systems of frequency distributions for water and environmental engineering abstract a wide spectrum of frequency distributions are used in hydrologic hydraulic environmental and water resources engineering these distributions may have different origins are based on different hypotheses and belong to different generating systems review of literature suggests that different systems of frequency distributions employed in science and engineering in general and environmental and water engineering in particular have been derived using different approaches which include 1 differential equations 2 distribution elasticity 3 genetic theory 4 generating functions 5 transformations 6 bessel function 7 expansions and 8 entropy maximization this paper revisits these systems of distributions and discusses the hypotheses that are used for deriving these systems it also proposes based on empirical evidence another general system of distributions and derives a number of distributions from this general system that are used in environmental and water engineering

tails of lorenz curves the lorenz dominance criterion is the centre piece of inequality analysis yet the appeal of this criterion which requires considering lorenz curves in their entirety is undermined by the practical problem that many sample lorenz curves intersect in the tails the commonly used inferential methods based on central limit theorem arguments do not apply to the tails since these contain too few observations by contrast we propose a test procedure for distributions whose tails lie in the domain of attraction of the frechet distribution which fully takes into account the tail behaviour of lorenz curves our experiments and empirical examples demonstrate the good performance of the proposed test in many cases we are able to infer that despite sample tail crossings the population lorenz curves do in fact exhibit lorenz dominance

using parametric classification trees for model selection with applications to financial risk management abstract we describe two parametric classification tree methods which allow formal selection of a member of a class of generalised distributions in the paper we consider generalised beta distributions for non negative random variables and the generalised skew student distribution for random variables distributed on the real line we introduce a class of symmetric generalised multivariate student distributions members of which may also be selected using the classification trees we present two versions of the parametric classification tree specific to general and general to specific we apply the classification methods to daily returns on stocks from a selection of 15 major mid cap and emerging markets the results show that the majority of return distributions follow student s t but that a non negligible minority follow a symmetric generalised student distribution we confirm a well known stylised fact about skewness it tends not to be persistent by contrast kurtosis is persistent using the symmetric generalised multivariate student distribution we present a risk management study based on efficient portfolios constructed from ukftse250 stocks and specifically concerned with the computation of value at risk the case study demonstrates that the model selection procedures based on the classification trees lead to more accurate computation of var than those based on the normal distribution or on non parametric approaches the study also shows that the normal distribution may be used for var computations for larger portfolios when the holding period is longer

comparisons of log normal mixture and pareto tails gb2 or log normal body of romania s all cities size distribution abstract modeling demographic data has been on the agenda of statisticians for many years some of the distributions used are pareto reverse pareto q exponential and log normal models an approach to this problem is to consider three statistical models one for the upper tail one for the middle range and another for the lower tail this paper deals with the size distribution of urban and rural agglomerations in romania for the 1992 2017 period by comparing the recently introduced three log normal mixture 3ln pareto tails log normal ptln and threshold double pareto generalized beta of second kind tdpgb2 models the tdpgb2 statistical model has the ptln distribution as a limiting case the maximum likelihood estimates of the distributions are computed and goodness of fit tests are performed using the kolmogorov smirnov ks cramer von mises cm and anderson darling ad statistics also we use the vuong and bayes factor log likelihood tests using both graphical and formal statistical tests our results rigorously confirm that the 3ln model is statistically equivalent to ptln and tdpgb2 distributions the preferred model being the ptln probability law both the ptln and tdpgb2 distributions have pareto tails but the 3ln model does not all the three models prove to be very well suited parameterizations of romania s city size data

complexity entropy causality plane a useful approach for distinguishing songs nowadays we are often faced with huge databases resulting from the rapid growth of data storage technologies this is particularly true when dealing with music databases in this context it is essential to have techniques and tools able to discriminate properties from these massive sets in this work we report on a statistical analysis of more than ten thousand songs aiming to obtain a complexity hierarchy our approach is based on the estimation of the permutation entropy combined with an intensive complexity measure building up the complexity entropy causality plane the results obtained indicate that this representation space is very promising to discriminate songs as well as to allow a relative quantitative comparison among songs additionally we believe that the here reported method may be applied in practical situations since it is simple robust and has a fast numerical implementation

on the upper tail of italian firms size distribution abstract in this paper we analyze the upper tail of the size distribution of italian companies with limited liability belonging to the cebi database size is defined in terms of net worth in particular we show that the largest firms follow a power law distribution according to the well known pareto law for which we give estimates of the shape parameter such a behavior seems to be quite persistent over time view that for almost 20 years of observations the shape parameter is always in the vicinity of 1 8 the power law hypothesis is also positively tested using graphical and analytical methods

a general functional form for approximating the lorenz curve abstract the purpose of this paper is to present several descriptive approximations of empirical lorenz curves each of these hypothetical forms is less restrictive than those considered before we compare these alternative forms by goodness of fit f tests of nested forms and measurement of the gini coefficient

elliptical lorenz curves abstract a family of elliptic lorenz curves is proposed for fitting grouped income data the associated distribution and density functions are displayed together with the gini indices estimation procedures are discussed comparisons are made with alternative models using australian 1967 68 income data

statistical inference for poverty measures with relative poverty lines relative poverty lines such as one half median income have been increasingly used in poverty studies this paper contributes to the literature by developing statistical inference for testing decomposable poverty measures with relative poverty lines the poverty lines we consider are percentages of mean income and percentages of quantiles we show that the estimates of poverty indices with relative poverty lines are asymptotically normally distributed and that the covariance structure can be consistently estimated as a consequence asymptotically distribution free statistical inference can be established in a straightforward manner

the exponentiated generalized birnbaum saunders distribution a new distribution called the exponentiated generalized birnbaum saunders distribution is proposed and studied the new distribution is quite flexible to model survival data reliability problems fatigue life studies and hydrological data it can have decreasing increasing upside down bathtub bathtub shaped increasing decreasing increasing and decreasing increasing decreasing hazard rate functions we provide a comprehensive account of the mathematical properties of the new distribution and various structural quantities are derived we discuss maximum likelihood estimation of the model parameters for uncensored and censored data two empirical applications of the new model to real data are presented for illustrative purposes

exponentiated beta distributions beta distributions are usually defined on the unit interval 0 1 for wider applicability we introduce six exponentiated beta distributions and derive several of their properties including the moment generating function expectation variance skewness kurtosis shannon entropy and the renyi entropy

heavy tailed distributions and the distribution of wealth evidence from rich lists in canada 1999 2017 abstract i examine whether a power law distribution fits the top wealth distribution i e the rich 100 from canada i use a maximum likelihood framework that considers the fit of the estimated distribution as well as allowing tests against alternative heavy tailed and skewed distributions i find that while a power law distribution does fit the data well it is not possible to distinguish it from a lognormal distribution truncated to the upper tail

asymptotic analysis of the optimal filtering problem for two dimensional diffusions measured in a low noise channel we consider the problem of filtering one dimensional diffusions with nonlinear drift coefficients transmitted through a nonlinear fow noise channel we construct an asymptotic solution to zakai s equation for the unnormalized conditional probability density of the signal given the noisy measurements this expansion is used to find the asymptotic expansion of the minimum error variance filter and its mean square estimation error msee we construct approximate filters whose msee agrees with that of the optimal one to a given degree of accuracy the dimension of the approximate filter increases with the required degree of accuracy similarly we expand the maximum a posteriori probability estimator and the minimum energy estimator and compare their performance we also discuss some extended kalman filters and present some examples

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

an algorithmic proof theory for hypergeometric ordinary and q multisum integral identities it is shown that every proper hypergeometric multisum integral identity orq identity with a fixed number of summations and or integration signs possesses a short computer constructible proof we give a fast algorithm for finding such proofs most of the identities that involve the classical special functions of mathematical physics are readily reducible to the kind of identities treated here we give many examples of the method including computer generated proofs of identities of mehta dyson selberg hille hardy q saalschutz and others the prospect of using the method for proving multivariate identities that involve an arbitrary number of summations integrations is discussed

maximum entropy formalism fractals scaling phenomena and 1 f noise a tale of tails in this report on examples of distribution functions with long tails we a show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function b review several models that yield log normal distributions c show that log normal distributions may mimic 1 f noise over a certain range and d present an amplification model to show how log normal personal income distributions are transformed into inverse power pareto distributions in the high income range

geometrically concave univariate distributions in this paper our aim is to show that if a probability density function is geometrically concave convex then the corresponding cumulative distribution function and the survival function are geometrically concave convex too under some assumptions the proofs are based on the so called monotone form of l hospital s rule and permit us to extend our results to the case of the concavity convexity with respect to holder means to illustrate the applications of the main results we discuss in details the geometrical concavity of the probability density function cumulative distribution function and survival function of some common continuous univariate distributions moreover at the end of the paper we present a simple alternative proof to schweizer s problem related to the mulholland s generalization of minkowski s inequality

general model of subtraction of stochastic variables attractor and stability analysis abstract we introduce a general process designed to model stochastic systems in which the dependence of random variables is not through addition only but combined addition and subtraction with bounded ranges and whose probabilistic factors have compact support we show that still retaining much of the general essence of the central limit theorem this process presents a functional attractor which is neither gaussian nor levy like and is precisely akin numerically to a probability density function shown in previous works to have ubiquitous character namely the two parameter beta distribution

statistical inference for testing inequality indices with dependent samples this paper develops asymptotically distribution free inference for testing inequality indices with dependent samples it considers the interpolated gini coefficient and the generalized entropy class which includes several commonly used inequality indices we first establish inference tests for changes in inequality indices with completely dependent samples i e matched pairs and then generalize the inference procedures to cases with partially dependent samples the effects of sample dependency on standard errors of inequality changes are examined through simulation studies as well as through applications to the cps and psid data

universality in the tail of musical note rank distribution abstract although power laws have been used to fit rank distributions in many different contexts they usually fail at the tails languages as sequences of symbols have been a popular subject for ranking distributions and for this purpose music can be treated as such here we show that more than 1800 musical compositions are very well fitted by the first kind two parameter beta distribution which arises in the ranking of multiplicative stochastic processes the parameters a and b are obtained for classical jazz and rock music revealing interesting features specially we have obtained a clear trend in the values of the parameters for major and minor tonal modes finally we discuss the distribution of notes for each octave and its connection with the ranking of the notes

streamflow record extension using power transformations and application to sediment transport to obtain a representative set of flow rates for a stream it is often desirable to fill in missing data or extend measurements to a longer time period by correlation to a nearby gage with a longer record linear least squares regression of the logarithms of the flows is a traditional and still common technique however its purpose is to generate optimal estimates of each day s discharge rather than the population of discharges for which it tends to underestimate variance maintenance of variance extension move equations hirsch 1982 were developed to correct this bias this study replaces the logarithmic transformation by the more general box cox scaled power transformation generating a more linear constant variance relationship for the move extension combining the box cox transformation with the move extension is shown to improve accuracy in estimating order statistics of flow rate particularly for the nonextreme discharges which generally govern cumulative transport over time this advantage is illustrated by prediction of cumulative fractions of total bed load transport

tail universalities in rank distributions as an algebraic problem the beta like function abstract although power laws of the zipf type have been used by many workers to fit rank distributions in different fields like in economy geophysics genetics soft matter networks etc these fits usually fail at the tail some distributions have been proposed to solve the problem but unfortunately they do not fit at the same time the body and the tail of the distribution we show that many different data in rank laws like in granular materials codons author impact in scientific journal etc can be very well fitted by the integrand of a beta function that we call beta like function then we propose that such universality can be due to the fact that systems made from many subsystems or choices present stretched exponential frequency rank functions which qualitatively and quantitatively are well fitted with the beta like function distribution in the limit of many random variables we give a plausibility argument for this observation by transforming the problem into an algebraic one finding the rank of successive products of numbers which is basically a multinomial process from a physical point of view the observed behavior at the tail seems to be related with the onset of different mechanisms that are dominant at different scales providing crossovers and finite size effects

an ordered family of lorenz curves abstract a general method for building parametric functional families of lorenz curves generated from an initial lorenz curve which satisfies some regularity conditions is presented it is shown that these families can be ordered in a manner which leads to a hierarchy of lorenz curves the method starts from a generating lorenz curve l 0 p and builds the family by increasing the number of parameters which can be easily interpreted in terms of the elasticities of l 0 p the method is applied to a family we term the pareto family since they use the pareto lorenz curves as their generating curves the family is shown to contain an important number of classical lorenz curves used in the existing literature several properties of this family are analyzed these include the population function inequality measures and lorenz orderings a general method for the estimation of these family is given and applied to the pareto family finally an application is presented for data from various countries the results are very robust across data sources the pareto models fit very well in a number of applications

using incomplete moments to measure inequality abstract incomplete moments of the income distribution form natural building blocks for measuring inequality for example income quintiles the lorenz curve pietra and gini measures of inequality all depend upon the incomplete moments of the income distribution this paper discusses these relationships and shows that they are easily calculated for a very broad family of distributions because they possess a closure property empirical illustrations are given for the white family income distribution for the 1948 to 1980 period

edgeworth expansions and normalizing transforms for inequality measures abstract finite sample distributions of studentized inequality measures differ substantially from their asymptotic normal distribution in terms of location and skewness we study these aspects formally by deriving the second order expansion of the first and third cumulant of the studentized inequality measure we state distribution free expressions for the bias and skewness coefficients in the second part we improve over first order theory by deriving edgeworth expansions and normalizing transforms these normalizing transforms are designed to eliminate the second order term in the distributional expansion of the studentized transform and converge to the gaussian limit at rate o n 1 this leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order o n 3 2 we illustrate our procedure with an application to regional inequality measurement in cote d ivoire

systems of frequency curves generated by transformations of logistic variables abstract the lognormal s sub l s sub b and s sub u systems of frequency curves are obtained by three simple transformations of the normal curve by applying the same transformations to the logistic systems l sub l l sub b and l sub u are obtained some properties of these systems and their relation to the s systems are set out tables to facilitate fitting l sub u curves by moments are presented author

a generalization of the beta distribution with applications abstract this paper introduces a five parameter beta distribution gb which nests the generalized beta and gamma distributions and includes more than thirty distributions as limiting or special cases the generalized beta leads to an exponential generalized beta egb distribution which includes generalized forms of the logistics exponential gompertz and gumbell distributions and the normal as special cases the egb family of distributiosn provides the basis for partially adaptive estimation of econometric models with possibly skewed and leptokurtic error distributions applications of the models to investigating the distribution of income stock returns and in regression analysis are considered

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

a 21st century proof of dougall s hypergeometric sum identity abstract my guess is that within fifty or hundred years or it might be one hundred and fifty computers will successfully compete with the human brain in doing mathematics and that their mathematical style will be rather different from ours fairly long computational verifications numerical or combinatorical will not bother them at all and this should lead not just to different sorts of proofs but more importantly to different sorts of theorems being proved

explicit expressions of the pietra index for the generalized function for the size distribution of income abstract the importance of the pietra index in socioeconomic systems and econophysics has been highlighted by eliazar and sokolov 2010 in this paper we obtain closed expressions for the pietra index for the generalized function for the size of income proposed by mcdonald 1984 this family is composed of three classes of distributions the generalized gamma distribution gg the generalized beta of the first kind gb1 and the generalized beta of the second kind gb2 for the different distributions we obtain closed and simple expressions of the pietra index which can be easily computed we also obtain the pietra index for other relevant income models including finite mixtures of distributions and the generalized distribution clementi et al 2008 finally two empirical applications with real income data are given

entropy maximization under the constraints on the generalized gini index and its application in modeling income distributions abstract in economics and social sciences the inequality measures such as gini index pietra index etc are commonly used to measure the statistical dispersion there is a generalization of gini index which includes it as special case in this paper we use principle of maximum entropy to approximate the model of income distribution with a given mean and generalized gini index many distributions have been used as descriptive models for the distribution of income the most widely known of these models are the generalized beta of second kind and its subclass distributions the obtained maximum entropy distributions are fitted to the us family total money income in 2009 2011 and 2013 and their relative performances with respect to generalized beta of second kind family are compared

income distribution and inequality measurement the problem of extreme values we examine the statistical performance of inequality indices in the presence of extreme values in the data and show that these indices are very sensitive to the properties of the income distribution estimation and inference can be dramatically affected especially when the tail of the income distribution is heavy even when standard bootstrap methods are employed however use of appropriate semiparametric methods for modelling the upper tail can greatly improve the performance of even those inequality indices that are normally considered particularly sensitive to extreme values

inverted rank distributions macroscopic statistics universality classes and critical exponents abstract an inverted rank distribution is an infinite sequence of positive sizes ordered in a monotone increasing fashion interlacing together lorenzian and oligarchic asymptotic analyses we establish a macroscopic classification of inverted rank distributions into five socioeconomic universality classes communism socialism criticality feudalism and absolute monarchy we further establish that i communism and socialism are analogous to a disordered phase feudalism and absolute monarchy are analogous to an ordered phase and criticality is the phase transition between order and disorder ii the universality classes are characterized by two critical exponents one governing the ordered phase and the other governing the disordered phase iii communism criticality and absolute monarchy are characterized by sharp exponent values and are inherently deterministic iv socialism is characterized by a continuous exponent range is inherently stochastic and is universally governed by continuous power law statistics v feudalism is characterized by a continuous exponent range is inherently stochastic and is universally governed by discrete exponential statistics the results presented in this paper yield a universal macroscopic socioeconophysical perspective of inverted rank distributions

on the estimation of long tailed skewed distributions with actuarial applications abstract very long tailed skewed distributions often arise in practice in particular we find that size of loss distributions in casualty insurance are mainly of this type compounding explains why many of these losses have approximate pareto generalized pareto burr and log t distributions an adaptation of the empirical mean residual life function helps the statistician select the correct model to fit in these cases it is then discovered that minimum distance estimates in particular that of cramer von mises and minimum chi square estimates are extremely valuable and easy to use in the case of grouped data two substantial examples are given one involving hurricane losses and the other dealing with malpractice claims

a filter with a guaranteed asymptotic performance in the case of low noise levels the optimal probability density function summarizing the available information about the state of a system can be accurately approximated by the product of a gaussian function and a linear function the approximation preserves the ability to estimate to an accuracy of o 2 the expected value of any twice continuously differentiable function defined on the state space the parameter depends on the noise level if the noise level in the system is low then is large a new filtering method based on this approximation is described the approximating function is updated recursively as the system evolves with time and as new measurements of the system state are obtained the updates preserve the ability to estimate the expected values of functions to an accuracy of o 2 the new filter does not store previous measurements or previous approximations to the optimal probability density function the new filter is called the asymptotic filter because the definition of the filter and the analysis of its properties are based on the theory of asymptotic expansion of integrals of laplace type an analysis of the state propagation equations shows that the asymptotic filter performs better than a particular widely used suboptimal approximation to the optimal filter the extended kalman filter the extended kalman filter does not in general preserve the ability to estimate expected values to an accuracy of o 2 the computational cost of the asymptotic filter is comparable to that of the iterated extended kalman filter

two flexible functional form approaches for approximating the lorenz curve abstract this paper introduces two flexible form approaches to approximate lorenz curves the first approach expands the inverse function of an income distribution in an exponential polynomial series and derives the lorenz curve from it the required convexity condition can be imposed using a bayesian method the second approach approximates the lorenz curve with a sequence of berstein polynomial functions the required convexity condition is automatically established in this approach we compare these approaches with other well known fixed functional form approaches we evaluate the performance of these functional forms by comparing approximation errors maximum error and the estimates of the gini coefficient produced by various approaches

beyond the zipf mandelbrot law in quantitative linguistics abstract in this paper the zipf mandelbrot law is revisited in the context of linguistics despite its widespread popularity the zipf mandelbrot law can only describe the statistical behaviour of a rather restricted fraction of the total number of words contained in some given corpus in particular we focus our attention on the important deviations that become statistically relevant as larger corpora are considered and that ultimately could be understood as salient features of the underlying complex process of language generation finally it is shown that all the different observed regimes can be accurately encompassed within a single mathematical framework recently introduced by c tsallis

order disorder transition in conflicting dynamics leading to rank frequency generalized beta distributions abstract the behavior of rank ordered distributions of phenomena present in a variety of fields such as biology sociology linguistics finance and geophysics has been a matter of intense research often power laws have been encountered however their validity tends to hold mainly for an intermediate range of rank values in a recent publication martinez mekler et al 2009 7 a generalization of the functional form of the beta distribution has been shown to give excellent fits for many systems of very diverse nature valid for the whole range of rank values regardless of whether or not a power law behavior has been previously suggested here we give some insight on the significance of the two free parameters which appear as exponents in the functional form by looking into discrete probabilistic branching processes with conflicting dynamics we analyze a variety of realizations of these so called expansion modification models first introduced by wentian li 1989 10 we focus our attention on an order disorder transition we encounter as we vary the modification probability p we characterize this transition by means of the fitting parameters our numerical studies show that one of the fitting exponents is related to the presence of long range correlations exhibited by power spectrum scale invariance while the other registers the effect of disordering elements leading to a breakdown of these properties in the absence of long range correlations this parameter is sensitive to the occurrence of unlikely events we also introduce an approximate calculation scheme that relates this dynamics to multinomial multiplicative processes a better understanding through these models of the meaning of the generalized beta fitting exponents may contribute to their potential for identifying and characterizing universality classes

an alternative method for modeling the size distribution of top wealth abstract the pareto distribution has been widely applied in modeling the distribution of wealth as well as top incomes cities and firms however recent evidence has shown that the pareto distribution is not consistent with many situations in which it was previously considered applicable we propose an alternative method for estimating the upper tail distribution of wealth and suggest a new lorenz curve for building models to provide such estimates applying our new models to the forbes world s billionaire lists we show that they significantly outperform the pareto lorenz curve as well as some other popular alternatives

maximum entropy lorenz curves abstract for describing the distribution of income a family of maximum entropy density functions is derived under optional side conditions on the distance of the mean income from the minimum income and the absolute gini index of concentration many of the resulting lorenz curves obtained from the reciprocal of the density function by repeated quadrature turn out to be well known models

efficiency of the extended kalman filter for nonlinear systems with small noise the problem of nonlinear filtering is studied asymptotically as the noise tends to zero detectability conditions ensuring that the filtering error tends to zero are derived and under these conditions it is proved that the extended kalman filter provides a good suboptimal filter the smoothing problem is also studied the proofs use some estimates for the linearization of stochastic systems some changes of probability and stochastic differentiation techniques