A product Pareto distribution

The Pareto distributions are becoming increasing prominent in several applied areas. In this note, a new Pareto distribution is introduced. It takes the form of the product of two Pareto probability density functions. Various structural properties of this distribution are derived, including its cumulative distribution function, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, method of moments estimates, maximum likelihood estimates and the Fisher information matrix. The calculations involve the use of several special functions.     

Information theoretic methods of spatial model building: A guide to the unbiased estimation of the form of probability distributions2

The methods of maximum entropy and minimum information are employed in order to provide unbiased a priori estimates of the form of probability distributions, given limited prior knowledge. These a priori estimates, in turn, can be cast in the form of theoretical models of the phenomena of interest; models to be tested against real world data. As such, the methods of maximum entropy and minimum information have relevance to any scientific or social-scientific discipline wherein the mathematical modelling of probability distributions is a concern. At the same time, however, the methods are mathematically difficult and conceptually demanding. This paper therefore presents a pedagogic introduction to the methods of maximum entropy and minimum information in the hope of making them more accessible to those social scientists who might find them relevant to their work. Using a simple example, the paper works through the two methods, one at a time, adding more and more information to the estimation problem as the discussion proceeds. The focus herein is a geographic one, yet the example employed is easily transferable to alternative contexts.     

Minimum kolmogorov distance estimates of parameters and parametrized distributions

Minimum Kolmogorov distance estimates of arbitrary parameters are considered. They are shown to be strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric can be locally uniformly upper-bounded by the induced metric then these estimates are shown to be consistent of ordern ^−1/2. Similar results are proved for minimum Kolmogorov distance estimates of densities from parametrized families where the consistency is considered in theL ₁-norm. The presented conditions for the existence, consistency, and consistency of ordern ^−1/2 are much weaker than those established in the literature for estimates with similar properties. It is shown that these assumptions are satisfied e.g. by all location and scale models with parent distributions different from Dirac, and by all standard exponential models. Supported by the scientific exchange program between the Hungarian Academy of Sciences and the Royal Belgian Academy of Sciences, and by GACR grant 201/93/0232.     

A general recurrence relation for moments of order statistics in a class of probability distributions and characterizations

Summary In a class of distribution functions, including exponential, power function, Pareto, Lomax, and logistic distributions, a general recurrence relation for moments of order statistics is given. The validity of this identity for certain constants and some sequence of order statistics leads to characterizations of probability distributions. Several recurrence relations and characterization results known from the literature are particular cases of the theorems stated.     

Moment inequalities for order statistics with applications to characterizations of distributions

General inequalities of Hölder type between moments of order statistics and moments of record values respectively are derived. Special choices of the involved sample sizes and ranks and discussions of when equality is attained in these inequalities yield several characterizations of well known distributions, such as the uniform, polynomial, Pareto, reflected Pareto, exponential, Weibull distribution and some others.     

Model comparison of coordinate-free multivariate skewed distributions with an application to stochastic frontiers

We consider classes of multivariate distributions which can model skewness and are closed under orthogonal transformations. We review two classes of such distributions proposed in the literature and focus our attention on a particular, yet quite flexible, subclass of one of these classes. Members of this subclass are defined by affine transformations of univariate (skewed) distributions that ensure the existence of a set of coordinate axes along which there is independence and the marginals are known analytically. The choice of an appropriate m-dimensional skewed distribution is then restricted to the simpler problem of choosing m univariate skewed distributions. We introduce a Bayesian model comparison setup for selection of these univariate skewed distributions. The analysis does not rely on the existence of moments (allowing for any tail behaviour) and uses equivalent priors on the common characteristics of the different models. Finally, we apply this framework to multi-output stochastic frontiers using data from Dutch dairy farms.     

Universal Pareto dominance and welfare for plausible utility functions

We study Pareto efficiency in a setting that involves two kinds of uncertainty: Uncertainty over the possible outcomes is modeled using lotteries whereas uncertainty over the agents’ preferences over lotteries is modeled using sets of plausible utility functions. A lottery is universally Pareto undominated if there is no other lottery that Pareto dominates it for all plausible utility functions. We show that, under fairly general conditions, a lottery is universally Pareto undominated iff it is Pareto efficient for some vector of plausible utility functions, which in turn is equivalent to affine welfare maximization for this vector. In contrast to previous work on linear utility functions, we use the significantly more general framework of skew-symmetric bilinear (SSB) utility functions as introduced by Fishburn (1982). Our main theorem generalizes a theorem by Carroll (2010) and implies the ordinal efficiency welfare theorem. We discuss three natural classes of plausible utility functions, which lead to three notions of ordinal efficiency, including stochastic dominance efficiency, and conclude with a detailed investigation of the geometric and computational properties of these notions.     

Approximate confidence and tolerance limits for the discrete Pareto distribution for characterizing extremes in count data

Statistical tolerance intervals for discrete distributions are widely employed for assessing the magnitude of discrete characteristics of interest in applications like quality control, environmental monitoring, and the validation of medical devices. For such data problems, characterizing extreme counts or outliers is also of considerable interest. These applications typically use traditional discrete distributions, like the Poisson, binomial, and negative binomial. The discrete Pareto distribution is an alternative yet flexible model for count data that are heavily right‐skewed. Our contribution is the development of statistical tolerance limits for the discrete Pareto distribution as a strategy for characterizing the extremeness of observed counts in the tail. We discuss the coverage probabilities of our procedure in the broader context of known coverage issues for statistical intervals for discrete distributions. We address this issue by applying a bootstrap calibration to the confidence level of the asymptotic confidence interval for the discrete Pareto distribution's parameter. We illustrate our procedure on a dataset involving cyst formation in mice kidneys.     

One-stage structural models to explain city size

Several papers have estimated the parameters of Pareto distributions for city sizes in different countries, but only one has attempted to explain the differing magnitudes of these parameters with a set of country-specific explanatory variables. While it is reassuring that there has been some research which advances beyond simple “curve-fitting” to explore the determinants of city size distributions, the existing research uses a two-stage OLS method which yields invalid second-stage standard errors (and, consequently, questionable hypothesis tests). In this paper, we develop candidate one-stage structural models with normal and non-normal errors which accommodate truncated size distributions, potentially Pareto-like shapes, and city-level variables. In general, these new models are nonlinear in parameters. We illustrate with data on U.S. urban areas.     

Characterization of optima in smooth Pareto economic systems

Simple techniques of calculus and geometry are used to study and characterize the optima of pure exchange economies in which the utility functions are smooth but not necessarily convex. It is also shown how one can reduce the problem of optimizing p functions on the manifold of states to that of maximizing a single function on a submanifold of this space. Two models are described: one in which a person cannot trade to an optimum unless he starts at one; and one in which a person cannot even get near a local Pareto optimum along continuous ‘trade curves’ from most initial distributions. Finally, the set of optima is described for a generic set of utility mappings.     

Yet another breakdown point notion: EFSBP

The breakdown point in its different variants is one of the central notions to quantify the global robustness of a procedure. We propose a simple supplementary variant which is useful in situations where we have no obvious or only partial equivariance: Extending the Donoho and Huber (The notion of breakdown point, Wadsworth, Belmont, 1983) Finite Sample Breakdown Point?, we propose the Expected Finite Sample Breakdown Point to produce less configuration-dependent values while still preserving the finite sample aspect of the former definition. We apply this notion for joint estimation of scale and shape (with only scale-equivariance available), exemplified for generalized Pareto, generalized extreme value, Weibull, and Gamma distributions. In these settings, we are interested in highly-robust, easy-to-compute initial estimators; to this end we study Pickands-type and Location-Dispersion-type estimators and compute their respective breakdown points.     

Characterizations of the Pareto and related distributions

In this note, a class of Pareto distributions is characterized based on the Shannon entropy of k-record statistics. As a consequence of that characterizations of the uniform and exponential distributions are given. Received: October 1999     

Multivariate regression models for panel data

The paper examines the relationship between heterogeneity bias and strict exogeneity in a distributed lag regression of y on x. The relationship is very strong when x is continuous, weaker when x is discrete, and non-existent as the order of the distributed lag becomes infinite. The individual specific random variables introduce nonlinearity and heteroskedasticity; so the paper provides an appropriate framework for the estimation of multivariate linear predictors. Restrictions are imposed using a minimum distance estimator. It is generally more efficient than the conventional estimators such as quasi-maximum likelihood. There are computationally simple generalizations of two- and three-stage least squares that achieve this efficiency gain. Some of these ideas are illustrated using the sample of Young Men in the National Longitudinal Survey. The paper reports regressions on the leads and lags of variables measuring union coverage, SMSA, and region. The results indicate that the leads and lags could have been generated just by a random intercept. This gives some support for analysis of covariance type estimates; these estimates indicate a substantial heterogeneity bias in the union, SMSA, and region coefficients.     

Quasi goodness of fit tests for lifetime distributions

In this paper we propose a general method for the construction of tests that can be used for testing goodness of fit of lifetime distributions. The method is the following: first find an identity which holds for the survival function or the cumulative hazard function of the null distribution. Then replace the function by a consistent estimate. The resulting statistic is asymptotically normal. Estimating its asymptotic variance then gives a test statistic which is underH ₀ asymptotically chi². The method can be used for randomly (right) censored and single type-I (right) censored data. We apply this method to the following distributions: Weibull, Log-logistic, Log-normal, Half-normal, Rayleigh, Gompertz, Pareto.     

On elementary functions of Pareto variables

Pareto variables are widely used. It is useful to be able to obtain the distribution of some simple functions of Pareto variables in a convenient manner. The Mellin transform with its convolution and exponentiation properties is utilized to that end. Specifically, expressions are written for products, quotients, and sums of products of Pareto variables. These include the distribution of the geometric mean and the product of minimum values of Pareto variables.The Office of Naval Research partially supported the work under Contract No. N000-14-75-C-0254.     

Bayesian inference for pareto populations

Bayesian techniques for samples from classical, generalized and multivariate Pareto distributions are described. We place emphasis on choosing proper prior distributions that do not lead to anomalous posterior densities.     

Early warning systems for currency crises: The Turkish case

Different severe financial crises episodes occurred in the Turkish economy in the last two decades. These crises led to severe economic and social consequences for Turkey in terms of increasing interest rates, large reserves losses, considerable currency depreciations, high output losses and high unemployment rates. This paper aims to illustrate the essential determinants of these crises by developing a multivariate logit model which estimates the predictive ability of sixteen economic and financial indicators in a sample that covers the period from January 1990 to December 2008. The empirical findings show that the Turkish crises are mainly due to excessive fiscal deficits, high money supply growths, sharp rises in short-term external debt, growing riskiness of the banking system (in particular currency and liquidity mismatches), and external adverse shocks.     

Planning for education in the health services: a multicriterion approach.

Administrators are often confronted with problems for which there exist several distinct measures of success. Such problems can be expressed in terms of linear programming models with several linear “criterion” functions instead of a single objective function. Although a variety of techniques are available for the solution of multicriterion problems, there exists a need for one which does not assume technical sophistication on the part of the decision maker and which provides valid solutions with minimum effort. “Efficient Manifold Presentation”, the approach used here, is based on the concept that the ideal solution must be a Pareto optimal solution. A method for finding an expression for the finite set of all Pareto optimal solutions to a linear program with multiple linear criteria is presented. Two processes are involved; first, the discovery of all Pareto optimal vertices of the feasible region, and secondly a grouping of these into sets each of which defines a convex polyhedron of Pareto optimal possibilities. Alternate versions of the second process are suggested for use under varying circumstances. An example of the applicability of the method for modeling enrollment and staffing policy in an educational institution is provided.     

Understanding the economic determinants of the severity of operational losses: A regularized generalized Pareto regression approach

We investigate a novel database of 10,217 extreme operational losses from the Italian bank UniCredit. Our goal is to shed light on the dependence between the severity distribution of these losses and a set of macroeconomic, financial, and firm‐specific factors. To do so, we use generalized Pareto regression techniques, where both the scale and shape parameters are assumed to be functions of these explanatory variables. We perform the selection of the relevant covariates with a state‐of‐the‐art penalized‐likelihood estimation procedure relying on L₁‐penalty terms. A simulation study indicates that this approach efficiently selects covariates of interest and tackles spurious regression issues encountered when dealing with integrated time series. Lastly, we illustrate the impact of different economic scenarios on the requested capital for operational risk. Our results have important implications in terms of risk management and regulatory policy.     

The size distribution of cities: An examination of the Pareto law and primacy

This paper examines the Pareto and primacy measures of the size distribution of cities. The mean Pareto exponent for a sample of 44 countries is 1.136, somewhat greater than the exponent of one implied by the rank-size rule. We find that value of the Pareto exponent is quite sensitive to the definition of the city and the choice of city sample size. The significance of non-linear terms in variants of the Pareto distribution also indicate that the rank-size rule is only a first approximation to a complete characterization of the size distribution of cities within a country. The relatively low correlation between primacy and Pareto measures confirms the need for a variety of measures of city size distributions. This paper also suggests that large cities are growing faster than small cities in most of the countries in our sample. This is indicated by the positive coefficient on the first non-linear term introduced into the Pareto equation. Finally, variations in the Pareto exponent and measures of primacy are partly explained by economic, demographic, and geographic factors.