Non independent continuous-time random walks

The usual development of the continuous-time random walk (CTRW) assumes that jumps and time intervals are a two-dimensional set of independent and identically distributed random variables. In this paper, we address the theoretical setting of nonindependent CTRWs where consecutive jumps and/or time intervals are correlated. An exact solution to the problem is obtained for the special but relevant case in which the correlation solely depends on the signs of consecutive jumps. Even in this simple case, some interesting features arise, such as transitions from unimodal to bimodal distributions due to correlation. We also develop the necessary analytical techniques and approximations to handle more general situations that can appear in practice.

HSU-Robbins and Spitzer's theorems for the variations of fractional Brownian motion

Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein's method, we prove Hsu-Robbins and Spitzer's theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.

Solvency Capital estimation and Risk Measures

This paper examines why a financial entity’s solvency capital estimation might be underestimated if the total amount required is obtained directly from a risk measurement. Using Monte Carlo simulation we show that, in some instances, a common risk measure such as Value-at-Risk is not subadditive when certain dependence structures are considered. Higher risk evaluations are obtained for independence between random variables than those obtained in the case of comonotonicity. The paper stresses, therefore, the relationship between dependence structures and capital estimation.

Compositional analysis of bivariate discrete probabilities

A joint distribution of two discrete random variables with finite support can be displayed as a two way table of probabilities adding to one. Assume that this table hasn rows and m columns and all probabilities are non-null. This kind of table can beseen as an element in the simplex of n · m parts. In this context, the marginals areidentified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclideanelements of the Aitchison geometry of the simplex can also be translated into the tableof probabilities: subspaces, orthogonal projections, distances.Two important questions are addressed: a) given a table of probabilities, which isthe nearest independent table to the initial one? b) which is the largest orthogonalprojection of a row onto a column? or, equivalently, which is the information in arow explained by a column, thus explaining the interaction? To answer these questionsthree orthogonal decompositions are presented: (1) by columns and a row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independenttwo-way tables and fully dependent tables representing row-column interaction. Animportant result is that the nearest independent table is the product of the two (rowand column)-wise geometric marginal tables. A corollary is that, in an independenttable, the geometric marginals conform with the traditional (arithmetic) marginals.These decompositions can be compared with standard log-linear models.Key words: balance, compositional data, simplex, Aitchison geometry, composition,orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure,contingency table

Probability distribution of the radionuclide half-lives

Power law distributions, a well-known model in the theory of real random variables, characterize a wide variety of natural and man made phenomena. The intensity of earthquakes, the word frequencies, the solar ares and the sizes of power outages are distributed according to a power law distribution. Recently, given the usage of power laws in the scientific community, several articles have been published criticizing the statistical methods used to estimate the power law behaviour and establishing new techniques to their estimation with proven reliability. The main object of the present study is to go in deep understanding of this kind of distribution and its analysis, and introduce the half-lives of the radioactive isotopes as a new candidate in the nature following a power law distribution, as well as a \canonical laboratory" to test statistical methods appropriate for long-tailed distributions.

Monitoring procedures in environmental geochemistry and compositional data analysis theory

First discussion on compositional data analysis is attributable to Karl Pearson, in 1897. However, notwithstanding the recent developments on algebraic structure of the simplex, more than twenty years after Aitchison’s idea of log-transformations of closed data, scientific literature is again full of statistical treatments of this type of data by using traditional methodologies. This is particularly true in environmental geochemistry where besides the problem of the closure, the spatial structure (dependence) of the data have to be considered. In this work we propose the use of log-contrast values, obtained by asimplicial principal component analysis, as LQGLFDWRUV of given environmental conditions. The investigation of the log-constrast frequency distributions allows pointing out the statistical laws able togenerate the values and to govern their variability. The changes, if compared, for example, with the mean values of the random variables assumed as models, or other reference parameters, allow definingmonitors to be used to assess the extent of possible environmental contamination. Case study on running and ground waters from Chiavenna Valley (Northern Italy) by using Na+, K+, Ca2+, Mg2+, HCO3-, SO4 2- and Cl- concentrations will be illustrated

Quantifying rock fabrics: a test of autocorrelation of the spatial distribution of cristals

A novel test of spatial independence of the distribution of crystals or phases in rocksbased on compositional statistics is introduced. It improves and generalizes the commonjoins-count statistics known from map analysis in geographic information systems.Assigning phases independently to objects in RD is modelled by a single-trial multinomialrandom function Z(x), where the probabilities of phases add to one and areexplicitly modelled as compositions in the K-part simplex SK. Thus, apparent inconsistenciesof the tests based on the conventional joins{count statistics and their possiblycontradictory interpretations are avoided. In practical applications we assume that theprobabilities of phases do not depend on the location but are identical everywhere inthe domain of de nition. Thus, the model involves the sum of r independent identicalmultinomial distributed 1-trial random variables which is an r-trial multinomialdistributed random variable. The probabilities of the distribution of the r counts canbe considered as a composition in the Q-part simplex SQ. They span the so calledHardy-Weinberg manifold H that is proved to be a K-1-affine subspace of SQ. This isa generalisation of the well-known Hardy-Weinberg law of genetics. If the assignmentof phases accounts for some kind of spatial dependence, then the r-trial probabilitiesdo not remain on H. This suggests the use of the Aitchison distance between observedprobabilities to H to test dependence. Moreover, when there is a spatial uctuation ofthe multinomial probabilities, the observed r-trial probabilities move on H. This shiftcan be used as to check for these uctuations. A practical procedure and an algorithmto perform the test have been developed. Some cases applied to simulated and realdata are presented.Key words: Spatial distribution of crystals in rocks, spatial distribution of phases,joins-count statistics, multinomial distribution, Hardy-Weinberg law, Hardy-Weinbergmanifold, Aitchison geometry

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Centralnotations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform.In this way very elaborated aspects of mathematical statistics can be understoodeasily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating,combination of likelihood and robust M-estimation functions are simple additions/perturbations in A2(Pprior). Weighting observations corresponds to a weightedaddition of the corresponding evidence.Likelihood based statistics for general exponential families turns out to have aparticularly easy interpretation in terms of A2(P). Regular exponential families formfinite dimensional linear subspaces of A2(P) and they correspond to finite dimensionalsubspaces formed by their posterior in the dual information space A2(Pprior).The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P.The discussion of A2(P) valued random variables, such as estimation functionsor likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Cumulative dominance and heuristic performance in binary multi-attribute choice

Several studies have reported high performance of simple decision heuristics multi-attribute decision making. In this paper, we focus on situations where attributes are binary and analyze the performance of Deterministic-Elimination-By-Aspects (DEBA) and similar decision heuristics. We consider non-increasing weights and two probabilistic models for the attribute values: one where attribute values are independent Bernoulli randomvariables; the other one where they are binary random variables with inter-attribute positive correlations. Using these models, we show that good performance of DEBA is explained by the presence of cumulative as opposed to simple dominance. We therefore introduce the concepts of cumulative dominance compliance and fully cumulative dominance compliance and show that DEBA satisfies those properties. We derive a lower bound with which cumulative dominance compliant heuristics will choose a best alternative and show that, even with many attributes, this is not small. We also derive an upper bound for the expected loss of fully cumulative compliance heuristics and show that this is moderateeven when the number of attributes is large. Both bounds are independent of the values ofthe weights.

Exact tests for correlation and for the slope in simple linear regressions without making assumptions

We present an exact test for whether two random variables that have known bounds on their support are negatively correlated. The alternative hypothesis is that they are not negatively correlated. No assumptions are made on the underlying distributions. We show by example that the Spearman rank correlation test as the competing exact test of correlation in nonparametric settings rests on an additional assumption on the data generating process without which it is not valid as a test for correlation.We then show how to test for the significance of the slope in a linear regression analysis that invovles a single independent variable and where outcomes of the dependent variable belong to a known bounded set.

Estimating learning models from experimental data

We study the statistical properties of three estimation methods for a model of learning that is often fitted to experimental data: quadratic deviation measures without unobserved heterogeneity, and maximum likelihood withand without unobserved heterogeneity. After discussing identification issues, we show that the estimators are consistent and provide their asymptotic distribution. Using Monte Carlo simulations, we show that ignoring unobserved heterogeneity can lead to seriously biased estimations in samples which have the typical length of actual experiments. Better small sample properties areobtained if unobserved heterogeneity is introduced. That is, rather than estimating the parameters for each individual, the individual parameters are considered random variables, and the distribution of those random variables is estimated.

Buy-and-Hold Strategies and Comonotonic Approximations

[cat] En aquest article estudiem estratègies “comprar i mantenir” per a problemes d’optimitzar la riquesa final en un context multi-període. Com que la riquesa final és una suma de variables aleatòries dependents, on cadascuna d’aquestes correspon a una quantitat de capital que s’ha invertit en un actiu particular en una data determinada, en primer lloc considerem aproximacions que redueixen l’aleatorietat multivariant al cas univariant. A continuació, aquestes aproximacions es fan servir per determinar les estratègies “comprar i mantenir” que optimitzen, per a un nivell de probabilitat donat, el VaR i el CLTE de la funció de distribució de la riquesa final. Aquest article complementa el treball de Dhaene et al. (2005), on es van considerar estratègies de reequilibri constant.

Second-order processes driven by dichotomous noise

We study free second-order processes driven by dichotomous noise. We obtain an exact differential equation for the marginal density p(x,t) of the position. It is also found that both the velocity ¿(t) and the position X(t) are Gaussian random variables for large t.

Buy-and-Hold Strategies and Comonotonic Approximations

[cat] En aquest article estudiem estratègies “comprar i mantenir” per a problemes d’optimitzar la riquesa final en un context multi-període. Com que la riquesa final és una suma de variables aleatòries dependents, on cadascuna d’aquestes correspon a una quantitat de capital que s’ha invertit en un actiu particular en una data determinada, en primer lloc considerem aproximacions que redueixen l’aleatorietat multivariant al cas univariant. A continuació, aquestes aproximacions es fan servir per determinar les estratègies “comprar i mantenir” que optimitzen, per a un nivell de probabilitat donat, el VaR i el CLTE de la funció de distribució de la riquesa final. Aquest article complementa el treball de Dhaene et al. (2005), on es van considerar estratègies de reequilibri constant.

Migration Flows and Quality of Life in a Metropolitan Area: the Case of Barcelona-Spain

The phenomenon of human migration is certainly not new and it has been studied from a variety of perspectives. Yet, the attention on human migration and its determinant has not been fading over time as confirmed by recent contributions (see for instance Cushing and Poot 2004 and Rebhun and Raveh 2006). In this paper we combine the recent theoretical contributions by Douglas (1997) and Wall (2001) with the methodological advancements of Guimarães et al. (2000, 2003) to model inter-municipal migration flows in the Barcelona area. In order to do that, we employ two different types of count models, i.e. the Poisson and negative binomial and compare the estimations obtained. Our results show that, even after controlling for the traditional migration factors, QoL (measured with a Composite Index which includes numerous aspects and also using a list of individual variables) is an important determinant of short distance migration movements in the Barcelona area.