Pareto-type Distributions Research Articles

An Asymptotic Result on Catastrophe Insurance Losses

Consider an insurer who both sells catastrophe insurance policies and makes risky investments. Suppose that insurance claims arrive according to a Poisson process and the price of the investment portfolio evolves according to a general stochastic process independent of the insurance claims. In the focus of catastrophe risk management are catastrophe insurance losses. For the case of heavy-tailed claims, we derive a simple asymptotic formula for the tail probability of the present value of future claims. The transparent expression of our formula explicitly reflects the different roles of the various underlying risks in driving catastrophic losses. Our work is distinguished from most other works in this strand of research in that we carry out the asymptotic study over the whole class of subexponential distributions. Thus, our work allows both very heavy-tailed distributions such as Pareto-type distributions and moderately heavy-tailed distributions such as Lognormal and Weibull distributions.

Tail index estimation in the presence of covariates: Stock returns’ tail risk dynamics

This paper provides novel theoretical results for the estimation of the conditional tail index of Pareto and Pareto-type distributions in a time series context. We show that both the estimators and relevant test statistics are normally distributed in the limit, when independent and identically distributed or dependent data are considered. Simulation results provide support for the theoretical findings and highlight the good finite sample properties of the approach in a time series context. The proposed methodology is then used to analyse stock returns’ tail risk dynamics. Two empirical applications are provided. The first consists in testing whether the time-varying tail exponents across firms follow Kelly and Jiang’s (2014) assumption of common firm level tail dynamics. The results obtained from our sample seem not to favour this hypothesis. The second application, consists of the evaluation of the impact of two market risk indicators, VIX and Expected Shortfall (ES) and two firm specific covariates, capitalization and market-to-book on stocks tail risk dynamics. Although all variables seem important drivers of firms’ tail risk dynamics, it is found that ES and firms’ capitalization seem to have overall wider impact.

Semiparametric tail-index estimation for randomly right-truncated heavy-tailed data

PurposeThe purpose of this paper is to propose a semiparametric estimator for the tail index of Pareto-type random truncated data that improves the existing ones in terms of mean square error. Moreover, we establish its consistency and asymptotic normality.Design/methodology/approachTo construct a root mean squared error (RMSE)-reduced estimator of the tail index, the authors used the semiparametric estimator of the underlying distribution function given by Wang (1989). This allows us to define the corresponding tail process and provide a weak approximation to this one. By means of a functional representation of the given estimator of the tail index and by using this weak approximation, the authors establish the asymptotic normality of the aforementioned RMSE-reduced estimator.FindingsIn basis on a semiparametric estimator of the underlying distribution function, the authors proposed a new estimation method to the tail index of Pareto-type distributions for randomly right-truncated data. Compared with the existing ones, this estimator behaves well both in terms of bias and RMSE. A useful weak approximation of the corresponding tail empirical process allowed us to establish both the consistency and asymptotic normality of the proposed estimator.Originality/valueA new tail semiparametric (empirical) process for truncated data is introduced, a new estimator for the tail index of Pareto-type truncated data is introduced and asymptotic normality of the proposed estimator is established.

A Note on Pareto-Type Distributions Parameterized by Its Mean and Precision Parameters

Pareto-type distributions are well-known distributions used to fit heavy-tailed data. However, the standard parameterizations used for Pareto-type distributions are poorly suited to modeling. On this note, we suggest new parameterizations that are better suited to the purpose. In addition, we propose many regression models where the response variable is Pareto-type distributed using new parameterizations that are indexed by mean and precision parameters. The main motivation for these new parametrizations is the useful interpretation of the regression coefficients in terms of the mean and precision, as is usual in the context of regression models. The parameter estimation of these new models is performed, based on the maximum likelihood paradigm. Some numerical illustrations of the estimators are presented with a discussion of the obtained results. Finally, we illustrate the practicality of the new models by means of two applications to real data sets.

DOES THE PARETO PRINCIPLE APPLY IN THE INNOVATION FIELD IN ROMANIA?

The technological transfer entities can obtain income by establishing robust partnerships with the business environment and using the fiscal facilities offered by the region where they produce the innovation, applying for non-refundable funds, attracting individual donors, charging a membership contribution to the organization members, exploitation of innovation produced by the sale of intellectual property. A robust innovation ecosystem is based on public and private investments which complement each other to reduce the risks and to divide profits by increasing the capacity of innovation and technological transfer of the technological transfer entities. The Gauss distributions are continuous probability distributions. These are” normal” distributions which illustrate the likelihood that a certain event takes place. A gauss distribution has the shape of a bell with most values distributed around the average value and can be fully described by variance and mean. A classic example is the distribution of patents value or profit obtained because of sale of intellectual property rights. The Pareto-type distributions are found in other phenomena around us, such as the size of economic entities follows a Pareto distribution. Pareto-type distributions are more suitable to describe the economic phenomena than the Gauss distributions, for instance, 80% of the world richness is held by 20% of population (principle 80-20 which means 80% of results are caused by 20% of causes). The innovation phenomenon is deeply systemic, multifactorial, multifunctional, dynamic and complex, determined by the whole socio-economic background which influences the innovation ecosystem. Therefore, its study for the highlighting of the intervention opportunities at level of fiscal and financial policies, but also at operational level, requires an adequate, multilateral and complete systemic approach.

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.

Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications

In this paper, we introduce a new robust estimator for the extreme value index of Pareto-type distributions under randomly right-truncated data and establish its consistency and asymptotic normality. Our considerations are based on the Lynden-Bell integral and a useful huberized M-functional and M-estimators of the tail index. A simulation study is carried out to evaluate the robustness and the nite sample behavior of the proposed estimator. Extreme quantiles estimation is also derived and applied to real data-set of lifetimes of automobile brake pads.

Additive models for extremal quantile regression with Pareto-type distributions

Estimating conditional quantiles in the tail of a distribution is an important problem for several applications. However, data sparsity indicates that the predictions of tail behavior are more difficult compared with those for the mean or center quantiles, in particular, when a multivariate covariate is used. As additive models are known to be an efficient approach for multiple regression, this study considers an additive model for extremal quantile regression. The conditional quantile function is first estimated using a two-stage estimation method for the intermediate-order (not too extreme) quantile. Subsequently, the extreme-order quantile estimator is constructed by extrapolating from the intermediate-order quantile estimator. By combining the asymptotic and extreme value theories, the theoretical properties of the intermediate- and extreme-order quantile estimators are evaluated. A simulation study is conducted to confirm the performance of the estimators, and an application using real data is provided.

A Note on Exact Laws of Large Numbers for Asymmetric Pareto-type Distributions with Applications to Ratios of Random Variables

A Note on Exact Laws of Large Numbers for Asymmetric Pareto-type Distributions with Applications to Ratios of Random Variables

The laws of large numbers for Pareto-type random variables with infinite means

In this paper, we consider the laws of large numbers for NSD random variables satisfying Pareto-type distributions with infinite means. Based on the Pareto-Zipf distributions, some weak laws of large numbers for weighted sums of NSD random variables are obtained. Meanwhile, we show that a weak law for Pareto-Zipf distributions cannot be extended to a strong law. Furthermore, based on the two tailed Pareto distribution, a strong law of large numbers for weighed NSD random variables is presented. Our results extend the corresponding earlier ones.

Bias-corrected estimation for conditional Pareto-type distributions with random right censoring

We consider bias-reduced estimation of the extreme value index in conditional Pareto-type models with random covariates when the response variable is subject to random right censoring. The bias-correction is obtained by fitting the extended Pareto distribution locally to the relative excesses over a high threshold using the maximum likelihood method. Convergence in probability and asymptotic normality of the estimators are established under suitable assumptions. The finite sample behaviour is illustrated with a simulation experiment and the method is applied to two real datasets.

Estimating the Second-Order Parameter of Regular Variation and Bias Reduction in Tail Index Estimation Under Random Truncation

In this paper, we proposed an estimator of the second-order parameter of randomly truncated Pareto-type distributions data and establish its consistency and asymptotic normality. Moreover, we derive an asymptotically unbiased estimator for the tail index and study its limit distribution. We show, by simulation, that the proposed estimators behave well, in terms of bias, root mean square error and standard error.

Ridge regression estimators for the extreme value index

We consider bias reduced estimators of the extreme value index (EVI) in case of Pareto-type distributions and under all max-domains of attraction. To this purpose we revisit the regression approach started in Feuerverger and Hall (Ann. Stat. 27, 760–781, 1999) and Beirlant et al. (Extremes 2, 177–200, 1999) in the case of a positive EVI, and in Beirlant et al. (2005) for real-valued EVI. We generalize these approaches using ridge regression exploiting the mathematical fact that the bias tends to 0 when the number of top data points used in the estimation is decreased. The penalty parameter is selected by minimizing the asymptotic mean squared error of the proposed estimator. The accuracy and utility of the ridge regression estimators are studied using simulations and are illustrated with case studies on reinsurance claim size data as well as daily wind speed data.

Shannon Information in K-records for Pareto-type Distributions

Pareto distributions provides models for many applications in the social, natural and physical sciences. In this paper, we derive the Shannon information contained in upper (lower) k-record values and associated k-record times of Pareto-type distributions for a finite sample of fixed size and for an inverse sampling plan. Properties of the Shannon information of the k-record values associated with Pareto-type distributions are investigated, both analytically and numerically.

Word frequencies: A comparison of Pareto type distributions

Mehri and Jamaati (2017) [18] used Zipf's law to model word frequencies in Holy Bible translations for one hundred live languages. We compare the fit of Zipf's law to a number of Pareto type distributions. The latter distributions are shown to provide the best fit, as judged by a number of comparative plots and error measures. The fit of Zipf's law appears generally poor.

Tail index varying coefficient model

ABSTRACTThis paper deals with a new class of tail index varying coefficient models with the random covariate under Pareto-type distributions. To estimate the unknown coefficient functions, we develop an estimation procedure via a local polynomial maximum likelihood techniques. The asymptotic normality of the estimated coefficient functions under some mild regularity conditions are established. Two numerical examples and one application are used to illustrate the performance of the proposed procedure.

A Characterization of Exponential Distribution in Risk Model

In the general risk model (or the Sparre-Andersen model), it is well-known that the following assertion holds: if the claim size is exponentially distributed then the non-ruin probability distribution is a mixture of exponential distributions. In this paper, under some general conditions, we prove that the converse statement of the previous assertion is also true. Besides, we define a new non-ruin measure associated with the aggregate logarithms of the claim-over-profit ratios and obtain a result on Pareto-type distributions.

Fitting tails affected by truncation

In several applications, ultimately at the largest data, truncation effects can be observed when analysing tail characteristics of statistical distributions. In some cases truncation effects are forecasted through physical models such as the Gutenberg-Richter relation in geophysics, while at other instances the nature of the measurement process itself may cause under recovery of large values, for instance due to flooding in river discharge readings. Recently Beirlant et al. (2016) discussed tail fitting for truncated Pareto-type distributions. Using examples from earthquake analysis, hydrology and diamond valuation we demonstrate the need for a unified treatment of extreme value analysis for truncated heavy and light tails. We generalise the classical Peaks over Threshold approach for the different max-domains of attraction with shape parameter $\xi>-1/2$ to allow for truncation effects. We use a pseudo-maximum likelihood approach to estimate the model parameters and consider extreme quantile estimation and reconstruction of quantile levels before truncation whenever appropriate. We report on some simulation experiments and provide some basic asymptotic results.

Tail fitting for truncated and non-truncated Pareto-type distributions

In extreme value analysis, natural upper bounds can appear that truncate the probability tail. At other instances ultimately at the largest data, deviations from a Pareto tail behaviour become apparent. This matter is especially important when extrapolation outside the sample is required. Given that in practice one does not always know whether the distribution is truncated or not, we consider estimators for extreme quantiles both under truncated and non-truncated Pareto-type distributions. We make use of the estimator of the tail index for the truncated Pareto distribution first proposed in Aban et al. (J. Amer. Statist. Assoc. 101(473), 270–277, 2006). We also propose a truncated Pareto QQ-plot and a formal test for truncation in order to help deciding between a truncated and a non-truncated case. In this way we enlarge the possibilities of extreme value modelling using Pareto tails, offering an alternative scenario by adding a truncation point T that is large with respect to the available data. In the mathematical modelling we hence let T→∞ at different speeds compared to the limiting fraction (k/n→0) of data used in the extreme value estimation. This work is motivated using practical examples from different fields, simulation results, and some asymptotic results.

Bias reduced tail estimation for censored Pareto type distributions

We consider bias reduced estimators for the tail index and tail probabilities under random right censoring in case of Pareto-type distributions. The solution is based on second-order refined peaks-over-threshold modelling as developed in Beirlant et al. (2009).