Generalized Beta Distribution Research Articles

A generalized beta copula with applications in modeling multivariate long-tailed data

This work proposes a new copula class that we call the MGB2 copula. The new copula originates from extracting the dependence function of the multivariate GB2 distribution (MGB2) whose marginals follow the univariate generalized beta distribution of the second kind (GB2). The MGB2 copula can capture non-elliptical and asymmetric dependencies among marginal coordinates and provides a simple formulation for multi-dimensional applications. This new class features positive tail dependence in the upper tail and tail independence in the lower tail. Furthermore, it includes some well-known copula classes, such as the Gaussian copula, as special or limiting cases. To illustrate the usefulness of the MGB2 copula, we build a trivariate MGB2 copula model of bodily injury liability closed claims. Extended GB2 distributions are chosen to accommodate the right-skewness and the long-tailedness of the outcome variables. For the regression component, location parameters with continuous predictors are introduced using a nonlinear additive function. For comparison purposes, we also consider the Gumbel and t copulas, alternatives that capture the upper tail dependence. The paper introduces a conditional plot graphical tool for assessing the validation of the MGB2 copula. Quantitative and graphical assessment of the goodness of fit demonstrate the advantages of the MGB2 copula over the other copulas.

A generalization of the bivariate Beta-Binomial distribution

This paper presents a new bivariate discrete distribution that generalizes the bivariate Beta-Binomial distribution. It is generated by Appell hypergeometric function F 1 and can be obtained as a Binomial mixture with an Exton's Generalized Beta distribution. The model has different marginal distributions which are, together with the conditional distributions, more flexible than the Beta-Binomial distribution. It has non-linear regression curves and is useful for random variables with positive correlation. These features make the model very adequate to fit observed data as the two applications included show.

Bayesian Monte Carlo method for monotonic models applying the Generalized Beta distribution

A novel Bayesian Monte Carlo method for monotonic models (BMCM) is described in this paper. The BMCM method enjoys the advantages of the recently developed method of Dynamic Bounds [1] for the reliability assessment of monotonic models, and incorporates weighted logical dependence between neighboring points of the limit state equation (LSE) as prior information for improving computational speed. The BMCM benefits from the global uncertainty concept that links global and local uncertainties. This integrated approach is superior to other techniques used in safety assessment of monotonic models as it substantially increases the efficiency of Monte Carlo method. The formulation of BMCM is provided in this paper with an example showing its ability to dramatically improve efficiency of simulations. This is achieved by employing prior information obtained from monotonic models and outcomes of the preceding simulations. The theory and numerical algorithms of the BMCM method for multi-dimensional problems, and integration with a probabilistic finite element model, are discussed in this paper. These coupled models are applied to the 17th Street Flood Wall in New Orleans to assess reliability of a flood defence system.

The inclusive and simplified forms of Bayesian interpolation for general and monotonic models using Gaussian and Generalized Beta distributions with application to Monte Carlo simulations

A recently developed Bayesian interpolation method (BI) and its application to safety assessment of a flood defense structure are described in this paper. We use a one-dimensional Bayesian Monte Carlo method (BMC) that has been proposed in (Rajabalinejad 2009) to develop a weighted logical dependence between neighboring points. The concept of global uncertainty is adequately explained and different uncertainty association models (UAMs) are presented for linking the local and global uncertainty. Based on the global uncertainty, a simplified approach is introduced. By applying the global uncertainty, we apply the Guassian error estimation to general models and the Generalized Beta (GB) distribution to monotonic models. Our main objective in this research is to simplify the newly developed BMC method and demonstrate that it can dramatically improve the simulation efficiency by using prior information from outcomes of the preceding simulations. We provide theory and numerical algorithms for the BI method geared to multi-dimensional problems, integrate it with a probabilistic finite element model, and apply the coupled models to the reliability assessment of a flood defense for the 17th Street Flood Wall system in New Orleans.

Order–disorder transition in conflicting dynamics leading to rank–frequency generalized beta distributions

The behavior of rank-ordered distributions of phenomena present in a variety of fields such as biology, sociology, linguistics, finance and geophysics has been a matter of intense research. Often power laws have been encountered; however, their validity tends to hold mainly for an intermediate range of rank values. In a recent publication (Martínez-Mekler et al., 2009 [7]), a generalization of the functional form of the beta distribution has been shown to give excellent fits for many systems of very diverse nature, valid for the whole range of rank values, regardless of whether or not a power law behavior has been previously suggested. Here we give some insight on the significance of the two free parameters which appear as exponents in the functional form, by looking into discrete probabilistic branching processes with conflicting dynamics. We analyze a variety of realizations of these so-called expansion–modification models first introduced by Wentian Li (1989) [10]. We focus our attention on an order–disorder transition we encounter as we vary the modification probability p . We characterize this transition by means of the fitting parameters. Our numerical studies show that one of the fitting exponents is related to the presence of long-range correlations exhibited by power spectrum scale invariance, while the other registers the effect of disordering elements leading to a breakdown of these properties. In the absence of long-range correlations, this parameter is sensitive to the occurrence of unlikely events. We also introduce an approximate calculation scheme that relates this dynamics to multinomial multiplicative processes. A better understanding through these models of the meaning of the generalized beta-fitting exponents may contribute to their potential for identifying and characterizing universality classes.

Univariate input models for stochastic simulation

Techniques are presented for modelling and then randomly sampling many of the continuous univariate probabilistic input processes that drive discrete-event simulation experiments. Emphasis is given to the generalized beta distribution family, the Johnson translation system of distributions, and the Bézier distribution family because of the flexibility of these families to model a wide range of distributional shapes that arise in practical applications. Methods are described for rapidly fitting these distributions to data or to subjective information (expert opinion) and for randomly sampling from the fitted distributions. Also discussed are applications ranging from pharmaceutical manufacturing and medical decision analysis to smart-materials research and health-care systems analysis.

Equilibrium preference free pricing of derivatives under the generalized beta distributions

This paper demonstrates that the risk neutral valuation relationship (RNVR) exists when the aggregate wealth and the underlying variable for derivatives follow a distribution from the family of transformed beta distributions. Specifically, the asset specific pricing kernel (ASPK) is solved for the generalized beta (GB) distribution class, which is extremely flexible to describe various shapes of underlying distributions. With the ASPK in hand, preference free call option formulas are obtained for rescaled and shifted beta distribution of the first kind (RSB1) and for the second kind (RSB2). These distributions include many well known important distributions as special cases. If the preference free formula does not exist under the GB distribution class, then the call price is shown to be numerically calculated without information of preference parameters once the spot price of the underlying is given.

DYNAMIC ORDER SUBMISSION AND HERDING BEHAVIOR IN ELECTRONIC TRADING

AbstractI analyze the dynamic trading behavior of market participants by developing a bivariate modeling framework for describing the arrival process of buy and sell orders in a limit order book. The model contains an extended autoregressive conditional duration model with a flexible generalized Beta distribution to explain the duration process, combined with a dynamic logit model to capture the traders’ order submission strategy. I find that the state of the order book as well as the speed of the order arrival have a significant influence on the order placement, inducing temporal asymmetric market movements.

Distance Distributions in Finite Uniformly Random Networks: Theory and Applications

In wireless networks, knowledge of internode distances is essential for performance analysis and protocol design. When determining distance distributions in random networks, the underlying nodal arrangement is almost universally taken to be a stationary Poisson point process. While this may be a good approximation in some cases, there are also certain shortcomings to this model, such as the fact that, in practical networks, the number of nodes in disjoint areas is not independent. This paper considers a more-realistic network model where a known and fixed number of nodes are independently distributed in a given region and characterizes the distribution of the Euclidean internode distances. The key finding is that, when the nodes are uniformly randomly placed inside a ball of arbitrary dimensions, the probability density function (pdf) of the internode distances follows a generalized beta distribution. This result is applied to study wireless network characteristics such as energy consumption, interference, outage, and connectivity.

Generalized Beta-Generated Distributions

This article introduces generalized beta-generated (GBG) distributions. Sub-models include all classical beta-generated, Kumaraswamy-generated and exponentiated distributions. They are maximum entropy distributions under three intuitive conditions, which show that the classical beta generator skewness parameters only control tail entropy and an additional shape parameter is needed to add entropy to the centre of the parent distribution. This parameter controls skewness without necessarily differentiating tail weights. The GBG class also have tractable properties: we present various expansions for moments, generating function, quantiles, deviations and reliability. The model parameters are estimated by maximum likelihood and the usefulness of the new class is illustrated by means of some real data sets.

Estimation of Peaked Densities Over the Interval [0,1] Using Two-Sided Power Distribution: Application to Lottery Experiments

This paper deals with estimating peaked densities over the interval [0,1] using two-sided power distribution (Kotz, van Dorp, 2004). Such data were encountered in experiments determining certainty equivalents of lotteries (Kontek, 2010). This paper summarizes the basic properties of the two-sided power distribution (TP) and its generalized form (GTP). The GTP maximum likelihood estimator, a result not derived by Kotz and van Dorp, is presented. The TP and GTP are used to estimate certainty equivalent densities in two data sets from lottery experiments. The obtained results show that even a two-parametric TP distribution provides more accurate estimates than the smooth three-parametric generalized beta distribution GBT (Libby, Novick, 1982) in one of the considered data sets. The three-parametric GTP distribution outperforms GBT for these data. The results are, however, the very opposite for the second data set, in which the data are greatly scattered. The paper demonstrates that the TP and GTP distributions may be extremely useful in estimating peaked densities over the interval [0,1] and in studying the relative utility function.

Universality of Rank-Ordering Distributions in the Arts and Sciences

Searching for generic behaviors has been one of the driving forces leading to a deep understanding and classification of diverse phenomena. Usually a starting point is the development of a phenomenology based on observations. Such is the case for power law distributions encountered in a wealth of situations coming from physics, geophysics, biology, lexicography as well as social and financial networks. This finding is however restricted to a range of values outside of which finite size corrections are often invoked. Here we uncover a universal behavior of the way in which elements of a system are distributed according to their rank with respect to a given property, valid for the full range of values, regardless of whether or not a power law has previously been suggested. We propose a two parameter functional form for these rank-ordered distributions that gives excellent fits to an impressive amount of very diverse phenomena, coming from the arts, social and natural sciences. It is a discrete version of a generalized beta distribution, given by f(r) = A(N+1-r)b/ra, where r is the rank, N its maximum value, A the normalization constant and (a, b) two fitting exponents. Prompted by our genetic sequence observations we present a growth probabilistic model incorporating mutation-duplication features that generates data complying with this distribution. The competition between permanence and change appears to be a relevant, though not necessary feature. Additionally, our observations mainly of social phenomena suggest that a multifactorial quality resulting from the convergence of several heterogeneous underlying processes is an important feature. We also explore the significance of the distribution parameters and their classifying potential. The ubiquity of our findings suggests that there must be a fundamental underlying explanation, most probably of a statistical nature, such as an appropriate central limit theorem formulation.

Estimating risk-neutral density with parametric models in interest rate markets

The departure in modelling terms from the log-normal distribution for option pricing has been largely driven by empirical observations on skewness. In recent years, the Weibull and generalized beta distributions have been used to fit the risk-neutral density from option prices. In this article, we also propose the use of the generalized gamma distribution for recovering the risk-neutral density. In terms of complexity, this distribution, having three parameters, falls between the Weibull and generalized beta distributions. New option pricing formulas for European call and put options are derived under the generalized gamma distribution. The empirical evidence based on a set of interest rate derivatives data indicates that this distribution is capable of producing the same type of performance as the Weibull, generalized beta, and Burr3 distributions. In addition, we analyze the effect of July 2005 bombings in London on interest rate markets under the best fitting distribution. Our results indicate that there was very little impact on the volatility of these markets.

MULTI-CHANNEL FUZZY QUEUING SYSTEMS AND MEMBERSHIP FUNCTIONS OF RELATED FUZZY SERVICES AND FUZZY INTER-ARRIVAL TIMES

Queuing models need well defined knowledge on arrivals and service times. However, in real applications, because of some measurement errors or some loss of information, it is hard to achieve deterministic knowledge. Non-deterministic knowledge interferes or complicates analysis of the queuing model. Additionally, when the customers are asked about their impressions on waiting times or service times, mostly the answers are linguistic expressions like "I waited too much", "service was fast", and that the responses are. Linguistic statements and ill defined data make the sense of imprecision in the model. In this study, arrivals and service times are defined as fuzzy numbers in order to represent this imprecision. Fuzzy multi-channel queuing systems and membership functions are introduced in defining the arrivals and service times. Besides, a new membership function based on a probability function is studied. Fuzzy queuing characteristics are calculated via different membership functions and the results are compared on simulations. Among models it is found that, Generalized Beta Distribution membership function is the one that minimized the queuing characteristics.

Super-fit control adaptation in memetic differential evolution frameworks

This paper proposes the super-fit memetic differential evolution (SFMDE). This algorithm employs a differential evolution (DE) framework hybridized with three meta-heuristics, each having different roles and features. Particle Swarm Optimization assists the DE in the beginning of the optimization process by helping to generate a super-fit individual. The two other meta-heuristics are local searchers adaptively coordinated by means of an index measuring quality of the super-fit individual with respect to the rest of the population. The choice of the local searcher and its application is then executed by means of a probabilistic scheme which makes use of the generalized beta distribution. These two local searchers are the Nelder mead algorithm and the Rosenbrock Algorithm. The SFMDE has been tested on two engineering problems; the first application is the optimal control drive design for a direct current (DC) motor, the second is the design of a digital filter for image processing purposes. Numerical results show that the SFMDE is a flexible and promising approach which has a high performance standard in terms of both final solutions detected and convergence speed.

Analysing liquidity and absorption limits of electronic markets with volume durations

This paper focuses on the liquidity of electronic stock markets applying a sequential estimation approach of models for volume duration with increasing threshold values. A modified ACD model with a Box–Tukey transformation and a flexible generalized beta distribution is proposed to capture the changing cluster structure of duration processes. The estimation results with German XETRA data reveal the market's absorption limit for high volumes of shares, expanding the time costs of illiquidity when trading these quantities.

Explicit expressions for the Leimkuhler curve in parametric families

In this paper we obtain the Leimkuhler curve in the case of some important statistical distributions proposed in the informetrics and econometrics literature. In this way, we complete the previous work of Burrell [Burrell, Q. L. (2005). Symmetry and other transformation features of Lorenz/Leimkuhler representations of informetric data. Information Processing and Management, 41, 1317–1329], where several open problems were stated. To do this, we use a recent and general definition of the Leimkuhler curve proposed by Sarabia [Sarabia, J. M. (2008a). A general definition of the Leimkuhler curve. Journal of Informetrics, 2, 156-163], and a new representation of the Leimkuhler curve in terms of the first-moment distribution of the population. Specifically, we obtain the Leimkuhler curve of the following distributions: classical and exponentiated Pareto distributions; three-parameter lognormal distribution; generalized gamma distribution, which includes to the exponential and classical gamma distributions among others; generalized beta distribution of the first kind and generalized beta distribution of the second kind, which includes as particular or limiting cases next important families like beta distribution of the second kind, Singh–Maddala, Dagum, Fisk or Lomax distributions. All the obtained Leimkuhler curves can be computed easily.

Equilibrium Preference Free Pricing of Derivatives Under Generalized Beta Distributions

This paper presents preference free derivatives pricing methodology in the representative agent framework when the aggregate wealth of economy and the underlying variable for derivative contracts follow a correlated transformed beta distributions. As an important example, the marginal distribution is specified as the generalized beta (GB) distribution, which is characterized by six parameters. To demonstrate the existence of the risk neutral valuation relationship in such an economy, we present the preference free closed form call option formula for rescaled and shifted beta distribution of the first kind (RSB1) and second kind (RSB2). As a special case of the RSB2 formula, Bookstaber and McDonald (1985)'s result is also obtained.

Estimating Trends in U.S. Income Inequality Using the Current Population Survey: The Importance of Controlling for Censoring

Using internal and public use March Current Population Survey (CPS) data, we analyze trends in US income inequality (1975-2004). We find that the upward trend in income inequality prior to 1993 significantly slowed thereafter once we control for top coding in the public use data and censoring in the internal data. Because both series do not capture trends at the very top of the income distribution, we use a multiple imputation approach in which values for censored observations are imputed using draws from a Generalized Beta distribution of the Second Kind (GB2) fitted to internal data. Doing so, we find income inequality trends similar to those derived from unadjusted internal data. Our trend results are generally robust to the choice of inequality index, whether Gini coefficient or other commonly-used indices. When we compare our best estimates of the income shares held by the richest tenth with those reported by Piketty and Saez (2003), our trends fairly closely match their trends, except for the top 1 percent of the distribution. Thus, we argue that if United States income inequality has been substantially increasing since 1993, such increases are confined to this very high income group.

Modelling Spikes in Electricity Prices*

During periods of market stress, electricity prices can rise dramatically. Electricity retailers cannot pass these extreme prices on to customers because of retail price regulation. Improved prediction of these price spikes therefore is important for risk management. This paper builds a time‐varying‐probability Markov‐switching model of Queensland electricity prices, aimed particularly at forecasting price spikes. Variables capturing demand and weather patterns are used to drive the transition probabilities. Unlike traditional Markov‐switching models that assume normality of the prices in each state, the model presented here uses a generalised beta distribution to allow for the skewness in the distribution of electricity prices during high‐price episodes.