Double Pareto Lognormal Distribution Research Articles

Global Self‐Similar Scaling of Terrestrial Carbon With Aridity

AbstractWhile it is well known that water availability controls vegetation growth and soil microbial activity, how aridity affects ecosystem carbon patterns is not completely understood. Toward a more quantitative assessment of terrestrial carbon stocks, here we apply dimensional analysis and scaling to the global joint distribution of terrestrial carbon stock, obtained from international survey data and harmonized global maps. The results show a remarkable self‐similar behavior of the global carbon stock with dryness index, whereby the key statistics (e.g., mean, quantiles, and standard deviation) of carbon stock tend to scale with the hydroclimatic regime (i.e., aridity) via a universal exponent. Such a scaling reflects the strong coupling between the hydrological cycle and biogeochemical process and enables robust predictions of carbon stocks as a function of aridity only. When normalized by its averages in the corresponding hydroclimatic regime, the carbon stock distributions collapse onto a single double Pareto lognormal distribution, often used in economics to describe income. The presence of this distribution in completely different physical contexts, such as biogeochemistry and economics, hints at generating mechanisms that transcend the details of the specific stock being considered.

Clariant sets new 2025 financial targets and announces step-up in growth and profitability

Gibrat’s Law – also called law of proportionate effect – states that the size of a company does not influence its economic growth. The present article aims at testing the validity of this law on all companies based in the autonomous region of Andalusia (Spain). The study uses data on employment compiled by the Instituto de Estadística y Cartografía de Andalucía for every company belonging to the two-digit NACE code, with yearly data for the 2000 to 2015 period: 7,046,539 firms in total. The results obtained allow to identify by sector the impact of size on business growth, to check compliance in this relevant region of the European Union and analyse the influence of the recent economic/financial crisis in terms of Gibrat’s Law validation. In general terms, the degree of compliance with the law is limited in the analysed period, even more so in the years of economic crisis. We have fitted the double Pareto log-normal distribution to our data. That distribution fits well the firm’ size distribution in many sectors of the Andalusian economy. Thus, these sectors would follow a Yule process. At the aggregate level, the sectors of the Andalusian economy that comply with Gibrat’s law also follow a Yule process.

Firm growth in the 21st century: Does the Andalusian economy comply with Gibrat’s Law?

Gibrat’s Law – also called law of proportionate effect – states that the size of a company does not influence its economic growth. The present article aims at testing the validity of this law on all companies based in the autonomous region of Andalusia (Spain). The study uses data on employment compiled by the Instituto de Estadística y Cartografía de Andalucía for every company belonging to the two-digit NACE code, with yearly data for the 2000 to 2015 period: 7,046,539 firms in total. The results obtained allow to identify by sector the impact of size on business growth, to check compliance in this relevant region of the European Union and analyse the influence of the recent economic/financial crisis in terms of Gibrat’s Law validation. In general terms, the degree of compliance with the law is limited in the analysed period, even more so in the years of economic crisis. We have fitted the double Pareto log-normal distribution to our data. That distribution fits well the firm’ size distribution in many sectors of the Andalusian economy. Thus, these sectors would follow a Yule process. At the aggregate level, the sectors of the Andalusian economy that comply with Gibrat’s law also follow a Yule process.

Analysis of an aggregate loss model in a Markov renewal regime

In this article we consider an aggregate loss model with dependent losses. The loss occurrence process is governed by a two-state Markovian arrival process (MAP2), a Markov renewal process that allows for (1) correlated inter-loss times, (2) non-exponentially distributed inter-loss times and, (3) overdisperse loss counts. Some quantities of interest to measure persistence in the loss occurrence process are obtained. Given a real OpRisk database, the aggregate loss model is estimated by fitting separately the inter-loss times and severities. The MAP2 is estimated via direct maximization of the likelihood function, and severities are modeled by the heavy-tailed, double-Pareto Lognormal distribution. In comparison with the fit provided by the Poisson process, the results point out that taking into account the dependence and overdispersion in the inter-loss times distribution leads to higher capital charges.

An EM Algorithm for Double-Pareto-Lognormal Generalized Linear Model Applied to Heavy-Tailed Insurance Claims

Generalized linear models might not be appropriate when the probability of extreme events is higher than that implied by the normal distribution. Extending the method for estimating the parameters of a double Pareto lognormal distribution (DPLN) in Reed and Jorgensen (2004), we develop an EM algorithm for the heavy-tailed Double-Pareto-lognormal generalized linear model. The DPLN distribution is obtained as a mixture of a lognormal distribution with a double Pareto distribution. In this paper the associated generalized linear model has the location parameter equal to a linear predictor which is used to model insurance claim amounts for various data sets. The performance is compared with those of the generalized beta (of the second kind) and lognorma distributions.

Research on fatigue life of steel wire ropes under impact loads based on double Pareto lognormal distribution

Fatigue damage is the most important failure mode of wire ropes, and models of cumulative fatigue damage of wire ropes under multiple impact loads have been proposed based on the principle of micro...

Fat-Tailed Regression with the DPLN Distribution

Traditional regression models might not be appropriate when the probability of extreme events is higher than that implied by the normal distribution. Extending the method for estimating the parameters of a double Pareto lognormal distribution (DPLN) in Reed and Jorgensen (2004), we develop an EM algorithm for heavy-tailed DPLN-regression. The DPLN distribution is obtained as a mixture of a lognormal distribution with a double Pareto distribution. In this paper the associated regression model has the location parameter equal to a linear predictor which is used to model insurance claim amounts for various data sets. The performance is compared with those of the generalised beta (of the second kind) and lognormal distributions.

A NOTE ON THE SIZE DISTRIBUTION OF CONSUMPTION: MORE DOUBLE PARETO THAN LOGNORMAL

The cross-sectional distribution of consumption is commonly approximated by the lognormal distribution. This note shows that consumption is better described by the double Pareto-lognormal distribution (dPlN), which has a lognormal body with two Pareto tails and arises as the stationary distribution in recently proposed dynamic general equilibrium models. dPlN outperforms other parametric distributions and is often not rejected by goodness-of-fit tests. The analytical tractability and parsimony of dPlN may be convenient for various economic applications.

Test of two hypotheses explaining the size of populations in a system of cities

Two classical hypotheses are examined about the population growth in a system of cities: Hypothesis 1 pertains to Gibrat's and Zipf's theory which states that the city growth–decay process is size independent; Hypothesis 2 pertains to the so-called Yule process which states that the growth of populations in cities happens when (i) the distribution of the city population initial size obeys a log-normal function, (ii) the growth of the settlements follows a stochastic process. The basis for the test is some official data on Bulgarian cities at various times. This system was chosen because (i) Bulgaria is a country for which one does not expect biased theoretical conditions; (ii) the city populations were determined rather precisely. The present results show that: (i) the population size growth of the Bulgarian cities is size dependent, whence Hypothesis 1 is not confirmed for Bulgaria; (ii) the population size growth of Bulgarian cities can be described by a double Pareto log-normal distribution, whence Hypothesis 2 is valid for the Bulgarian city system. It is expected that this fine study brings some information and light on other usually considered to be more pertinent countries of city systems.

Consumption Has Heavier Tails than You Think: The Double Pareto-Lognormal Stands Out

The cross-sectional distribution of consumption is commonly approximated by the lognormal distribution. This note shows that consumption is better described by the \emph{double Pareto-lognormal distribution} (dPlN), which has a lognormal body with two Pareto tails and arises as the stationary distribution in recently proposed dynamic general equilibrium models. dPlN outperforms other parametric distributions and is often not rejected by goodness-of-fit tests. The analytical tractability and parsimony of dPlN may be convenient for various economic applications.

Pareto–lognormal distributions: Inequality, poverty, and estimation from grouped income data

The (double) Pareto–lognormal is an emerging parametric distribution for income that has a sound underlying generating process, good theoretical properties, and some limited favorable evidence of its fit to data. We extend existing results for this distribution in 3 directions. (1) We derive closed form formula for its moment distribution functions, and for various inequality and poverty measures. (2) We show how it can be estimated from grouped data using GMM. (3) Using grouped data from ten countries, we compare its performance with that of another leading 4-parameter income distribution, the generalized beta-2 distribution. The results confirm that, when using grouped data, both distributions provide a good fit, with the double Pareto–lognormal distribution outperforming the beta distribution in 4 out of 10 cases.

Random phenotypic variation of yeast (Saccharomyces cerevisiae) single-gene knockouts fits a double pareto-lognormal distribution.

BackgroundDistributed robustness is thought to influence the buffering of random phenotypic variation through the scale-free topology of gene regulatory, metabolic, and protein-protein interaction networks. If this hypothesis is true, then the phenotypic response to the perturbation of particular nodes in such a network should be proportional to the number of links those nodes make with neighboring nodes. This suggests a probability distribution approximating an inverse power-law of random phenotypic variation. Zero phenotypic variation, however, is impossible, because random molecular and cellular processes are essential to normal development. Consequently, a more realistic distribution should have a y-intercept close to zero in the lower tail, a mode greater than zero, and a long (fat) upper tail. The double Pareto-lognormal (DPLN) distribution is an ideal candidate distribution. It consists of a mixture of a lognormal body and upper and lower power-law tails.Objective and MethodsIf our assumptions are true, the DPLN distribution should provide a better fit to random phenotypic variation in a large series of single-gene knockout lines than other skewed or symmetrical distributions. We fit a large published data set of single-gene knockout lines in Saccharomyces cerevisiae to seven different probability distributions: DPLN, right Pareto-lognormal (RPLN), left Pareto-lognormal (LPLN), normal, lognormal, exponential, and Pareto. The best model was judged by the Akaike Information Criterion (AIC).ResultsPhenotypic variation among gene knockouts in S. cerevisiae fits a double Pareto-lognormal (DPLN) distribution better than any of the alternative distributions, including the right Pareto-lognormal and lognormal distributions.Conclusions and SignificanceA DPLN distribution is consistent with the hypothesis that developmental stability is mediated, in part, by distributed robustness, the resilience of gene regulatory, metabolic, and protein-protein interaction networks. Alternatively, multiplicative cell growth, and the mixing of lognormal distributions having different variances, may generate a DPLN distribution.

The Size Distribution Across All 'Cities': A Unifying Approach

In this paper we show that the double Pareto lognormal (DPLN) parameterization provides an excellent fit to the overall US city size distribution, regardless of whether “cities” are administratively defined Census places or economically defined area clusters. We then consider an economic model that combines scale-independent urban growth (Gibrat’s law) with endogenous city creation. City sizes converge to a DPLN distribution in this model, which is much better in line with the data than previous urban growth frameworks that predict a lognormal or a Pareto city size distribution (Zipf’s law).

Generating Statistically Correct Random Topologies for Testing Smart Grid Communication and Control Networks

In order to design an efficient communication scheme and examine the efficiency of any networked control architecture in smart grid applications, we need to characterize statistically its information source, namely the power grid itself. Investigating the statistical properties of power grids has the immediate benefit of providing a natural simulation platform, producing a large number of power grid test cases with realistic topologies, with scalable network size, and with realistic electrical parameter settings. The second benefit is that one can start analyzing the performance of decentralized control algorithms over information networks whose topology matches that of the underlying power network and use network scientific approaches to determine analytically if these architectures would scale well. With these motivations, in this paper we study both the topological and electrical characteristics of power grid networks based on a number of synthetic and real-world power systems. The most interesting discoveries include: the power grid is sparsely connected with obvious <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">small-world</i> properties; its nodal degree distribution can be well fitted by a mixture distribution coming from the sum of a truncated geometric random variable and an irregular discrete random variable; the power grid has very distinctive graph spectral density and its algebraic connectivity scales as a power function of the network size; the line impedance has a heavy-tailed distribution, which can be captured quite accurately by a clipped double Pareto lognormal distribution. Based on the discoveries mentioned above, we propose an algorithm that generates random topology power grids featuring the same topology and electrical characteristics found from the real data.

On Modeling Random Topology Power Grids for Testing Decentralized Network Control Strategies

An electrical power grid is a critical infrastructure. Its reliable, robust, and efficient operation inevitably depends on underlying telecommunication networks. In order to design an efficient communication scheme and examine the efficiency of any networked control architecture, we need to characterize statistically its information source, namely the power grid itself. In this paper we studied both the topological and electrical characteristics of power grid networks based on a number of synthetic and real-world power systems. We made several interesting discoveries: the power grids are sparsely connected and the average nodal degree is very stable regardless of network size; the nodal degrees distribution has exponential tails, which can be approximated with a shifted Geometric distribution; the algebraic connectivity scales as a power function of network size with the power index lying between that of one-dimensional and two-dimensional lattice; the line impedance has a heavy-tailed distribution, which can be captured quite accurately by a Double Pareto LogNormal distribution. Based on the discoveries mentioned above, we propose an algorithm that generates random power grids featuring the same topology and electrical characteristics we found from the real data.

Competition and fragmentation: a simple model generating lognormal-like distributions

The current distribution of language size in terms of speaker population is generally described using a lognormal distribution. Analyzing the original real data we show how the double-Pareto lognormal distribution can give an alternative fit that indicates the existence of a power law tail. A simple Monte Carlo model is constructed based on the processes of competition and fragmentation. The results reproduce the power law tails of the real distribution well and give better results for a poorly connected topology of interactions.

Autoregressive processes with normal-Laplace marginals

The normal-Laplace distribution introduced by Reed and Jorgensen [Reed, W.J., Jorgensen, M., 2004. The double Pareto-lognormal distribution, A new parametric model for size distributions. Comm. Statist-Theory and Methods, 33, 1733–1753], is studied briefly. A first order autoregressive process with normal-Laplace stationary marginal distribution is introduced and various properties are discussed. The process gives a combination of Gaussian as well as non-Gaussian time series models for the first time and is free from the zero defect problem. Simulation studies are done and sample path properties are explored. Regression behaviour of the process is studied. The applications in modelling data from various contexts are also discussed.

Are Fluctuating Asymmetry Studies Adequately Sampled? Implications of a New Model for Size Distribution

Previous work on fluctuating asymmetry (FA) has highlighted its controversial relationship with environmental stress and genetic architecture. While size‐based measures of FA have been assumed to have half‐normal distributions within populations, studies of model developmental mechanisms have suggested other plausible distributions for FA. We investigated the distribution of FA in large empirical data sets of wing shape and wing size asymmetry from three species of insects (cotton aphid Aphis gossipyii Glover, honeybee Apis mellifera, and long‐legged fly Chrysosoma crinitus). Regardless of measurement method, FA was best described by a double Pareto–lognormal (DPLN) distribution or one of its limiting functional forms. To investigate convergence of mean sample FA to the population mean at various sample sizes, we sampled repeatedly under a DPLN distribution using parameter values that best fitted our data. Sample variances are much larger, and hence, convergence is slowed considerably with univariate or multivariate size‐based measures of FA in contrast to a multivariate shape‐based measure of FA. We suggest that much of the past work on FA may be undersampled, and we recommend using multivariate shape‐based approaches or collecting larger data sets in future studies. We also discuss the implications of the DPLN distribution for understanding the developmental mechanisms underlying FA.

Are Fluctuating Asymmetry Studies Adequately Sampled? Implications of a New Model for Size Distribution

Previous work on fluctuating asymmetry (FA) has highlighted its controversial relationship with environmental stress and genetic architecture. While size-based measures of FA have been assumed to have half-normal distributions within populations, studies of model developmental mechanisms have suggested other plausible distributions for FA. We investigated the distribution of FA in large empirical data sets of wing shape and wing size asymmetry from three species of insects (cotton aphid Aphis gossipyii Glover, honeybee Apis mellifera, and long-legged fly Chrysosoma crinitus). Regardless of measurement method, FA was best described by a double Pareto-lognormal (DPLN) distribution or one of its limiting functional forms. To investigate convergence of mean sample FA to the population mean at various sample sizes, we sampled repeatedly under a DPLN distribution using parameter values that best fitted our data. Sample variances are much larger, and hence, convergence is slowed considerably with univariate or multivariate size-based measures of FA in contrast to a multivariate shape-based measure of FA. We suggest that much of the past work on FA may be undersampled, and we recommend using multivariate shape-based approaches or collecting larger data sets in future studies. We also discuss the implications of the DPLN distribution for understanding the developmental mechanisms underlying FA.