Distribution Models Theory

The following sections are included:

• Preface

• Contents

This paper presents a new two parameter family of continuous distribution on a bounded domain which has an elevated but finite density value at its lower bound. Such a characteristic appears to be useful, for example, when representing income distributions at lower income ranges. The family generalizes the one parameter Topp and Leone distribution originated in the 1950's and recently rediscovered. The family of beta distributions has been used for modeling bounded income distribution phenomena, but it only allows for an infinite and zero density values at its lower bound, and a constant density of 1 in case of its uniform member. The proposed family alleviates this apparent jump discontinuity at the lower bound. The U.S. Income distribution data for the year 2001 is used to fit distributions for Caucasian (Non-Hispanic), Hispanic and African-American populations via a maximum likelihood procedure. The results reveal stochastic ordering when comparing the Caucasian (Non-Hispanic) income distribution to that of the Hispanic or African-American population. The latter indicates that although substantial advances have reportedly been made in reducing the income distribution gap amongst different ethnic groups in the U.S. during the last 20 years or so, these differences still exist.

This paper is based in the MTDF methodology, which lies in obtaining the value of an asset from value of a specific index (Ballestero, 1973). The topics of this paper are to apply this methodology in the case of two indexes under uncertainty, the construction of a copula FGM using marginal TSP given the classical values (a, m, b) and the application in an empirical case. Under an uncertainty environment a high correlation exists between the indexes, which imply the impossibility to apply the FGM copula that is restricted to the weak correlations case. This article overcomes this disadvantage presenting an alternative that later is applied in a practical case.

In this paper, we discuss a new application of the survival functions in asset pricing from quality indexes. Thus, we propose the valuation method based on the two survival functions (VMTS) to find, under uncertainty, the market value from a quality index. Within this framework, from a one-dimensional quality index, VMTS is equivalent to the valuation method of the two distribution functions (VMTD), which produces loss with respect to the assessments from each component of a multidimensional quality index; nevertheless, VMTS provides profit with respect to these assessments from each component. Finally, we motivate the use of VMTS, as tools for the valuation of an asset, through a practical application on land pricing.

In risk analysis, different procedures based on weighted probability models are usual tools to reduce loss of the assessments in multivariate scenarios. In particular, the weighted distribution functions have been widely used to correct and fit the market value of an asset, through the valuation methods of the two functions, with respect to the appraisals from each component of the multidimensional quality index, in the field of the Valuation Theory.

In this context, the weighting procedures are of interest to find the weights and consequently, to generate these weighted probability models.

The main objective of this paper is to analyze the different weighting techniques used in the Valuation Theory, as well as to propose an alternative to calculate the weights and a new tool to generate these weighted probability models.

First, the well-known weighting techniques to generate the weights are introduced, under both independence and dependence presence of the components of the quality index. Secondly, we expand these weighting techniques by the survival functions, which allows us to generate other weighted probability models.

Likewise, we discuss a new tool to determine the weights of the components of the quality index, modal mean technique, based on the mode values of its marginal distribution functions, which extend the size of the possible weighted probability models to approach the market value of the asset.

Finally, we give an application of these weighting techniques to generate weighted probability models in one example of land pricing, and thus we obtain the assessments of the land property according to each weighted probability model.

The main aim of this paper is to generate compound distributions, discrete or continuous, from Binomial conditional distributions by means of Bayesian techniques. Besides, the authors extend Kowar's paper (1975) by characterizing some discrete and continuous distributions, in the context of some well-known distributions, from the conditional distribution of a random variable (r. v.) and by the linear regression of the latter given the former.

Univariate stochastic and dispersive ordering have extensively been characterized by many authors over the last two decades. Stochastic orderings are also applied in Economic. In particular, it is interesting to compare situations where one utility function (or one distribution function) is obtained from the other by means of some operation that has an economic meaning. To this end, stochastic properties for distributions associated to normal distribution in sampling are studied in this paper. An application of the multivariate dispersion order in the problem of detection and characterization of influential observations in regression analysis is also shown. This problem can often be associated to compare two multivariate t-distributions.

In this paper we apply the generating function to obtain the density of the overall sample. This density is called mix density and is proportional to the geometric mean of the subgroups densities. This approach can be use to measure the polarization when it is understood as an economic distance between distributions. An empirical illustration is provided using the data from the Spanish Household Expenditure Survey corresponding to the regions of Andalucía and Cataluña, elaborated by the Instituto Nacional de Estadística (INE) for the year 1999.

This study introduces a new measure of dissimilarity between distributions, related to Gini's mean difference, and applies it to analyse the convergence between personal income distributions within the 15 EU member states, during the period 1993–2000. According to this measure of dissimilarity, relationships of proximity between these distributions during that period of time constitute the basis of the analysis. Multidimensional scaling techniques are used to construct the temporal trajectories of such distributions in a factor space, optimally reduced for the analysis of their differences. Data are taken from the European Community Household Panel.

Analysis of the evolution of specific fecundity rates, by the age of the mother, i.e. fecundity curves, and their modelling, is of vital importance when we seek to obtain projections or forecasts of the behaviour of this demographic phenomenon. Indeed, on some occasions these estimates do not need to be reasonable from the populational standpoint, but may have the goal of establishing hypothetical scenarios. The present study includes an analysis of the observed data for total births (without taking into account the order of birth) by age and by female population. These data, for the period 1975–2001, were provided by the Statistical Institute of Andalusia (IEA) and were used to construct synthetic fecundity indicators, which are the most basic and the most effective means of accounting for the global behaviour pattern of the phenomenon within a given period. Subsequently, the observed fecundity curves were fitted using a Gammatype distribution. This distribution is one of the most commonly used, for two main reasons: it provides very good quality fits, and the parameters of the distribution are identified perfectly with the indicators of fecundity. Finally, various behaviour hypotheses are proposed, on the basis of the information obtained during the period of analysis.

The present paper is a brief survey of the classes of bivariate distributions with normal and lognormal conditionals. Basic properties including conditional moments, marginal distributions, characterizations, parameterizations, dependence and modality are revised. Estimation and applications of these models are studied. Finally, some extensions of the bivariate conditional normal model are reviewed.

This paper studies the foundations of income inequality measures and its relations with Lorenz curves, the Pigou-Dalton transfer principle and majorization relations among income vectors. So, the historic development of these concepts is surveyed to see how the actual set of properties and axioms was generated, in order to define when an inequality measure has a good perform. Finally, this work includes an analysis studying the problem associated with inequality orders and dominance relations among income vectors.

The aim of this paper is to obtain a bivariate distribution that extends the Bivariate generalized Waring distribution (BGWD) and that preserves some of its properties, such as the partition of the variance into three distinguishable components due to randomness, proneness and liability. Finally, an example in the context of accident theory is included in order to illustrate the versatility of this new distribution.

In a standard Bayesian model, a prior distribution is elicited for the structure parameter in order to obtain an estimate of this unknown parameter. The hierarchical model is a two way Bayesian one which incorporates a hyperprior distribution for some of the hyperparameters of the prior. In this way and under the Poisson-Gamma-Gamma model, a new distribution is obtained by computing the unconditional distribution of the random variable of interest. This distribution seems to provide a better fit to the data, given a policyholders' portfolio. Furthermore, Bayes premiums are thus obtained under a bonus-malus system and solve some of the problems of surcharges which appear in these systems when they are applied in a simple manner.

In studying the behaviour of human phenomena, it is of interest to examine the patterns that remain more or less stable, whether the comparison is made of different populations at a given moment or at different times, or of the same population in different situations. Such regularities have long been modelled, and this has enabled researchers to discover aspects and properties that are inherent to the phenomenon being studied. In the present paper, various techniques, some of which are relatively modern, are applied to the analysis of the empirical distribution of the residuals derived from fitting the Heligman and Pollard curve to mortality data. Firstly, we perform a graphical illustration from the ime perspective (curves fitted over various periods) and then a static one for the ages (i.e. obtaining fits to different ages). The aim of this study is to explore the different distributions of the residuals at each age and thus to evaluate the correspondence between models (such as the Heligman and Pollard curve) and reality (the observed rates of mortality). For this purpose, we use graphical techniques, non-parametric techniques such as kernel smoothing, splines and weighted local fit, and generalised additive models, together with bootstrap sampling techniques to describe distributions of statistical measures of the residuals.

We analyse the dependent relationship between technical efficiency and profitability of commercial banks in Spain, using multivariate techniques, such as factorial, cluster and discrimination analyses. Efficiency measurements are obtained by Data Envelopment Analysis (DEA), incorporating size and management-related variables as inputs and outputs. Efficiency and super-efficiency coefficients are obtained for each bank and conclusions are made concerning the existence of differing levels of profitability according to the efficiency level with which the banks are managed.