scholarly journals A Method for Simulating Nonnormal Distributions with Specified L-Skew, L-Kurtosis, and L-Correlation

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Todd C. Headrick ◽  
Mohan D. Pant
Keyword(s):  
Monte Carlo ◽  
Pearson Correlation ◽  
Theoretical Development ◽  
Nonnormal Distributions ◽  
Heavy Tailed Distributions ◽  
Moment Estimates ◽  
Monte Carlo Studies ◽  
Asymmetric Distributions ◽  
Heavy Tailed ◽  
Primary Focus

This paper introduces two families of distributions referred to as the symmetric κ and asymmetric - distributions. The families are based on transformations of standard logistic pseudo-random deviates. The primary focus of the theoretical development is in the contexts of L-moments and the L-correlation. Also included is the development of a method for specifying distributions with controlled degrees of L-skew, L-kurtosis, and L-correlation. The method can be applied in a variety of settings such as Monte Carlo studies, simulation, or modeling events. It is also demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when moderate-to-heavy-tailed distributions are of concern.

scholarly journals Characterizing Tukey and -Distributions through -Moments and the -Correlation

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Todd C. Headrick ◽  
Mohan D. Pant
Keyword(s):  
Monte Carlo ◽  
Risk Analysis ◽  
Extreme Events ◽  
Pearson Correlation ◽  
Simulation Studies ◽  
Nonnormal Distributions ◽  
Heavy Tailed Distributions ◽  
Moment Estimates ◽  
Asymmetric Distributions ◽  
Heavy Tailed

This paper introduces the Tukey family of symmetric and asymmetric -distributions in the contexts of univariate -moments and the -correlation. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of -skew, -kurtosis, and -correlations. The procedure can be applied in a variety of settings such as modeling events (e.g., risk analysis, extreme events) and Monte Carlo or simulation studies. Further, it is demonstrated that estimates of -skew, -kurtosis, and -correlation are substantially superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when heavy-tailed distributions are of concern.

scholarly journals A Doubling Method for the Generalized Lambda Distribution

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Todd C. Headrick ◽  
Mohan D. Pant
Keyword(s):  
Monte Carlo ◽  
Pearson Correlation ◽  
Relative Bias ◽  
Simulation Studies ◽  
Generalized Lambda Distribution ◽  
New Family ◽  
Heavy Tailed Distributions ◽  
Moment Estimates ◽  
Doubling Method ◽  
Heavy Tailed

This paper introduces a new family of generalized lambda distributions (GLDs) based on a method of doubling symmetric GLDs. The focus of the development is in the context of L-moments and L-correlation theory. As such, included is the development of a procedure for specifying double GLDs with controlled degrees of L-skew, L-kurtosis, and L-correlations. The procedure can be applied in a variety of settings such as modeling events and Monte Carlo or simulation studies. Further, it is demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are substantially superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when heavy tailed distributions are of concern.

A Monte Carlo Investigation of the Fisher Z Transformation for Normal and Nonnormal Distributions

2000 ◽  
Vol 87 (3_suppl) ◽  
pp. 1101-1114 ◽  
Author(s):  
Kenneth J. Berry ◽  
Paul W. Mielke
Keyword(s):  
Monte Carlo ◽  
Correlation Coefficient ◽  
Bivariate Normal Distribution ◽  
Small Samples ◽  
Bivariate Distributions ◽  
Nonnormal Distributions ◽  
Monte Carlo Investigation ◽  
Sample Correlation ◽  
Heavy Tailed ◽  
Z Transformation

The Fisher transformation of the sample correlation coefficient r (1915, 1921) and two related techniques by Gayen (1951) and Jeyaratnam (1992) are examined for robustness to nonnormality. Monte Carlo analyses compare combinations of sample sizes and population parameters for seven bivariate distributions. The Fisher, Gayen, and Jeyaratnam approaches are shown to provide useful results for a bivariate normal distribution with any population correlation coefficient ρ and for nonnormal bivariate distributions when ρ = 0. In contrast, the techniques are virtually useless for nonnormal bivariate distributions when ρ#0.0. Surprisingly, small samples are found to provide better estimates than large samples for skewed and symmetric heavy-tailed bivariate distributions.

scholarly journals Bayesian Estimation of Random Parameter Models of Responses with Normal and Skew-t Distibutions Evidence from Monte Carlo Simulation

2018 ◽  
Vol 24 (1) ◽  
pp. 27-50
Author(s):  
Mohammad Masjkur ◽  
Henk Folmer
Keyword(s):  
Monte Carlo ◽  
Dose Response ◽  
Random Parameter ◽  
Response Models ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Response Modeling ◽  
T Distribution ◽  
Valuation Criteria ◽  
Skew Normal

Random parameter models have been found to outperform xed pa-rameter models to estimate dose-response relationships with independent errors. Amajor restriction, however, is that the responses are assumed to be normally andsymmetrically distributed. The purpose of this paper is to analyze Bayesian infer-ence of random parameter response models in the case of independent responseswith normal and skewed, heavy-tailed distributions by way of Monte Carlo simu-lation. Three types of Bayesian estimators are considered: one applying a normal,symmetrical prior distribution, a second applying a Skew-normal prior and, a thirdapplying a Skew-t-distribution. We use the relative bias (RelBias) and Root MeanSquared Error (RMSE) as valuation criteria. We consider the commonly applied lin-ear Quadratic and the nonlinear Spillman-Mitscherlich dose-response models. Onesimulation examines the performance of the estimators in the case of independent,normally and symmetrically distributed responses; the other in the case of indepen-dent responses following a heavy-tailed, Skew-t-distribution. The main nding isthat the estimator based on the Skew-t prior outperforms the alternative estima-tors applying the normal and Skew-normal prior for skewed, heavy-tailed data. Fornormal data, the Skew-t prior performs approximately equally well as the Skew-normal and the normal prior. Furthermore, it is more ecient than its alternatives.Overall, the Skew-t prior seems to be preferable to the normal and Skew-normal fordose-response modeling.

scholarly journals A Method for Simulating Burr Type III and Type XII Distributions through -Moments and -Correlations

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Mohan D. Pant ◽  
Todd C. Headrick
Keyword(s):  
Relative Efficiency ◽  
Simulation Studies ◽  
Distribution Fitting ◽  
Fracture Roughness ◽  
Nonnormal Distributions ◽  
Type Iii ◽  
Burr Distributions ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Pearson Correlations

This paper derives the Burr Type III and Type XII family of distributions in the contexts of univariate -moments and the -correlations. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of -skew, -kurtosis, and -correlations. The procedure can be applied in a variety of settings such as statistical modeling (e.g., forestry, fracture roughness, life testing, operational risk, etc.) and Monte Carlo or simulation studies. Numerical examples are provided to demonstrate that -moment-based Burr distributions are superior to their conventional moment-based analogs in terms of estimation and distribution fitting. Evaluation of the proposed procedure also demonstrates that the estimates of -skew, -kurtosis, and -correlation are substantially superior to their conventional product moment-based counterparts of skew, kurtosis, and Pearson correlations in terms of relative bias and relative efficiency—most notably when heavy-tailed distributions are of concern.

scholarly journals BEYOND THE PEARSON CORRELATION: HEAVY-TAILED RISKS, WEIGHTED GINI CORRELATIONS, AND A GINI-TYPE WEIGHTED INSURANCE PRICING MODEL

2017 ◽  
Vol 47 (3) ◽  
pp. 919-942 ◽  
Author(s):  
Edward Furman ◽  
Ričardas Zitikis
Keyword(s):  
Pearson Correlation ◽  
Correlation Coefficients ◽  
Capital Asset Pricing ◽  
Pricing Model ◽  
Insurance Pricing ◽  
Asset Pricing Model ◽  
Research Areas ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
Theoretical Considerations

AbstractGini-type correlation coefficients have become increasingly important in a variety of research areas, including economics, insurance and finance, where modelling with heavy-tailed distributions is of pivotal importance. In such situations, naturally, the classical Pearson correlation coefficient is of little use. On the other hand, it has been observed that when light-tailed situations are of interest, and hence when both the Gini-type and Pearson correlation coefficients are well defined and finite, these coefficients are related and sometimes even coincide. In general, understanding how these correlation coefficients are related has been an illusive task. In this paper, we put forward arguments that establish such a connection via certain regression-type equations. This, in turn, allows us to introduce a Gini-type weighted insurance pricing model that works in heavy-tailed situations and thus provides a natural alternative to the classical capital asset pricing model. We illustrate our theoretical considerations using several bivariate distributions, such as elliptical and those with heavy-tailed Pareto margins.

scholarly journals A Logistic -Moment-Based Analog for the Tukey -, , , and - System of Distributions

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Todd C. Headrick ◽  
Mohan D. Pant
Keyword(s):  
Monte Carlo ◽  
Standard Error ◽  
Relative Bias ◽  
Relative Standard Error ◽  
Moment Estimators ◽  
Heavy Tailed Distributions ◽  
Relative Standard ◽  
Standard Normal ◽  
Heavy Tailed

This paper introduces a standard logistic L-moment-based system of distributions. The proposed system is an analog to the standard normal conventional moment-based Tukey g-h, g, h, and h-h system of distributions. The system also consists of four classes of distributions and is referred to as (i) asymmetric -, (ii) log-logistic , (iii) symmetric , and (iv) asymmetric -. The system can be used in a variety of settings such as simulation or modeling events—most notably when heavy-tailed distributions are of interest. A procedure is also described for simulating -, , , and - distributions with specified L-moments and L-correlations. The Monte Carlo results presented in this study indicate that estimates of L-skew, L-kurtosis, and L-correlation associated with the -, , , and - distributions are substantially superior to their corresponding conventional product-moment estimators in terms of relative bias and relative standard error.

Probability and Random Processes

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Random Processes ◽  
Distribution Functions ◽  
Basic Logic ◽  
Heavy Tailed Distributions ◽  
Random Components ◽  
Classical Distribution ◽  
Heavy Tailed ◽  
Basic Facts ◽  
Stochastic Events ◽  
Stochastic Event

Phenomena, systems, and processes are rarely purely deterministic, but contain stochastic,probabilistic, or random components. For that reason, a probabilistic descriptionof most phenomena is necessary. Probability theory provides us with the tools for thistask. Here, we provide a crash course on the most important notions of probabilityand random processes, such as odds, probability, expectation, variance, and so on. Wedescribe the most elementary stochastic event—the trial—and develop the notion of urnmodels. We discuss basic facts about random variables and the elementary operationsthat can be performed on them. We learn how to compose simple stochastic processesfrom elementary stochastic events, and discuss random processes as temporal sequencesof trials, such as Bernoulli and Markov processes. We touch upon the basic logic ofBayesian reasoning. We discuss a number of classical distribution functions, includingpower laws and other fat- or heavy-tailed distributions.

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