Search

The paper explores the frequency and size distributions of firm-size in a novel dataset for the mid-Victorian era from a recent extraction of the England and Wales population censuses of 1851, 1861, 1871, and 1881. The paper contrasts the hypothesis of the Power Laws against the Lognormal model for the tails of the distributions using maximum likelihood estimation, log likelihood ratio, clipped sample coefficient of variation UMPU-Wilks test, Kolmogorov–Smirnov statistic, among other state-of-the-art statistical methods. Our results show that the Power Law hypothesis is accepted for the size distribution for the years 1851 and 1861, while 1871 is marginally non-significant, but for 1881 the test is inconclusive. The paper discusses the process that generates these distributions citing recent literature that shows how after adding an i.i.d. noise to the Gibrat's multiplicative model one can recreate a Power Law behaviour. Overall, the paper provides, describes and statistically tests for the very first time a unique historical dataset confirming that the tails of the distributions at least for 1851 and 1861 follow a Pareto model and that the Lognormal model is firmly rejected. • We test the tail behaviour of mid-Victorian firm-size distributions. • Pareto model fits the tail of the size distribution of firm size for 1851 and 1861. • The tests are inconclusive for the frequency distributions. • Modern data show a fattening of the tails. [ABSTRACT FROM AUTHOR]

Abstract Based on the probability integral transform statistic, Finkelstein et al. (2006) have proposed a robust estimator for the shape parameter of Pareto distribution. In this paper, based on the same method, a robust and efficient estimator for the shape parameter of the inverse Pareto distribution is developed assuming that the threshold parameter is known. To study the robustness properties of this new estimator, we derive the asymptotic variance, breakdown point and gross error sensitivity. However, since the inverse Pareto distribution is literally an inverse of the Pareto distribution, some derivations and proofs for the probability integral transform statistic estimator presented in this paper are found closely related to those provided by Finkelstein et al. (2006). The performance of this new estimator and the maximum likelihood estimator is assessed through a simulation study. For the application, an inverse Pareto distribution is fitted to the lower tail data of Malaysian household incomes for the year of 2014, involving the proposed estimator in order to allow for the presence of outliers. Based on the inverse Pareto model, the parametric Lorenz curve is fitted and the Gini coefficient is estimated to measure the income inequality of poor households in Malaysia. Highlights • This study proposes a robust and efficient estimator for the shape parameter of inverse Pareto distribution based on the probability integral transform statistic. • The performance of the new estimator and the maximum likelihood estimator is assessed through a simulation study. • It is found the new estimator gives a desirable protection against outliers especially for large sample size. • For the application, the inverse Pareto distribution is fitted to the lower tail data of the Malaysian household income involving the proposed estimator. [ABSTRACT FROM AUTHOR]

Abstract In this study, a robust semi-parametric approach which involves combining the empirical distribution and Pareto model is applied for measuring the income distribution in Malaysia based on the survey data of household incomes for the years of 2007, 2009, 2012 and 2014. From our analysis, it is found that the Pareto model is well fitted to the top household incomes for all years considered, suggesting that this model is a suitable parametric distribution to describe the top household incomes in Malaysia. Based on the semi-parametric Gini, generalized entropy and Atkinson indices, it is found that the income inequality shows a decreasing trend over the period, indicating an improvement of income distribution in Malaysia. Moreover, the fitted semi-parametric Lorenz curves show that the economic pie of the high income group reduced slightly after being taken up by the low income group while there is no clear change for the middle household income group. Highlights • A robust semi-parametric approach provides good results for measuring income inequality. • PITSE method provides good parameter estimation of Pareto model in the presence of outliers. • Various inequality indices based on semi-parametric model are provided. [ABSTRACT FROM AUTHOR]

The Pareto distribution is widely applied in many areas of studies such as economics and sciences. An important issues related to Pareto tail modelling is to determine the optimal threshold of the Pareto distribution. One of the methods used for determining the optimal threshold of Pareto distribution is by choosing the threshold that minimizes the goodness-of-fit statistics found based on empirical distribution function (EDF). This study involves determination of the shape parameter of the Pareto distribution using the maximum likelihood method and robust method based on the probability integral transform statistics. In addition, given the particular estimates of the shape parameter, comparison of the performance of several EDF statistics, namely, Kolmogorov–Smirnov, Kuiper, Anderson–Darling, Cramer–von Misses and Watson statistics in determining the optimal threshold in the presence of outliers is studied based on Monte Carlo simulation. Since the EDF statistics are found smallest for Kolmogorov–Smirnov or Kuiper statistics, these two EDF statistics outperformed other EDF statistics considered. The findings are illustrated using a sample of household income data of the Malaysian population. The optimal threshold found can be used to classify the high income earners in Malaysia since Pareto distribution is one of the most frequently used model to describe the upper tail of income distribution. [ABSTRACT FROM AUTHOR]

Understanding of the evolutionary mechanism of online social networks is greatly significant for the development of network science, but up to now most present researches on this topic have not enough insight. In this study, firstly we empirically showed the essential evolution characteristics of Renren online social network. The evolution mechanism of online social networks is explained by the perspective of Pareto wealth distribution and bidirectional preferential attachment. Then a novel model is proposed to reproduce the evolution characteristics which are consistent with the ones of Renren online social network, and the evolutionary analytical solution to the proposed model was presented. The results suggest that both Pareto wealth distribution and bidirectional preferential attachment play an important role in the evolution process of online social networks and can help us to understand the evolutionary origin of online social networks. The model has significant implications for dynamic simulation researches of social networks, especially in information diffusion through online communities and infection spreading in real societies. [ABSTRACT FROM AUTHOR]

We present a statistical model with 8 parameters that combines three components, namely a truncated lognormal distribution central body with a lower and an upper tail, both consisting of Generalized Pareto (GP) distributions. We fit the model to a dataset of more than 85 million records of the logarithm of fire radiative power (FRP) at the global scale, as derived from satellite observations covering a 19-year period (2002–2021). We use the model to characterize fire activity associated to three fire macro-regimes, designated Wild, Tamed and Domesticated, and to identify a global decreasing trend in fire intensity with moderate release of FRP (10–100 MW) during 2002–2021 that is consistent with agricultural expansion and intensification. The model can be applied to the calibration of meteorological fire danger indices that are used to assist fire prevention and suppression activities. • The distribution consists of a lognormal body and two generalized Pareto tails. • We apply the distribution to fire radiative power released by vegetation fires. • The dataset consists of 85 million records at the global level covering 19 years. • There is an excellent fit of the distribution to the dataset. • We use the model to assess changes in fire radiative power during the study period. [ABSTRACT FROM AUTHOR]

Abstract We examine the extreme behaviors at both the lower 5% and 1% quantile levels of the three exchange rates series (Japanese yen, Korean won, and New Taiwan dollar) against the US dollar between 2002 and 2017. We employ two econometric tools for specific purposes: (1) to determine the bootstrap goodness of fit test for the Generalized Pareto distribution (GPD) of the tail behaviors proposed by Villaseñor-Alva and González-Estrada (2009); and (2) to identify, estimate, and test the hidden Markov model (HMM). The testing outcomes mostly reject the GPD assumption for this study. HMM estimation outcomes provide the detailed pictures. They evidence the multiple structural breaks in the returns and the gaps between the qualified extremes. The respective properties of the exchange markets are revealed in the comparisons of the estimation outcomes in terms of transition matrix, response parameter, and bivariate analysis. In general, Japanese yen is evaluated as a resilient safe haven currency. Korean won exhibits longer distress duration. New Taiwan dollar occurs with heavier losses with longer gaps. Highlights • We employ two econometric tools for the specific purposes: (1) to determine the bootstrap goodness of fit test for the Generalized Pareto distribution (GPD) of the tail behaviors proposed by Villaseñor-Alva and González-Estrada (2009); and (2) to identify, estimate, and test the hidden Markov model (HMM). • The testing outcomes mostly reject the GPD assumption for this study. HMM estimation outcomes provide the detailed pictures. • They evidence the multiple structural breaks in the returns and the gaps between the qualified extremes. The respective properties of the exchange markets are revealed in the comparisons of the estimation outcomes in terms of transition matrix, response parameter, and bivariate analysis. [ABSTRACT FROM AUTHOR]

This study provides a systematic review of the existing literature on Zipf’s law for city size distribution. Existing empirical evidence suggests that Zipf’s law is not always observable even for the upper-tail cities of a territory. However, the controversy with empirical findings arises due to sample selection biases, methodological weaknesses and data limitations. The hypothesis of Zipf’s law is more likely to be rejected for the entire city size distribution and, in such case, alternative distributions have been suggested. On the contrary, the hypothesis is more likely to be accepted if better empirical methods are employed and cities are properly defined. The debate is still far from to be conclusive. In addition, we identify four emerging areas in Zipf’s law and city size distribution research including the size distribution of lower-tail cities, the size distribution of cities in sub-national regions, the alternative forms of Zipf’s law, and the relationship between Zipf’s law and the coherence property of the urban system. [ABSTRACT FROM AUTHOR]

We present a study of the capital income distribution and of its contribution to the total income (capital income share) using individual tax income data in Romania, for 2013 and 2014. Using a parametric representation we show that the capital income is Pareto distributed in the upper tail, with a Pareto coefficient α ∼ 1 . 44 which is much smaller than the corresponding coefficient for wage- and non-wage-income (excluding capital income), of α ∼ 2 . 53 . Including the capital income contribution has the effect of increasing the overall inequality measures. [ABSTRACT FROM AUTHOR]

The proper characterization of the size distribution and growth of firms represents an important issue in economics and business. We use the Maximum Entropy approach to assess the plausibility of the assumption that firm size follows Lognormal or Pareto distributions, which underlies most recent works on the subject. A comprehensive dataset covering the universe of French firms allows us to draw two major conclusions. First, the Pareto hypothesis for the whole distribution should be rejected. Second, by discriminating across firms based on the number of products sold and markets served, we find that, within the class of multi-product companies active in multiple markets, the distribution converges to a Zipf’s law. Conversely, Lognormal distribution is a good benchmark for small single-product firms. The size distribution of firms largely depends on firms’ diversification patterns. [ABSTRACT FROM AUTHOR]

We apply five distributions–lognormal, double-Pareto lognormal, lognormal-upper tail Pareto, Pareto tails-lognormal, and Pareto tails-lognormal with differentiability restrictions–to estimate the size distribution of all Indian cities. Since India contains numerous small cities, it is important to explicitly model the lower-tail behavior for studying the distribution of all Indian cities. Our results rigorously confirm, using both graphical and formal statistical tests, that among these five distributions, Pareto tails-lognormal is a better suited parametrization of the Indian city size data, verifying that the Indian city size distribution exhibits a strong reverse Pareto in the lower tail, lognormal in the mid-range body, and Pareto in the upper tail. [ABSTRACT FROM AUTHOR]

Using the empirical data from the Norwegian Tax Administration, we analyze the wealth and income of the richest individuals in Norway during the period 2010–2013. We find that both annual income and wealth level of the richest individuals are describable using the Pareto law. We find that the robust mean Pareto exponent over the four-year period to be ≈2.3 for income and ≈1.5 for wealth. [ABSTRACT FROM AUTHOR]

In recent years, with a growing interest in big or large datasets, there has been a rise in the application of large graphs and networks to financial big data. Much of this research has focused on the construction and analysis of the network structure of stock markets, based on the relationships between stock prices. Motivated by Boginski et al. (2005), who studied the characteristics of a network structure of the US stock market, we construct network graphs of the UK stock market using same method. We fit four distributions to the degree density of the vertices from these graphs, the Pareto I, Fréchet, lognormal, and generalised Pareto distributions, and assess the goodness of fit. Our results show that the degree density of the complements of the market graphs, constructed using a negative threshold value close to zero, can be fitted well with the Fréchet and lognormal distributions. [ABSTRACT FROM AUTHOR]

We present a study of the distribution of the gross personal income and income inequality in Romania, using individual tax income data, and both non-parametric and parametric methods. Comparing with official results based on household budget surveys (the Family Budgets Survey and the EU-SILC data), we find that the latter underestimate the income share of the high income region, and the overall income inequality. A parametric study shows that the income distribution is well described by an exponential distribution in the low and middle incomes region, and by a Pareto distribution in the high income region with Pareto coefficient α = 2.53 . We note an anomaly in the distribution in the low incomes region ( ∼ 9,250 RON), and present a model which explains it in terms of partial income reporting. [ABSTRACT FROM AUTHOR]

The increased uncertainty and complexity of nonlinear systems have motivated investigators to consider generalized approaches to defining an entropy function. New insights are achieved by defining the average uncertainty in the probability domain as a transformation of entropy functions. The Shannon entropy when transformed to the probability domain is the weighted geometric mean of the probabilities. For the exponential and Gaussian distributions, we show that the weighted geometric mean of the distribution is equal to the density of the distribution at the location plus the scale (i.e. at the width of the distribution). The average uncertainty is generalized via the weighted generalized mean, in which the moment is a function of the nonlinear source. Both the Rényi and Tsallis entropies transform to this definition of the generalized average uncertainty in the probability domain. For the generalized Pareto and Student’s t-distributions, which are the maximum entropy distributions for these generalized entropies, the appropriate weighted generalized mean also equals the density of the distribution at the location plus scale. A coupled entropy function is proposed, which is equal to the normalized Tsallis entropy divided by one plus the coupling. [ABSTRACT FROM AUTHOR]

Despite Mexican peso crisis in 1994 followed by a severe economic recession, individual and household income distributions in the period 1992–2008 always exhibit a two-class structure; a highly fluctuating high-income class adjusted to a Pareto power-law distribution, and a low-income class (including poor and middle classes) adjusted to either Log-normal or Gamma distributions, where poor agents are defined as those with income below the maximum of the uni-modal distribution. Then the effects of crisis on the income distributions of the three classes are briefly analysed. [ABSTRACT FROM AUTHOR]

The degree distribution of a simulated Barabási–Albert graph under linear preferential attachment is investigated. Specifically, the parameters of the power law distribution are estimated and compared against the theoretical values derived using mean field theory. Least squares method and MLE-nonparametric method were utilized to estimate the distribution parameters on 1000 simulated Barabási–Albert graphs for edge parameter m ∈ { 2 , 4 , 6 } and size n ∈ { 2 k : k = 5 , 6 , … , 14 , 15 } . Goodness of fit metrics were computed on a second set of simulated graphs for the median of the estimated parameters and other hypothetical values for the distribution parameters. The results suggest that the distribution of the parameters from simulated graphs are significantly different from the theoretical distribution and is also dependent on m . Further results confirm the finding that the parameter of the power law distribution, β , increases as m increases. [ABSTRACT FROM AUTHOR]

We consider a distribution, which consists of lower tail Pareto, lognormal body, and upper tail Pareto, to estimate the size distribution of all US cities. This distribution fits the data more accurately than a distribution that comprises of only lognormal and the upper tail Pareto. [ABSTRACT FROM AUTHOR]

We consider a version of D. Price's model for the growth of a bibliographic network, where in each iteration, a constant number of citations is randomly allocated according to a weighted combination of the accidental (uniformly distributed) and the preferential (rich-get-richer) rule. Instead of relying on the typical master equation approach, we formulate and solve this problem in terms of the rank–size distribution. We show that, asymptotically, such a process leads to a Pareto-type 2 distribution with a new, appealingly interpretable parametrisation. We prove that the solution to the Price model expressed in terms of the rank–size distribution coincides with the expected values of order statistics in an independent Paretian sample. An empirical analysis of a large repository of academic papers yields a good fit not only in the tail of the distribution (as it is usually the case in the power law-like framework), but also across a significantly larger fraction of the data domain. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A new Pareto-type distribution is introduced and studied. This new model is a generalization of the well-known Pareto distribution. We derive some of its probabilistic and inferential properties. We deduce the mathematical form of the Lorenz curve and the Gini index associated with the new model. The maximum likelihood estimators are derived and their performance are evaluated through a Monte Carlo simulation study. Finally, we illustrate the flexibility of the new distribution by means of three applications to real data sets. [ABSTRACT FROM AUTHOR]

Studies that analyzed the size distribution of U.S. cities have mainly focused on the upper tail and showed that these cities adhere to Zipf’s law. However, even though a large number of cities are in the lower tail, very few studies have examined the distribution of these small cities because of data limitations. We apply reverse Pareto and reverse general Pareto distributions to analyze U.S. lower tail cities. Our results show the power law behavior of lower tail U.S. cities is accurately represented by both the reverse Pareto and general Pareto. [ABSTRACT FROM AUTHOR]

The circular inverse of the tangent function is used to simply derive a generalization of the Pareto distribution, the Pareto ArcTan (PAT) distribution. This model includes as limiting cases Pareto and Zipf distributions. This new probabilistic family is used to describe city size data for Australia and New Zealand. Urban agglomerations of these two countries presents similar features, a few large metropolitan areas with a steadily increasing population in the last years and many small cities. The PAT distribution improves the performance of other traditionally used models in urban agglomeration economics such as the classical Pareto, lognormal and the recently proposed Pareto positive stable. [ABSTRACT FROM AUTHOR]

The increasing inequality in income and wealth in recent years, and the associated excessive pay packages of CEOs in the US and elsewhere, is of growing concern among policy makers as well as the common person. However, there seems to be no satisfactory answer, in conventional economic theories and models, to the fundamental questions of what kind of income distribution we ought to see, at least under ideal conditions, in a free market environment, and whether this distribution is fair. We propose a novel microeconomic game theoretic framework that addresses these questions and proves that the lognormal distribution is the fairest inequality of pay in an organization comprising of homogeneous agents, under ideal free market conditions at equilibrium. We also show that for a population of two different classes of agents, the equilibrium distribution is a combination of two different lognormal distributions where one of them, corresponding to the top ∼3–5% of the population, can be misidentified as a Pareto distribution. We compare our predictions with empirical data on global income inequality trends provided by Piketty and others. Our analysis suggests that the Scandinavian countries, and to a lesser extent Switzerland, Netherlands and Australia, have managed to get close to the ideal distribution for the bottom ∼99% of the population, while the US and UK remain less fair at the other extreme. Other European countries such as France and Germany, and Japan and Canada, are in the middle. Our theory also shows the deep and direct connection between potential game theory and statistical mechanics through entropy, which we identify as a measure of fairness in a distribution. This leads us to propose the fair market hypothesis , that the self-organizing dynamics of the ideal free market, i.e., Adam Smith’s “invisible hand”, not only promotes efficiency but also maximizes fairness under the given constraints. [ABSTRACT FROM AUTHOR]

Predicting the occurrence of tail events is of great importance in financial risk management. By employing the method of peak-over-threshold (POT) in extreme value theory (EVT) to identify the financial extremes, we perform a recurrence interval analysis (RIA) on these extremes. We find that the waiting time between consecutive extremes (recurrence interval) follows a q -exponential distribution, and the sizes of extremes above the thresholds (exceeding size) conform to a generalized Pareto distribution. We also find that there is a significant correlation between recurrence intervals and exceeding sizes. We thus model the joint distribution of recurrence intervals and exceeding sizes through connecting the two corresponding marginal distributions with the Frank and Ali-Mikhail-Haq (AMH) copula functions, and apply this joint distribution to estimate the hazard probability to observe another extreme in Δ t time since the last extreme happened t time ago. Furthermore, an extreme predicting model based on RIA-EVT-Copula is proposed by applying a decision-making algorithm on the hazard probability. Both in-sample and out-of-sample tests reveal that this new extreme forecasting framework has better predicting performance than the forecasting model based on the hazard probability only estimated from the distribution of recurrence intervals. Our results not only shed a new light on understanding the occurring pattern of extremes in financial markets, but also improve the accuracy of predicting financial extremes for risk management. • We confirm the negative correlation between recurrence intervals and exceeding sizes. • The correlation between recurrence interval and exceeding size is modeled by copula. • We propose a RIA-EVT-Copula framework to predict the future extremes. [ABSTRACT FROM AUTHOR]

We first analyze the distribution of metropolitan (city) size, the distribution of metropolitan (city) GDP and the relation of both distributions. It is found that (1) the tails of distributions of size and GDP both obey Pareto Law with the Pareto exponent 1; (2) compared with Pareto exponent in GDP, Pareto exponent in size is bigger. Then an agent model is built to study the underlying formation mechanism of distributions of region size and GDP. Our model presents the mechanism how economic factors flow between regions to reproduce the tail behavior and the difference between the Pareto exponents of size and those of GDP. At last, the simulated results agree with the real empirical well. [ABSTRACT FROM AUTHOR]

In this article, we propose a new approach for studying the patterns of size distribution in settlement systems, based on the analysis of the shape of the Pareto curve (PC). To study the shape of the PC, we used the Gini coefficient, the asymmetry coefficient, and, by analogy with the physics of phase transitions, critical exponent — the index of the PC degree in the neighborhood of zero. An empirical analysis of the PC of various levels of aggregation in the US settlement system has been performed. The form of size distribution of states was studied by decades from 1790 to 2010. The spatial analysis of the PC shape for counties and cities was performed for 2010. The results of an empirical study showed that the PC of the states had left-hand asymmetry over 220 years. The PC of counties and cities had both right-hand and left-hand asymmetries. The obtained results explain in which cases the Pareto distribution having a PC with right-hand asymmetry, and the lognormal distribution with a symmetric PC may not correspond objectively to real settlement systems. As an alternative to power-series distribution and lognormal distribution, we considered an analytically simple two-parameter model with a wide range of PC asymmetry that combines the properties of power-series distribution and lognormal distribution. Verification of the model showed that it adequately described the size of settlements in homogeneous settlement systems. • The Pareto curves of the settlement system by state are left-handedly asymmetric. • The average asymmetry of settlement systems across districts and cities is zero. • Increased aggregation of systems reduces the concentration of inequalities. • The primate appears in the systems at the value of the critical index β < 0.585. [ABSTRACT FROM AUTHOR]

Abstract: According to previous studies that applied a popular goodness-of-fit test, the wealth of the world’s billionaires does not follow a Pareto distribution. The test applied by those studies assumes that wealth is measured without error, yet, if different sources of data on the wealthiest people in the world are compared, then wealth appears to be measured with error. This paper shows that the conclusions drawn from the goodness-of-fit test can change when the test is modified to account for measurement errors. [Copyright &y& Elsevier]

In a recently published paper in Physica A it is claimed the statistical evidence of having a power law distribution in the upper tail of the world billionaires' data in the period 2010–20. We address (almost) the same data sets, with the same truncation points in the data, and obtain two main conclusions. Firstly, both the truncated lognormal and the power law or Pareto distributions might be improved in principle to describe the upper tails' data to the level of quality of recent studies for other types of data, as the Anderson–Darling test yields rejection of both distributions almost always. Secondly, the truncated lognormal is statistically comparable or even better to the Pareto by different criteria, formal and graphical. [ABSTRACT FROM AUTHOR]

Abstract: This paper formulates a random-growth urban model with a notion of geographical fitness. Using techniques of complex-network theory, we study our system as a type of preferential-attachment model with fitness, and we analyze its macro behavior to clarify the properties of the city-size distributions it predicts. First, restricting the geographical fitness to take positive values and using a continuum approach, we show that the city-size distributions predicted by our model asymptotically approach Pareto distributions with coefficients greater than unity. Then, allowing the geographical fitness to take negative values, we perform local coefficient analysis to show that the predicted city-size distributions can deviate from Pareto distributions, as is often observed in actual city-size distributions. As a result, the model we propose can generate a generic class of city-size distributions, including but not limited to Pareto distributions. For applications to city-population projections, our simple model requires randomness only when new cities are created, not during their subsequent growth. This property leads to smooth trajectories of city population growth, in contrast to other models using Gibrat’s law. In addition, a discrete form of our dynamical equations can be used to estimate past city populations based on present-day data; this fact allows quantitative assessment of the performance of our model. Further study is needed to determine appropriate formulas for the geographical fitness. [Copyright &y& Elsevier]

Abstract: The key idea of this model is that firms are the result of an evolutionary process. Based on demand and supply considerations the evolutionary model presented here derives explicitly Gibrat’s law of proportionate effects as the result of the competition between products. Applying a preferential attachment mechanism for firms, the theory allows to establish the size distribution of products and firms. Also established are the growth rate and price distribution of consumer goods. Taking into account the characteristic property of human activities to occur in bursts, the model allows also an explanation of the size–variance relationship of the growth rate distribution of products and firms. Further the product life cycle, the learning (experience) curve and the market size in terms of the mean number of firms that can survive in a market are derived. The model also suggests the existence of an invariant of a market as the ratio of total profit to total revenue. The relationship between a neo-classic and an evolutionary view of a market is discussed. The comparison with empirical investigations suggests that the theory is able to describe the main stylized facts concerning the size and growth of firms. [Copyright &y& Elsevier]

Abstract: The allometric relationship between the basal metabolic rate B and total body mass M is , where the allometry coefficient a and the allometry exponent b have been fit to various data sets for over 150 years. The best fit of the allometry exponent to 391 mammalian species is given by Heusner  to be midway between the leading theoretical values of 2/3 and 3/4. Most theoretical investigations have focused on determining the proper value of b entailed by an appropriate biological model and with some notable exceptions ignored the allometry coefficient a altogether. Herein, we shift the focus and use the above data to settle on an empirical value of b that gives rise to an allometry coefficient with random variability described by a Pareto distribution. This new perspective suggests an interesting biological interpretation of the statistical fluctuations of the allometry coefficient. The time distribution of the intermittent fluctuations in a are determined to be of the same statistical form as those of speciation found in the punctuated equilibrium theory of macroevolution (Eldredge and Gould , Sneppen et al.  and Rikvold and Zia ). [Copyright &y& Elsevier]

Abstract: This work presents a comprehensive study of the evolution of the expenditure distribution in India. The consumption process is theoretically modeled based on certain physical assumptions. The proposed statistical model for the expenditure distribution may follow either a double Pareto distribution or a mixture of log-normal and Pareto distribution. The goodness-of-fit tests with the Indian data, collected from the National Sample Survey Organisation Reports for the years of 1983–2007, validate the proposal of a mixture of log-normal and Pareto distribution. The relative weight of the Pareto tail has a remarkable magnitude of approximately 10%–20% of the population. Moreover, though the Pareto tail is widening over time for the rural sector only, there is no significant change in the overall inequality measurement across the entire period of study. [ABSTRACT FROM AUTHOR]

Abstract: We study the validity of Zipf’s Law in a data set of Chinese city sizes for the years 1999–2004, when the numbers of cities remain almost constant after a rapid urbanization process during the period of the market-oriented economy and reform-open policy. Previous investigations are restricted to log–log rank–size regression for a fixed sample. In contrast, we use rolling sample regression methods in which the sample is changing with the truncation point. The intuition is that if the distribution is Pareto with a coefficient one (Zipf’s law holds), rolling sample regressions should yield a constant coefficient regardless of what the sample is. We find that the Pareto exponent is almost monotonically decreasing in the truncation point; the mean estimated coefficient is 0.84 for the full dataset, which is not so far from 1. [Copyright &y& Elsevier]

Abstract: The Pareto-positive stable (PPS) distribution is introduced as a new model for describing city size data in a country. The PPS distribution provides a flexible model for fitting the entire range of a set of city size data, where zero and unimodality are possible, and the classical Pareto and Zipf distributions are included as a particular case. Expressions for the shape, moments and other descriptive probabilistic measures are given. Estimation methods are discussed and a simple graphical method for studying the adequacy of the data to model is given. Finally, we consider city size data for Spain for several different years. The new distribution is compared with three classical models: Pareto, lognormal and Tsallis distributions. In all the data sets considered, the PPS distribution outperforms the fits of these three previous distributions. [Copyright &y& Elsevier]

Abstract: Government regulators and other concerned citizens warily view multilevel marketing enterprises (MLM) because of their close operational resemblance to exploitative pyramid schemes. We analyze two types of MLM network architectures — the unilevel and binary, in terms of growth behavior and earning potential among members. We show that network growth decelerates after reaching a size threshold, contrary to claims of unrestricted growth by MLM recruiters. We have also found that the earning potential in binary MLM’s obey the Pareto “80–20” rule, implying an earning opportunity that is strongly biased against the most recent members. On the other hand, unilevel MLM’s do not exhibit the Pareto earning distribution and earning potential is independent of member position in the network. Our analytical results agree well with field data taken from real-world MLM’s in the Philippines. Our analysis is generally valid and can be applied to other MLM architectures. [Copyright &y& Elsevier]

Abstract: We present a universal mechanism for the temporal generation of power-law distributions with arbitrary integer-valued exponents. [Copyright &y& Elsevier]

Abstract: The daily financial volume of transaction on the New York Stock Exchange and its day-to-day fluctuations are analysed with respect to power-law tails as well to their long-term trends. We also model the transition to a Gaussian distribution for longer time intervals, like months instead of days. [Copyright &y& Elsevier]

Abstract: This study considers a model of the income distribution of agents whose pairwise interaction is asymmetric and price-invariant. Asymmetric transactions are typical for chain-trading groups who arrange their business such that commodities move from senior to junior partners and money moves in the opposite direction. The price-invariance of transactions means that the probability of a pairwise interaction is a function of the ratio of incomes, which is independent of the price scale or absolute income level. These two features characterize the hierarchical model. The income distribution in this class of models is a well-defined double-Pareto function, which possesses Pareto tails for the upper and lower incomes. For gross and net upper incomes, the model predicts definite values of the Pareto exponents, and , which are stable with respect to quantitative variation of the pair-interaction. The Pareto exponents are also stable with respect to the choice of a demand function within two classes of status-dependent behavior of agents: linear demand (, ) and unlimited slowly varying demand (). For the sigmoidal demand that describes limited returns, , with some satisfying a transcendental equation. The low-income distribution may be singular or vanishing in the neighborhood of the minimal income; in any case, it is -integrable and its Pareto exponent is given explicitly. The theory used in the present study is based on a simple balance equation and new results from multiplicative Markov chains and exponential moments of random geometric progressions. [Copyright &y& Elsevier]

We study Sutton''s ‘microcanonical’ model for the internal organization of firms, that leads to non-trivial scaling properties for the statistics of growth rates. We show that the growth rates are asymptotically Gaussian in this model, whereas empirical results suggest that the kurtosis of the distribution increases with size. We also obtain the conditional distribution of the number and size of sub-sectors in Sutton''s model. We formulate and solve an alternative model, based on the assumption that the sector sizes follow a power-law distribution. We find in this new model both anomalous scaling of the variance of growth rates and non-Gaussian asymptotic distributions. We give some testable predictions of the two models that would differentiate them further. We also discuss why the growth rate statistics at the country level and at the company level should be identical. [Copyright &y& Elsevier]

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. • The degree of compliance with Gibrat's Law in the Andalusian sectors is limited. • Sectors complying with Gibrat's Law are characteristically made up of large firms. • The economic situation appears to have affected the compliance with Gibrat's Law. • The double log-normal Pareto distribution fits well the firms' size distribution. [ABSTRACT FROM AUTHOR]

The Pareto distribution is known for its wide range of applications, including the distribution of firm sizes. Exhaustive databases on firm sizes showed deviations from the Pareto firm size distribution for the smallest and largest firm sizes. Therefore, stochastic models of firm dynamics reproducing a Pareto firm size distribution could be further generalized in order to also reproduce these deviations. Based on Córdoba (2008), we build a model of firm dynamics which can generate a wide variety of steady-state distributions and we make clear the relationship between the growth dynamics of the firms and the resulting shape of the firm size distribution. To the light of our model, we analyse the links between the observed firm dynamics and the shape of firm size distribution in Belgium between 2006 and 2013. [ABSTRACT FROM AUTHOR]

Campolieti (2020) modelled the number of members in different unions (union size) in different decades in Canada using the power-law, lognormal and stretched exponential distributions. Here, we show that the generalized Pareto distribution (which has the same number of parameters as the power-law, lognormal and stretched exponential distributions) can provide better fits for the data. • The generalized Pareto distribution provides better fits for the union size data. • Evidence of better fit given in terms of AIC, BIC and AICc. • Evidence of better fit given also in terms of the Kolmogorov–Smirnov statistic. [ABSTRACT FROM AUTHOR]

In practice, income distributions are often contaminated with outliers that can affect the estimation of the income inequality. In this study, we propose a robust method for measuring income inequality based on a semi-parametric approach that applicable to positive income data. The semi-parametric approach introduced here combines the inverse-Pareto, empirical, and Pareto distributions. In addition, a robust estimation method based on probability integral transform statistic is applied to estimate the shape parameters of the inverse-Pareto and Pareto models to allow for the presence of outliers in the lower and upper tails of an income distribution. Then, a semi-parametric form of the Lorenz curve and three inequality measures are derived, including the Gini coefficient, generalized entropy index, and Atkinson index. We conduct a simulation study to compare the performance of the proposed semi-parametric approach with that of the conventional non-parametric method for estimating income inequality in the presence of outliers. The results show that the income inequality measure based on the proposed semi-parametric approach outperforms the conventional non-parametric method. Lastly, we apply the proposed semi-parametric approach in the measurement of income inequality among households in Malaysia based on survey data of household incomes for the years 2007, 2009, 2012, and 2014. • A robust semi-parametric approach for measuring income inequality is proposed. • The PITSE is used to estimate the tail index of the inverse-Pareto and Pareto. • A semi-parametric form of several inequality indices are derived. • The proposed approach is applied for measuring income inequality in Malaysia. [ABSTRACT FROM AUTHOR]

Building on the two-class income model (Banerjee and Yakovenko, 2010), this paper proposes a new methodology to estimate the income distribution that accounts for hidden sources of income, such as retained earnings. We promote and illustrate this methodology with a case study of the Chilean economy. Our results reveal that after including the hidden sources of income, economic inequality in Chile has remained relatively constant over the last 15 years. • We conceptualize the idea of hidden incomes using Yakovenko et al.'s model. • We propose a new method to estimate income distributions that accounts for hidden incomes, such as retained earnings. • We illustrate this methodology with a case study of the Chilean economy. • Income inequality in Chile has remained relatively constant over the last 15 years. [ABSTRACT FROM AUTHOR]

It has been suggested that occupations where one is paid by the hour are not scalable, while scalable occupations allow one to make more money without an equivalent increase in labor and time. Non-scalable occupations are expected to have low income variance, whereas scalable ones show large income inequalities. This study examines the evidence for this suggested categorizing using personal earnings microdata for twelve candidate occupations of both types, scalable and not. We find the upper tails of all distributions decay as power laws. Moreover, we cannot reject the suggested categorizing for earnings above medians. • We examine the evidence for the distinction between scalable and non-scalable occupations. • We consider earnings data from 12 selected candidate occupations, scalable and not. • We find the distinction makes sense for the portion of data of extreme earnings above medians. • The upper tails of all distributions of earnings decay as power laws. • The distributions of the scalable occupations exhibit heavier tails. [ABSTRACT FROM AUTHOR]

This paper connects the Generalized Beta 2 distribution with a plausible stochastic model of economic exchange (Hertzler, 2003). The GB2 is the steady-state equilibrium distribution of this model of exchange which, along with unique properties of the GB2, provides an explanation for its success in describing both income and wealth distributions, and financial market volatility. This brings about a better understanding of the parametric dependencies of the distribution and quantities derived from it, such as the Gini, Hoover and Theil inequality measures. Our examination of the properties of these measures in relation to the GB2 motivates the proposal of an alternative measure of inequality which is less prone to extremes stemming from the fat tails of the GB2. Maximum likelihood estimation methods are applied to housing sales prices as an application and to validate more accurate measures of goodness of fit, such as Kolmogorov–Smirnov statistic. • Stochastic Model of Economic Exchange. • Generalized Beta Prime distribution. • Expansion, Simplification of Inequality Measures. • New Measure of Inequality. [ABSTRACT FROM AUTHOR]

This study proposes a reverse Pareto model to describe the power-law behavior for the lower tail data of income distribution and illustrates an application on Malaysian and Italian household income data. A robust method based on probability integral transform statistic is used for estimating the shape parameter of the reverse Pareto model to allow for the existence of outlying observations in the lower tail data. Besides that, the optimal threshold of reverse Pareto is determined by using Kolmogorov–Smirnov statistic. It is found that the fitted reverse Pareto adequately describes the lower tail data of both datasets, suggesting that the power-law behavior is obeyed. In addition, the estimated optimal threshold of reverse Pareto model can be utilized as an alternative measure for the relative poverty line. Based on the reverse Pareto model, the Lorenz curve, Gini and Theil coefficients are determined, and it is found that low income inequality is observed for the period of the study. The fitted Lorenz curve shows that nearly 80% of the total household income is owned by the bottom 80%, whereas the remaining total household income is owned by the top 20%. Finally, comparison of the reverse Pareto model with some alternative distributions such as shifted reverse exponential, shifted reverse stretched exponential and shifted reverse lognormal in terms of model fitting for the lower tail data is also conducted. The results show that the reverse Pareto model outperform all the other models. • The reverse Pareto model is proposed to describe the lower tail of income distribution. • Power-law behavior is obeyed in the lower tail data. • Robust method is applied since outliers are present. • The estimated threshold is interpreted as relative poverty line. [ABSTRACT FROM AUTHOR]

An asymmetric power autoregressive conditional heteroscedasticity with generalized Pareto distribution (APARCH-GPD) model is proposed in this study to determine the optimal margin level for the Hang Seng Index futures contracts on the Hong Kong Futures Exchange. This method requires two steps. First, the APARCH model is used to measure the time-varying volatility of futures contract returns. Then, the tail distribution of the residuals from APARCH model is estimated by the GPD on the basis of the extreme value theory. Value-at-risk is finally estimated and predicted by the APARCH-GPD model, and this is compared with the APARCH-t and EWMA models by backtesting historical return series. The experimental results show that the long trading position of the Hang Seng Index futures contract undertakes more risk than the short trading position. Moreover, the APARCH-GPD model offers better one-day forecasting on both positions than the other models. The findings of this study provide important implications for making decisions on financial risk management. • A hybrid model is proposed to measure the tail risk of futures for margin setting. • Compared the model with other methods based on different distribution assumption. • This model gives more accurate margin setting and guarantees the market liquidity. • This method performs well when residuals show asymmetrical light-tail distribution. [ABSTRACT FROM AUTHOR]

In order to further improve the performance of the Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA/D) in solving multi-objective optimization problems, this paper constructs a multi-objective optimization algorithm by taking MOEA/D as the multi-objective framework and the Squirrel Search Algorithm (SSA) as the core evolutionary strategy. Besides, both of them are modified and the Multi-objective Improved Squirrel Search Algorithm based on Decomposition with External Population and Adaptive Weight Vectors Adjustment (MOEA/D-EWA-ISSA) is proposed. MOEA/D-EWA-ISSA establishes an external population for every individual to retain the evolutionary information and maintain the population diversity, external individuals participate in evolution and produce better offspring, the convergence and distribution of Pareto Front (PF) are improved. As for SSA, the jumping search method and the progressive search method are introduced to it, different evolutionary strategies are provided to solve subproblems, which further improves the ability of core evolutionary strategy to solve subproblems and the convergence of the obtained PF. Furthermore, MOEA/D-EWA-ISSA adjusts every weight vector adaptively according to the population's actual evolutionary direction and the representative neighbor weight vectors, the distribution of the obtained PF is improved as well. The experimental results on multi-objective test functions show that the convergence and distribution of PF have obvious improvement when the improved SSA, the improved MOEA/D and the whole MOEA/D-EWA-ISSA are used to solve multi-objective optimization problems. • This paper proposes a multi-objective optimization algorithm to solve the multi-objective optimization problems. • The proposed algorithm improves the core evolutionary strategy Squirrel Search Algorithm and the multi-objective framework MOEA/D, the combination of them constitutes the proposed multi-objective optimization algorithm. • The proposed algorithm revised SSA and MOEA/D according to different evolutionary requirements of MOEA/D and the characters of SSA. • Three groups of experiments verify the effectiveness of the improved evolutionary strategy, the improved MOEA/D framework and the complete multi-objective optimization algorithm respectively. • The multi-objective optimization problems especially the complex ones can be solved much better. [ABSTRACT FROM AUTHOR]