Miguel A. Sordo

Universidad de Cádiz | UCA · Department of Statistics and Operative Research

In risk theory, financial asset returns often follow heavy-tailed distributions. Investors and risk managers used to compare risk measures as the value at risk or tail value at risk in order over the whole confidence levels to avoid the exposure to to large risks. In this paper we analyze the comparison between tail values at risk from a confidence...

Given a multivariate random vector, Efron’s marginal monotonicity (EMM) refers to the stochastic monotonicity of the variables given the value of their sum. Recently, based on the notion of total positivity of the joint density of the vector, Pellerey and Navarro (2021) obtained sufficient conditions for EMM when the monotonicity is in terms of the...

Let X be a random variable with distribution function F and let \(\mathcal {F}_X\) be the family of proportional reversed hazard rate distribution functions associated to F. Given the random vector (X, Y) with copula C and respective marginal distribution functions F and \(G\in \mathcal {F}_X,\) we obtain sufficient conditions for the existence of...

We obtain here sufficient conditions for increasing concave order and location independent more riskier order of lower record values based on stochastic comparisons of minimum order statistics. We further discuss stochastic orderings of lower record spacings. In particular, we show that increasing convex order of adjacent spacings between minimum o...

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure...

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an...

Co-risk measures and risk contributions measures are used in portfolio risk analysis to assess and quantify the risk of contagion, given that one or more assets in the portfolio are in distress. In this paper, given two random vectors X and Y that represent two portfolios of n assets (n≥2) and exhibit some kind of positive dependence, we give suffi...

New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the definition and description of these no...

We provide in this paper sufficient conditions for comparing, in terms of the increasing concave order, some income random variables based on linear combinations of order statistics that are relevant in the framework of social welfare. The random variables under study are weighted average incomes of the poorest and, for some particular weights, the...

The preservation of stochastic orders by distortion functions has become a topic of increasing interest in the reliability analysis of coherent systems. The reason of this interest is that the reliability function of a coherent system with identically distributed components can be represented as a distortion function of the common reliability funct...

The tail value at risk at level p, with p ∈ ( 0 , 1 ) , is a risk measure that captures the tail risk of losses and asset return distributions beyond the p quantile. Given two distributions, it can be used to decide which is riskier. When the tail values at risk of both distributions agree, whenever the probability level p ∈ ( 0 , 1 ) , about which...

We present a method for constructing and interpreting weighted premium principles. The method is based on modifying the underlying risk distribution in such a way that the risk-adjusted expected value (or premium) is greater than the expected value of some conveniently chosen function of claims, which defines the insurer’s perception of the risk. U...

In the context of robust Bayesian analysis for multiparameter distributions, we introduce a new class of priors based on stochastic orders, multivariate total positivity of order 2 (M T P 2) and weighted distributions. We provide the new definition, its interpretation and the main properties and we also study the relationship with other classical c...

In this paper, we explore a class of tail variability measures based on distances among proportional hazards models. Tail versions of some well-known variability measures, such as the Gini mean difference, the Wang right tail deviation and the cumulative residual entropy are, up to a scale factor, in this class. These tail variability measures are...

Given a set of n≥2 independent and identically distributed claims, the expected average of the n−i largest claims, with 0≤i≤n−1, is shown to be a distortion risk measure with concave distortion function that can be represented in terms of mixtures of tail value-at-risks with beta mixing distributions. This result allows to interpret the tail value-...

Three functional measures of the shape of univariate distributions are proposed which are consistent with respect to the convex transform order. The first two are weighted tail indices that characterize location-scale families of distributions, whilst the third is a skewness measure. Properties of the new measures are established for various classe...

We study the propagation of uncertainty from a class of priors introduced by Arias-Nicolás et al. [(2016) Bayesian Analysis , 11 (4), 1107–1136] to the premiums (both the collective and the Bayesian), for a wide family of premium principles (specifically, those that preserve the likelihood ratio order). The class under study reflects the prior unce...

Disparities in economic welfare, inequality and poverty across and within countries are of great interest to sociologists, economists, researchers, social organizations and political scientists. Information about these topics is commonly based on surveys. We present a package called rtip that implements techniques based on stochastic dominance to m...

We prove that different conditional distributions can be represented as distorted distributions. These representations are used to obtain stochastic comparisons and bounds for them based on properties of the underlying copula. These properties can be used to explain the meaning of mathematical properties of copulas connecting them with dependence c...

Conditional risk measures (or co-risk measures) and risk contribution measures are increasingly used in actuarial portfolio analysis to evaluate the systemic risk, which is related to the risk that the failure or loss of a component spreads to another component or even to the whole portfolio: while co-risk measures are risk-adjusted versions of mea...

In this paper we introduce the notion of relative spacings. We show the interest of this notion in several contexts like reliability and economy, and we provide several results for the comparison of relative spacings from two populations.

The Lorenz curve is the most widely used graphical tool for describing and comparing inequality of income distributions. In this paper, we show that the elasticity of this curve is an indicator of the effect, in terms of inequality, of a truncation of the income distribution. As an application, we consider tax returns as equivalent to the truncatio...

Actuarial risks and ?nancial asset returns are tipically heavy tailed. In this paper, we introduce two stochastic dominance criteria, called the right tail order and the left tail order, to compare these variables stochastically. The criteria are based on comparisons of expected utilities, for two classes of utility functions that give more weight...

Comparison of residual lives and inactivity times is an important topic in reliability. In this framework, our purpose is twofold. First, we interpret a family of stochastic orders, known in the literature as transform stochastic orderings, in terms of stochastic comparisons among residual lives and inactivity times at quantiles. Second, we introdu...

We provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times...

Se presenta el paquete estadístico rtip en lenguaje de programación R para el cálculo de índices y curvas de interés en el estudio de la pobreza, desigualdad y bienestar. Entre dichos índices se encuentran algunos de los "indicadores laeken" más utilizados. Además se implementan funciones que permiten comparar la dominancia de las curvas calcul...

Risk-adjusted distributions are commonly used in actuarial science to define premium principles. In this paper, we claim that an appropriate risk-adjusted distribution, besides of satisfying other desirable properties, should be well-behaved under conditioning with respect to the original risk distribution. Based on a sequence of such risk-adjusted...

R tools to measure and compare inequality, welfare and poverty using the EU statistics on income and living conditions surveys.

In this paper, we propose a generalization of the increasing convex order to the multivariate setting to compare vectors of risks that accounts for both the marginal impacts and the dependence structures of the vectors. This generalization is suitable for comparing vectors with heterogeneous components and extends some well-known properties of the...

The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the increasing convex order of spaci...

In this paper, we derive a measure of discrepancy based on the Gini’s mean difference to test the null hypothesis that two random variables, which are ordered in a variability-type stochastic order, are equally dispersive versus the alternative that one strictly dominates the other. We describe the test, evaluate its performance under a variety of...

The preservation of stochastic orders under the formation of coherent systems is a relevant topic in the reliability theory. Several properties have been obtained under the assumption of identically distributed components. In this paper we obtain ordering preservation results for generalized distorted distributions (GDD) which, in particular, can b...

Given a portfolio of risks, we study the marginal behavior of the -th risk under an adverse event, such as an unusually large loss in the portfolio or, in the case of a portfolio with a positive dependence structure, to an unusually large loss for another risk. By considering some particular conditional risk distributions, we formalize, in several...

In this paper, we introduce a new variability order that can be interpreted in terms of tail-heaviness which we will call the tail dispersive order. We provide the new definition, its interpretation and properties and the main characterization. We also study the relationship with other classical variability orders. Finally, we study the tail disper...

The preservation of reliability aging classes under the formation of coherent systems is a relevant topic in reliability theory. Thus, it is well known that the new better than used class is preserved under the formation of coherent systems with independent components. However, surprisingly, the increasing failure rate class is not preserved in the...

The economic literature contains many parametric models for the Lorenz curve. A number of these models can be obtained by distorting an original Lorenz curve $L$ L by a function $h$ h , giving rise to a distorted Lorenz curve ${\widetilde{L}}=h\circ L$ L ~ = h ∘ L . In this paper, we study, in a unified framework, this family of curves...

In this paper a new probability density function with bounded domain is presented. The new distribution arises from the generalized Lindley distribution proposed by Zakerzadeh and Dolati (2010). This new distribution that depends on two parameters can be considered as an alternative to the classical beta distribution. It presents the advantage of n...

Two basic ideas that give rise to global dependence stochastic orders were introduced and studied in Shaked et al. (Methodology and Computing in Applied Probability 14:617–648, 2012). Here these are reviewed, and two new ideas that give rise to new global dependence orders are then brought out and discussed. Two particular global dependence orders...

In this paper, we obtain ordering properties for coherent systems with possibly dependent identically distributed components. These results are based on a representation of the system reliability function as a distorted function of the common component reliability function. So, the results included in this paper can also be applied to general disto...

In this paper, we consider the comparison of parametric families of income distributions in terms of inequality and relative deprivation, which are two important aspects for understanding the concentration of incomes in a population. Based on inequalities of their parameters, we give sufficient, and in some cases necessary, conditions to show when...

In this paper, we consider a new criterion to compare risks based on the notion of expected proportional shortfall. This criterion is useful for comparing risks of different nature and does not depend on the base currency. We study its relationships with other criteria and provide some characterizations that highlight the role of this new criterion...

In actuarial theory, the Lp-metric is used to evaluate how well a probability distribution approximates another one. In the context of the distorted expectation hypothesis, the actuary replaces the original probability distribution by a distorted probability, so it makes sense to interpret the Lp-metric between them as a characteristic of the under...

Let X and Y be two randomvectors sharing the same dependence structure, that is, with a common copula. As many authors have pointed out, results of the following form are of interest: under which conditions, the stochastic comparison of the marginals of and is a sufficient condition for the comparison of the expected values for some transformations...

p>El orden de Lorenz es una herramienta adecuada para comparar la desigualdad de dos distribuciones de rentas. En este artículo obtenemos una condición suficiente para que dos distribuciones sean comparables en el orden de Lorenz y aplicamos el resultado para ordenar la familia de distribuciones Gamma triparamétricas. Lorenz ordering is an useful...

El orden de Lorenz es una herramienta adecuada para comparar la desigualdad de dos distribuciones de rentas. En este artículo obtenemos una condición suficiente para que dos distribuciones sean comparables en el orden de Lorenz y aplicamos el resultado para ordenar la familia de distribuciones Gamma triparamétricas. Lorenz ordering is an useful to...

Two basic ideas, that give rise to global dependence stochastic orders, are introduced and studied. The similarities and differences between the resulting global dependence orders, and the known common positive dependence orders, are discussed. Some desirable properties that global dependence orders may expected to satisfy are listed and checked. T...

In this paper, we consider the dispersive order and the excess wealth order to compare the variability of distorted distributions. We know from Sordo (2009a) that the excess wealth order can be characterized in terms of a class of variability measures associated to the tail conditional distribution which includes, as a particular measure, the tail...

In this paper, we establish some results for the increasing convex comparisons of generalized order statistics. First, we prove that if the minimum of two sets of generalized order statistics are ordered in the increasing convex order, then the remaining generalized order statistics are also ordered in the increasing convex order. This result is ex...

Non-intersection of TIP curves is recognized as a criterion to compare two income distributions in terms of poverty. The purpose of this paper it to obtain comparable poverty results for income distributions whose TIP curves intersect (possibly more than once). To deal with such situations, a sequence of higher-degree dominance criteria between TIP...

Li and Shaked (2007) introduced the family of generalized total time on test transform (TTT) stochastic orders, which is parameterized by a real function h that can be used to capture the preferences of a decision maker. It is natural to look for properties of these orders when there is an uncertainty in determining the appropriate function h. In t...

Li and Shaked (2007) introduced the family of generalized total time on test transform (TTT) stochastic orders, which is parameterized by a real function h that can be used to capture the preferences of a decision maker. It is natural to look for properties of these orders when there is an uncertainty in determining the appropriate function h . In...

There is a growing interest in the actuarial community in employing certain tail conditional characteristics as measures of risk, which are informative about the variability of the losses beyond the value-at-risk (one example is the tail conditional variance, introduced by Furman and Landsman (2006a, 2006b)). However, comparisons of tail risks base...

Location-independent riskier order and its dual version, excess wealth order, compare random variables in terms of dispersion. In this note, we derive the relationship of both orders to the usual stochastic order. Some new properties of these orders are obtained as a consequence.

In this paper, a class C1 of risk measures, which generalizes the class of risk measures for the right-tail deviation suggested by Wang [Wang, S., 1998. An actuarial index of the right-tail risk. North Amer. Actuarial J. 2, 88–101], is characterized in terms of dispersive order. If dispersive order does not hold, unanimous comparisons are still pos...

We examine the conditions under which unanimous poverty rankings of income distributions can be obtained for a general class of poverty indices. The �per-capita income gap� and the Shorrocks and Thon poverty measures are particular members of this class. The conditions of dominance are stated in terms of comparisons of the corresponding TIP curves...

Random variables may be compared with respect to their location by comparing certain functionals ad hoc, such as the mean or median, or by means of stochastic ordering based directly on the properties of the corresponding distribution functions. These alternative approaches are brought together in this paper. We focus on the class of L-functionals...

In this paper, the comparison of random variables according to the functionals of a general class of dispersion measures is characterized in terms of the dilation order. The Gini's mean difference is a particular member of this general class. In addition, a new and weaker order, called the second-order absolute Lorenz ordering, is introduced, and w...

The generalized Lorenz order and the absolute Lorenz order are used in economics to compare income distributions in terms of social welfare. In Section 2, we show that these orders are equivalent to two stochastic orders, the concave order and the dilation order, which are used to compare the dispersion of probability distributions. In Section 3, a...

In this paper, we introduce a new stochastic order between continuous non-negative random variables called the PLR (proportional likelihood ratio) order, which is closely related to the usual likelihood ratio order. The PLR order can be used to characterize random variables whose logarithms have log-concave (log-convex) densities. Many income rando...

Se prueba en este trabajo que el orden inducido por las curvas de Lorenz generalizadas de orden j sólo debe ser considerado un orden en desigualdad cuando j=1, caso en el que coincide con el orden clásico de Lorenz. Se propone como alternativa un nuevo orden, definido igualmente a partir de las curvas de Lorenz generalizadas de orden j, que satisfa...

A new method is proposed for the analysis of first price and all pay auctions, where bidding functions are written not as functions of values but as functions of the rank or quantile of the bidder’s value in the distribution from which it was drawn. This method gives new results in both symmetric and asymmetric cases with independent values. It is...

RESUMEN Se caracteriza la comparación de variables aleatorias de acuerdo con los funcionales de una familia general de medidas de dispersión en términos del orden en dilatación. Además, se introduce un orden nuevo, más débil que el anterior, que nos permite comparar variables aleatorias de acuerdo con ciertos funcionales de la citada familia cuando...

RESUMEN En el presente trabajo proponemos una familia de medidas de pobreza dentro de la clase de medidas basadas en el rango y estudiamos sus propiedades. Esta familia incluye como casos particulares algunas medidas de pobreza de frecuente aparición en la literatura como son las de Foster et al. (1984), Thon (1983) o Shorrocks (1995). Palabras y f...