Pareto-type Distribution Research Articles

Examining Tail Index Estimators in New Pareto Distribution: Monte Carlo Simulations and Income Data Applications

An evolved form of Pareto distribution, the new Pareto-type distribution, offers an alternative model for data with heavy-tailed characteristics. This investigation examines and discusses fourteen diverse estimators for the tail index of the new Pareto-type, including estimators such as maximum likelihood, method of moments, maximum product of spacing, its modified version, ordinary least squares, weighted least squares, percentile, Kolmogorov-Smirnov, Anderson-Darling, its modified version, Cramér-von Mises, and Zhang's variants of the previous three. Using Monte Carlo simulations, the effectiveness of these estimators is compared both with and without the presence of outliers. The findings show that, without outliers, the maximum product of spacing, its modified version, and maximum likelihood are the most effective estimators. In contrast, with outliers present, the top performers are Cramér-von Mises, ordinary least squares, and weighted least squares. The study further introduces a graphical method called the new Pareto-type quantile plot for validating the new Pareto-type assumptions and outlines a stepwise process to identify the optimal threshold for this distribution. Concluding the study, the new Pareto-type distribution is employed to model the high-end household income data from Italy and Malaysia, leveraging all the methodologies proposed.

Bayesian Estimation of a New Pareto-Type Distribution Based on Mixed Gibbs Sampling Algorithm

In this paper, based on the mixed Gibbs sampling algorithm, a Bayesian estimation procedure is proposed for a new Pareto-type distribution in the case of complete and type II censored samples. Simulation studies show that the proposed method is consistently superior to the maximize likelihood estimation in the context of small samples. Also, an analysis of some real data is provided to test the Bayesian estimation.

Conditional tail moment and reinsurance premium estimation under random right censoring

We propose an estimator of the conditional tail moment (CTM) when the data are subject to random censorship. The variable of main interest and the censoring variable both follow a Pareto-type distribution. We establish the asymptotic properties of our estimator and discuss bias-reduction. Then, the CTM is used to estimate, in case of censorship, the premium principle for excess-of-loss reinsurance. The finite sample properties of the proposed estimators are investigated with a simulation study and we illustrate their practical applicability on a dataset of motor third party liability insurance.

Inference Based on New Pareto-Type Records With Applications to Precipitation and Covid-19 Data

We consider estimation and prediction of future records based on observed records from the new Pareto type distribution proposed recently by Bourguignon et al. (2016), “M. Bourguignon, H. Saulo, R. N. Fernandez, A new Pareto-type distribution with applications in reliability and income data, Physica A, 457 (2016), 166-175.”. We derived several point predictors for a future record on the basis of the first n records. Two real data sets on precipitation and Covid 19 are analysed and a Monte Carlo simulation study has been performed to evaluate the statistical performance of point predictors presented in this paper.

Modeling the Incomes of the Upper-Class Group in Malaysia using New Pareto-Type Distribution

The new Pareto-type distribution has been previously introduced as an alternative to the conventional Pareto distribution in modeling income distribution. It is claimed to provide better flexibility for mathematical simplicity of probability functions and has a more straightforward mathematical form. In this study, the new Pareto-type distribution is used to model the income of the Malaysian upper-class group. The threshold is determined using the fixed proportion technique and the maximum likelihood estimator method is used to estimate the shape parameter. Then, the goodness-of-fit of the fitted new Pareto model is measured using the coefficient of determination, R2 and Kolmogorov–Smirnov statistics. We also measure the income inequality among the Malaysian top income earners using the Lorenz curve, Gini and Theil indices based on the fitted new Pareto model. Finally, the new Pareto distribution is compared to alternative distributions to analyze which model can give the best fit for the data. Our analysis shows that the Pareto type-1 and the new Pareto models are well fitted to the top income data for all years considered. However, the new Pareto model provides better flexibility which covering more incomes in the upper tail of the distribution than the Pareto type-1 model.

Identifiability of Asymmetric Circular and Cylindrical Distributions

Identifiability of statistical models is a fundamental and essential condition that is required to prove the consistency of maximum likelihood estimators. The identifiability of the skew families of distributions on the circle and cylinder for estimating model parameters has not been fully investigated in the literature. In this paper, a new method combining the trigonometric moments and the simultaneous Diophantine approximation is proposed to prove the identifiability of asymmetric circular and cylindrical distributions. Using this method, we prove the identifiability of general sine-skewed circular distributions, including the sine-skewed von Mises and sine-skewed wrapped Cauchy distributions, and that of a M\"{o}bius transformed cardioid distribution, which can be regarded as asymmetric distributions on the unit circle. In addition, we prove the identifiability of two cylindrical distributions wherein both marginal distributions of a circular random variable are the sine-skewed wrapped Cauchy distribution, and conditional distributions of a random variable on the non-negative real line given the circular random variable are a Weibull distribution and a generalized Pareto-type distribution, respectively.

Embodied GHG emissions of high speed rail stations: Quantification, data-driven prediction and cost-benefit analysis

PurposeThe embodied green-house gas (GHG) emissions from the high speed rail (HSR) stations are non-negligible given the considerable scale of modern HSR stations that serve as multifunctional complex. However, the magnitude of the GHG emissions is often underestimated owing to the limited use of reliable inventory data, as well as the simplified modelling that fails to capture the versatile structural/architectural features. To fill the knowledge gap, this paper reviewed the design documents of 117 HSR stations in China to provide more reliable quantification and reveal the factors associated with the GHG emissions. MethodsThe embodied GHG emissions of the HSR stations were quantified by use of the actual inventory data. Based on the quantification results, a data-driven prediction model was constructed by an optimized general regression neural network (GRNN), which identified a link between the seven design parameters of HSR stations and the embodied GHG emissions. In addition, a cost-benefit analysis was conducted considering the balance between the embodied GHG emissions and serviceability. Resultsand discussion The results revealed a non-negligible contribution of HSR stations to the embodied GHG emissions of rail infrastructure. For regional HSR lines, the embodied GHG emissions from the stations could account for over 26% of the whole line. The results also revealed the large stations to be the major source of emissions, for which the embodied emissions per area are 2.84 times of the smaller stations. A Pareto-type distribution exists as the ten largest stations (in station area) contribute nearly 60% of the total emissions of the 117 samples. A canonical correlation analysis (CCA) revealed the station area to be the most prominent factor to the embodied GHG emissions, and the optimized GRNN with CCA-processed input vector make more reliable prediction. The regional HSR lines are subject to higher risk of unsound carbon investment, owing to the low service level of some stations.

A note on exact laws of large numbers for the range of a sample from Pareto-type distributions

We consider a triangular array of i.i.d. random variables which are positive and have a Pareto-type distribution. Denote by Rn the range in the nth row i.e. the difference between the maximal and minimal order statistics in this row. We prove the strong law of large numbers for weighted sums of (Rn)n∈N. The obtained theorem extends and generalizes some of the results known so far for Pareto distributions and arrays with fixed or logarithmically growing length of the rows.

On Bayesian approach to composite Pareto models.

In data modelling using the composite Pareto distribution, any observations above a particular threshold value are assumed to follow Pareto type distribution, whereas the rest of the observations are assumed to follow a different distribution. This paper proposes on the use of Bayesian approach to the composite Pareto models involving specification of the prior distribution on the proportion of data coming from the Pareto distribution, instead of assuming the prior distribution on the threshold, as often done in the literature. Based on a simulation study, it is found that the parameter estimates determined when using uniform prior on the proportion is less biased as compared to the point estimates determined when using uniform prior on the threshold. Applications on income data and finance are included for illustrative examples.

Estimation and prediction for a new Pareto-type distribution under progressive type-II censoring

In this paper, we consider the problems of estimating the unknown parameters as well as predicting the failure times of the removed units in multiple stages of the progressively censored sample coming from a new Pareto-type distribution. First, maximum likelihood, least squares and Bayesian methods (Lindley’s approximation and Markov chain Monte Carlo) are applied for estimating the parameters involved in this model. For predicting the failure times of the removed units, different prediction methods including maximum likelihood, best unbiased, conditional median and Bayesian prediction methods are adopted. We also develop prediction intervals of future lifetimes using pivot and highest conditional density methods. Comprehensive Monte Carlo simulations are performed to assess the performance of the estimation methods as well as prediction methods. Finally, analysis of a real data set representing the duration of remission of leukemia patients who were treated by a specific drug is performed for illustration purposes.

Robust estimation of Pareto-type tail index through an exponential regression model

In this paper, we introduce a robust estimator of the tail index of a Pareto-type distribution. The estimator is obtained through the use of the minimum density power divergence with an exponential regression model for log-spacings of top order statistics. The proposed estimator is compared to existing minimum density power divergence estimators of the tail index based on fitting an extended Pareto distribution and exponential regression model on log-ratio of spacings of order statistics. We derive the influence function and gross error sensitivity of the proposed estimator of the tail index to study its robustness properties. In addition, a simulation study is conducted to assess the performance of the estimators under different contaminated samples from different distributions. The results show that our proposed estimator of the tail index has better mean square errors and is less sensitive to an increase in the number of top order statistics. In addition, the estimation of the exponential regression model yields estimates of second-order parameters that can be used for estimation of extreme events such as quantiles and exceedance probabilities. The proposed estimator is illustrated with practical datasets on insurance claims and calcium content in soil samples.

TEMPERED PARETO-TYPE MODELLING USING WEIBULL DISTRIBUTIONS

AbstractIn various applications of heavy-tail modelling, the assumed Pareto behaviour is tempered ultimately in the range of the largest data. In insurance applications, claim payments are influenced by claim management and claims may, for instance, be subject to a higher level of inspection at highest damage levels leading to weaker tails than apparent from modal claims. Generalizing earlier results of Meerschaert et al. (2012) and Raschke (2020), in this paper we consider tempering of a Pareto-type distribution with a general Weibull distribution in a peaks-over-threshold approach. This requires to modulate the tempering parameters as a function of the chosen threshold. Modelling such a tempering effect is important in order to avoid overestimation of risk measures such as the value-at-risk at high quantiles. We use a pseudo maximum likelihood approach to estimate the model parameters and consider the estimation of extreme quantiles. We derive basic asymptotic results for the estimators, give illustrations with simulation experiments and apply the developed techniques to fire and liability insurance data, providing insight into the relevance of the tempering component in heavy-tail modelling.

The Probability Distribution of Worldwide Forest Areas

This paper analyses the probability distribution of worldwide forest areas. We find moderate support for a Pareto-type distribution (power law) using FAO data from 1990 to 2015. Power laws are common features of many complex systems in nature. A power law is a plausible model for the world probability distribution of forest areas in all examined years, although the log-normal distribution is a plausible alternative model that cannot be rejected. The random growth of forest areas could generate a power law or log-normal distribution. We study the change in forest coverage using parametric and non-parametric methods. We identified a slight convergence of forest areas over the time reviewed; however, random forest area growth cannot be rejected for most of the distribution of forest areas. Therefore, our results give support to theoretical models of stochastic forest growth.

Prediction Methods for Future Failure Times Based on Type-II Right-Censored Samples from New Pareto-Type Distribution

In this paper, we consider the prediction problem of the future failure times based on observed data from two-parameter Pareto-type distribution under Type-II right-censored samples. Different point predictors including best unbiased, maximum likelihood and conditional median predictors of the future failure times are obtained. The corresponding prediction intervals using pivotal quantity, conditional argument and shortest-length based method are also developed. The so-obtained point predictors and prediction intervals are compared via experimental numerical simulation. The optimality criteria considered for comparison purposes are prediction bias and mean square prediction error for point predictors and average length and coverage probability for prediction intervals. Real-life data representing the duration of the failure times of mechanical components as reported by Murthy (in: Wiley series in probability and statistics, Wiley, Hoboken, 2004) are analyzed for illustrative purposes.

Comparisons of Methods of Estimation for a New Pareto-type Distribution

Bourguignon et al. (2016) introduced a new Pareto-type distribution to model income and reliability data. The aim of this paper is to estimate the parameters of this distribution from both frequentist and Bayesian view points. The maximum likelihood estimates, method of moment estimates, percentile estimates, least square and weighted least square estimates and maximum product of spacing estimates are considered as frequentist estimates. We have also considered the Bayes estimates of the unknown parameters and the associated credible intervals. The Bayes estimates are computed using an importance sampling method. To evaluate the performance of the different estimates, a Monte Carlo simulation study is carried out. Some real life data sets have been analyzed for illustrative purposes.

Is Gibrat’s “Economic Inequality” lognormal?

In the seminal book “Les Inegalites Economiques,” Gibrat (Les Inegalites Economiques, Librairie du Recueil Sirey, Paris, 2013) proposed the law of proportional effect and claimed that a variety of empirical size distributions—such as income, wealth, firm size, and city size—obey the lognormal distribution. Gibrat’s law went on to become a stylized result stimulating a voluminous subsequent research that has contributed to our understanding of stochastic growth processes and a statistical regularity of the size distribution. However, many of the motivating examples used by Gibrat in his original work were subject to various data issues, and Gibrat’s reasoning of lognormal fit was based solely on graphical analysis. In this paper, we revisit the original 24 data sets considered by Gibrat (Les Inegalites Economiques, Librairie du Recueil Sirey, Paris, 2013) and show that in the majority of cases, the Pareto-type distribution actually provides a better fit to the data than lognormal. We show that Gibrat’s erroneous conclusion is partly due to data binning, truncation, and failure to weight data points properly.

Local Robust Estimation of Pareto-Type Tails with Random Right Censoring

We propose a nonparametric robust estimator for the tail index of a conditional Pareto-type distribution in the presence of censoring and random covariates. The censored distribution is also of Pareto-type and the index is estimated locally within a narrow neighbourhood of the point of interest in the covariate space using the minimum density power divergence method. The main asymptotic properties of our robust estimator are derived under mild regularity conditions and its finite sample performance is illustrated on a small simulation study. A real data example is included to illustrate the practical applicability of the estimator.

On a new Pareto-type distribution with applications in the study of income inequality and risk analysis

A recent paper published in this journal (Bourguignon et al., 2016) introduces and studies a new Pareto-type distribution, deriving some of its probabilistic and inferential properties. In this paper, we obtain additional and important properties of this distribution and derive simpler expressions for relevant inequality indices and risk measures. One important feature of this distribution is that many of these expressions are simple functions of the incomplete beta function. Using this result, we obtain closed expressions for the moments and for several inequality measures including the Gini index and the Lorenz curve, the Donaldson–Weymark–Kakwani index, the Pietra index and the generalized entropy measures. Basic risk measures are also obtained. Finally, empirical applications with real income data are given.

A cylindrical distribution with heavy-tailed linear part

A cylindrical distribution whose linear part models heavy-tailedness is proposed. The conditional distribution of the linear variable given the circular variable is a generalized Pareto-type distribution. Therefore, it may not have any conditional moments; however, the mode and median have closed-form expressions. The circular marginal distribution is a wrapped Cauchy distribution, and the conditional distribution of the circular variable given the linear variable belongs to a family of symmetric distributions. These properties allow its application to cylindrical data, whose linear observations may take large values and whose circular observations are symmetric. As illustrative examples, the proposed distribution is fitted to two data sets, and the results are compared with those by other cylindrical distributions that cannot model heavy-tailedness for the linear parts.

Concentration measures in emerging banking

In this study, we investigate the sensitivity of different concentration measures, such as classical index concentration and spatial index concentration, to varying regimes for Zipf's exponent (Ɓ) for the Pareto-type distribution of bank sizes. We establish relationships between each concentration measure and Zipf's exponent by introducing the Riemann zeta function and calculating the elasticity for each index. We prove that the spatial concentration index is the most robust for varying regimes. Therefore, the choice of an appropriate concentration index must be carefully considered before drawing inferences about the relationship between concentration and banking fragility.