequently be used with the proposed methodology, which must be implemented computationally. Originality/value The proposed multivariate methodology allows it to describe demand dependence structures through inventory models applicable to components used to produce menus in food services.

The Continuous Hidden Threshold Mixed Skew-Symmetric Distribution

This paper explores a way to construct a new family of univariate probability distributions where the parameters of the distribution capture the dependence between the variable of interest and the continuous latent state variable (the regime). The distribution nests two well known families of distributions, namely, the skew normal family of Azzalini (1985) and a mixture of two Arnold et al. (1993) distribution. We provide a stochastic representation of the distribution which enables the user to easily simulate the data from the underlying distribution using generated uniform and normal variates. We also derive the moment generating function and the moments. The distribution comprises eight free parameters that make it very flexible. This flexibility allows the user to capture many stylized facts about the data such as the regime dependence, the asymmetry and fat tails as well as thin tails.

The Continuous Hidden threshold Mixed Skew-Symmetric Distribution

This paper explores a way to construct a new family of univariate probability distributions where the parameters of the distribution capture the dependence between the variable of interest and the continuous latent state variable (the regime). The distribution nests two well known families of distributions, namely, the skew normal family of Azzalini (1985) and a mixture of two Arnold et al. (1993) distribution. We provide a stochastic representation of the distribution which enables the user to easily simulate the data from the underlying distribution using generated uniform and normal variates. We also derive the moment generating function and the moments. The distribution comprises eight free parameters that make it very flexible. This flexibility allows the user to capture many stylized facts about the data such as the regime dependence, the asymmetry and fat tails as well as thin tails.

Candidate Distributions for Climatological Drought Indices (SPI and SPEI)

AbstractThe Standardized Precipitation Index (SPI), a well‐reviewed meteorological drought index recommended by the World Meteorological Organization (WMO), and its more recent climatic water balance variant, the Standardized Precipitation‐Evapotranspiration Index (SPEI), both rely on selection of a univariate probability distribution to normalize the index, allowing for comparisons across climates. Choice of an improper probability distribution may impart bias to the index values, exaggerating or minimizing drought severity. This study compares a suite of candidate probability distributions for use in SPI and SPEI normalization using the 0.5° × 0.5° gridded Watch Forcing Dataset (WFD) at the continental scale, focusing on Europe. Several modifications to the SPI and SPEI methodology are proposed, as well as an updated procedure for evaluating SPI/SPEI goodness of fit based on the Shapiro–Wilk test. Candidate distributions for SPI organize into two groups based on their ability to model short‐term accumulation (1–2 months) or long‐term accumulation (>3 months). The two‐parameter gamma distribution is recommended for general use when calculating SPI across all accumulation periods and regions within Europe, in agreement with previous studies. The generalized extreme value distribution is recommended when computing the SPEI, in disagreement with previous recommendations.

Comparison of mixed distribution with EV1 and GEV components for analyzing hydrologic data containing outlier

An outlier is a very large or very small value that does not follow the general trend of a given data set. Outliers in rainfall data cause uncertainty in water engineering studies and estimated design events. As such, an additional mathematical tool for dealing with outliers is needed. One of the main issues in hydrologic frequency analysis is the problem of mixed distributions or multiple populations in hydrologic time series. Univariate probability distributions are unsuitable for data sets with outliers, therefore three mixed distributions (mixed Gumbel, mixed GEV, EV1–GEV) were used in this paper. The mixed Gumbel distribution was found to be the best distribution to fit to the 24-h annual maximum rainfall data at all of the rainfall gauging stations used in this study, on the basis of the minimum standard error of fit.

A web-based tool for eliciting probability distributions from experts

We present a web-based probability distribution elicitation tool: The MATCH Uncertainty Elicitation Tool. The Tool is designed to help elicit probability distributions about uncertain model parameters from experts, in situations where suitable data is either unavailable or sparse. The Tool is free to use, and offers five different techniques for eliciting univariate probability distributions. A key feature of the Tool is that users can log in from different sites and view and interact with the same graphical displays, so that expert elicitation sessions can be conducted remotely (in conjunction with tele- or videoconferencing). This will make probability elicitation easier in situations where it is difficult to interview experts in person. Even when conducting elicitation remotely, interviewers will be able to follow good elicitation practice, advise the experts, and provide instantaneous feedback and assistance.