James B. McDonald

Some Generalized Functions for the Size Distribution of Income

Many distributions have been used as descriptive models for the size distribution of income. This paper considers two generalized beta distributions which include many of these models as special or limiting cases. These generalized distributions have not been used as models for the distribution of income and provide a unified method of comparing many models previously considered.

Partially Adaptive Estimation of Regression Models via the Generalized T Distribution

This paper considers M-estimators of regression parameters that make use of a generalized functional form for the disturbance distribution. The family of distributions considered is the generalized t (GT), which includes the power exponential or Box-Tiao, normal, Laplace, and t distributions as special cases. The corresponding influence function is bounded and redescending for finite “degrees of freedom.†The regression estimators considered are those that maximize the GT quasi-likelihood, as well as one-step versions. Estimators of the parameters of the GT distribution and the effect of such estimates on the asymptotic efficiency of the regression estimates are discussed. We give a minimum-distance interpretation of the choice of GT parameter estimate that minimizes the asymptotic variance of the regression parameters.

A General Distribution for Describing Security Price Returns

This paper introduces a generalized distribution, called the GB2 distribution, for describing security returns. The distribution is extremely flexible, containing a large number of well-known distributions, such as the lognormal, log-t, and log-Cauchy distribu tions, as special or limiting cases and allowing large, even infinite, higher moments. This flexibility allows a direct representation of different degrees of fat tails in the distribution. The properties of the GB2 make it useful in empirical estimation of security returns and in facilitating the development of option-pricing models and other models that depend on the specification and mathematical manipulation of distributions. Copyright 1987 by the University of Chicago.

Time Series Prediction With Genetic‐Algorithm Designed Neural Networks: An Empirical Comparison With Modern Statistical Models

Neural networks whose architecture is determined by genetic algorithms outperform autoregressive integrated moving average forecasting models in six different time series examples. Refinements to the autoregressive integrated moving average model improve forecasting performance over standard ordinary least squares estimation by 8% to 13%. In contrast, neural networks achieve dramatic improvements of 10% to 40%. Additionally, neural networks give evidence of detecting patterns in data which remain hidden to the autoregression and moving average models. The consequent forecasting potential of neural networks makes them a very promising addition to the variety of techniques and methodologies used to anticipate future movements in time series.

Some Generalized Mixture Distributions with an Application to Unemployment Duration

Compounding or mixture distributions provide a rich class of models for applications ranging from models of heterogeniety, measurement error, distribution of stock returns and income to models of unemployment duration. Some very general mixtures are considered which include many new mixture models and also provide a unified method of organizing and comparing previously considered models as well as a test of heterogeneity. These models are used to analyze CPS unemployment duration data. A heterogeniety interpretation of the mixture models explains the discrepancy between implications of search theory and patterns observed in aggregate unemployment data. Copyright 1987 by MIT Press.

Applications of the GB2 family of distributions in modeling insurance loss processes

Abstract This paper investigates the use of a four parameter family of probability distributions, the generalized beta of the second kind (GB2), for modeling insurance loss processes. The GB2 family includes many commonly used distributions such as the lognormal, gamma and Weibull. The GB2 also includes the Burr and generalized gamma distributions. Members of this family and their inverse distributions have significant potential for improving the distributional fit in many applications involving thin or heavy-tailed distributions. Members of the GB2 family can be generated as mixtures of well-known distributions and provide a model for heterogeneity in claims distributions. Examples are presented which consider models of the distribution of individual and of aggregate losses. The results suggest that seemingly slight differences in modeling the tails can result in large differences in reinsurance premiums and quantiles for the distribution of total insurance losses.

Model selection: some generalized distributions

Many models have been used to represent the distributions of random variables in statistics, engineering, business, and the physical and social science. This paper considers two, four-parameter generalized bea distributions that include nearly all the models actually used as special or limiting cases. Properties and the interrelationships among these distributions are considered. Expressions are reported that facilitate parameter estimation and the analysis of associated means, variances, hazard functions and other distributional characteristics. Estimation procedures corresponding to different data types are considered. Maximum likelihood estimation is used and the value of the likelihood function provides and important criterion for model selection. The relative performance of the various models is compared for several data sets.

The Generalized Beta Distribution as a Model for the Distribution of Income: Estimation of Related Measures of Inequality

The generalized beta (GB) is considered as a model for the distribution of income. It is well known that its special cases include Dagum’s distribution along with the Singh-Maddala distribution. Related measures of inequality such as the Gini Coefficient, Pietra Index, or Theil Index are expressed in terms of the parameters of the generalized beta. This paper also explores the use of numerical integration techniques for calculating inequality indexes. Numerical integration may be useful since in some cases it may be computationally very difficult to evaluate the equations that have been derived or the equations are not available. We provide examples from the distribution of family income in the United States for the year 2000.

Parametric models for partially adaptive estimation with skewed and leptokurtic residuals

Abstract This paper investigates distributional characteristics associated with two flexible parametric distribution which can be used as models for residuals in regression and time series models

Regression models for positive random variables

Abstract Many variables in economics, finance, and other disciplines are positive and nonnormally distributed. This paper considers a regression model with a specification of residuals which allows great flexibility for the distribution of the dependent variable. These distributions include the exponential, Weibull, gamma, lognormal, Burr, generalized gamma, generalized beta type 2, and others as special cases. The specification provides a general parametric form for modeling observed and unobserved heterogeneity as well as including the possibility of increasing, decreasing, constant, ∩- and ∪-shaped hazard functions. The models are used to study the distribution and hazard functions of welfare duration and the relationship between these spells and various demographic characteristics.