We provide general expressions for the joint distributions of the k most significant b-ary digits and of the k leading continued fraction (CF) coefficients of outcomes of arbitrary continuous random variables. Our analysis highlights the connections between the two problems. In particular, we give the general convergence law of the distribution of the jth significant digit, which is the counterpart of the general convergence law of the distribution of the jth CF coefficient (Gauss-Kuz’min law). We also particularise our general results for Benford and Pareto random variables. The former particularisation allows us to show the central role played by Benford variables in the asymptotics of the general expressions, among several other results, including the analogue of Benford’s law for CFs. The particularisation for Pareto variables—which include Benford variables as a special case—is especially relevant in the context of pervasive scale-invariant phenomena, where Pareto variables occur much more frequently than Benford variables. This suggests that the Pareto expressions that we produce have wider applicability than their Benford counterparts in modelling most significant digits and leading CF coefficients of real data. Our results may find practical application in all areas where Benford’s law has been previously used.

Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$, $a > 0$. As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

A set of data with positive values follows a Pareto distribution if the log–log plot of value versus rank is approximately a straight line. A Pareto distribution satisfies Zipf’s law if the log–log plot has a slope of $-1$. Since many types of ranked data follow Zipf’s law, it is considered a form of universality. We propose a mathematical explanation for this phenomenon based on Atlas models and first-order models, systems of strictly positive continuous semimartingales with parameters that depend only on rank. We show that the stationary distribution of an Atlas model will follow Zipf’s law if and only if two natural conditions, conservation and completeness, are satisfied. Since Atlas models and first-order models can be constructed to approximate systems of time-dependent rank-based data, our results can explain the universality of Zipf’s law for such systems. However, ranked data generated by other means may follow non-Zipfian Pareto distributions. Hence, our results explain why Zipf’s law holds for word frequency, firm size, household wealth, and city size, while it does not hold for earthquake magnitude, cumulative book sales, and the intensity of wars, all of which follow non-Zipfian Pareto distributions.

Marine is the oldest type of insurance coverage. Nevertheless, unlike cargo and hull covers, marine liability is a rather young line of business with claims that can have very heavy and long tails. For reinsurers, the accumulation of losses from an event insured by various Protection and Indemnity clubs is an additional source for very large claims in the portfolio. In this paper, we first describe some recent developments of the marine liability market and then statistically analyze a data set of large losses for this line of business in a detailed manner both in terms of frequency and severity, including censoring techniques and tests for stationarity over time. We further formalize and examine an optimization problem that occurs for reinsurers participating in XL on XL coverages in this line of business and give illustrations of its solution.

In this paper, we introduce the simplest exponential dispersion model containing the Pareto and exponential distributions. In this way, we obtain distributions with support (0, ∞) that in a long interval are equivalent to the Pareto distribution; however, for very high values, decrease like the exponential. This model is useful for solving relevant problems that arise in the practical use of extreme value theory. The results are applied to two real examples, the first of these on the analysis of aggregate loss distributions associated to the quantitative modelling of operational risk. The second example shows that the new model improves adjustments to the destructive power of hurricanes, which are among the major causes of insurance losses worldwide.

In this paper, a new methodology based on the use of the inverse of the circular tangent function that allows us to add a scale parameter (say α) to an initial survival function is presented. The latter survival function is determined as limiting case when α tends to zero. By choosing as parent the classical Pareto survival function, the Pareto ArcTan (PAT) distribution is obtained. After providing a comprehensive analysis of its statistical properties, theoretical results with reference to insurance are illustrated. Its performance is compared, by means of the well-known Norwegian fire insurance data, with other existing heavy-tailed distributions in the literature such as Pareto, Stoppa, Shifted Lognormal, Inverse Gamma and Fréchet distributions.

Pareto distributions are most popular for modeling heavy tailed data. Here, we obtain weak limits of a sequence of extremal and a sequence of additive processes constructed by a series of Bernoulli point processes with bivariate Pareto space components. For the limiting processes we derive the one dimensional distributions in explicit forms. Some of the main properties of these distributions are also proved.

Random networks were first used to model epidemic dynamics in the 1950s, but in the last decade it has been realized that scale-free networks more accurately represent the network structure of many real-world situations. Here we give an analytical and a Monte Carlo method for approximating the basic reproduction number ${R}_{0} $ of an infectious agent on a network. We investigate how final epidemic size depends on ${R}_{0} $ and on network density in random networks and in scale-free networks with a Pareto exponent of 3. Our results show that: (i) an epidemic on a random network has the same average final size as an epidemic in a well-mixed population with the same value of ${R}_{0} $; (ii) an epidemic on a scale-free network has a larger average final size than in an equivalent well-mixed population if ${R}_{0} \lt 1$, and a smaller average final size than in a well-mixed population if ${R}_{0} \gt 1$; (iii) an epidemic on a scale-free network spreads more rapidly than an epidemic on a random network or in a well-mixed population.

We analyze fire exposure rating for three types of risk profiles: policy profiles, top location profiles and location profiles. Location profiles offer more detailed information than top location profiles, which in turn are better than policy profiles. We prove criteria to ensure that a better quality of risk profile leads to a lower price. These criteria are discussed with respect to standard exposure rating and an alternative method called ‘burning cost-adjusted exposure rating’, where the loss ratio is adjusted by means of the burning cost of a low reference layer. Further, we introduce a family of analytic exposure curves, the so-called EP exposure curves. These curves are useful for practical application, since the criteria given in this paper can be checked easily for exposure curves that can be approximated by EP exposure curves. This is of particular interest for discrete exposure curves. Finally, we apply the results to the MBBEFD exposure curves.

In this paper a method for determining benchmark rates for the excess of loss reinsurance of a Motor Third Party Liability insurance portfolio will be developed based on observed market rates. The benchmark rates are expressed as a percentage of the expected premium income that is available to cover the whole risk of the portfolio. The rates are assumed to be based on a compound process with a heavy tailed severity, such as Burr or Pareto distributions. In the absence of claim data these assumptions propagate the theoretical benchmark rate component of the regression model.

Given the whole set of excess of loss reinsurance rates in a given market, the unknown parameters are estimated within the framework of quasi-likelihood estimation. This framework makes it possible to select a theoretical benchmark rate model and to choose a parsimonious submodel for describing the observed market rates over a 4-years observation period. This method is applied to the Belgian Motor Third Party Liability excess of loss rates observed during the years 2001 till 2004.

It is well established that resource variability generated by spatial patchiness and turbulence is an importantinfluence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulationsindicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment canbe formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly, if jumps are of a fixed sizeand occur as a Poisson process (embedded within a drift-diffusion), recruitment is effectively described by a diffusion process alone.Secondly, in the absence of diffusion, and for “patchy” jumps (of negative binomial size with Pareto inter-arrivals), the encounter processbecomes superdiffusive. To synthesise these results we conduct a strategic simulation study where “patchy” jumps are embedded in adrift-diffusion process. We conclude that increasingly Lévy-like predator foraging strategies can have a significantly positive effect onrecruitment at the population level.

This paper addresses the Bayesian estimation of the shape parameter of Pareto distributions, and its application to premium calculation of large claims excess of loss (XL) reinsurance contracts. It studies the use of the generalized inverse Gaussian (GIG) as a Pareto prior conjugate, a family that contains as a particular case the gamma distribution. An exact credibility formula is deduced allowing the calculation of individual reinsurance premiums. These are premiums suited to the excesses history of a sole portfolio. A family of predictive distributions for the excesses is derived. We apply our exact credibility model to a sample of excesses arisen in ten Spanish portfolios of liability motor insurance from year 1992 to year 2001.

The inability to predict the future growth rates and earnings of growth stocks (such as biotechnology and internet stocks) leads to the high volatility of share prices and difficulty in applying the traditional valuation methods. This paper attempts to demonstrate that the high volatility of share prices can nevertheless be used in building a model that leads to a particular cross-sectional size distribution. The model focuses on both transient and steady-state behavior of the market capitalization of the stock, which in turn is modeled as a birth-death process. Numerical illustrations of the cross-sectional size distribution are also presented.

In this paper we study the asymptotic behavior of the tail of the stationary backlog distribution in a single server queue with constant service capacity c, fed by the so-called M/G/∞ input process or Cox input process. Asymptotic lower bounds are obtained for any distribution G and asymptotic upper bounds are derived when G is a subexponential distribution. We find the bounds to be tight in some instances, e.g. when G corresponds to either the Pareto or lognormal distribution and c − ρ < 1, where ρ is the arrival rate at the buffer.

We consider the problem of forecasting the total cost of claims in excess-of-loss reinsurance. The number of claims reported to the direct insurer is assumed to follow a Poisson law, and the claim severities are modelled by a Pareto distribution. The Poisson frequency as well as the Pareto parameter will be considered as random parameters in a Bayesian setting. We derive the class of conjugate joint prior distributions, which turn out to specify a (prior) dependence between the two parameters. The use of conjugate priors facilitates the mathematical analysis, and it also makes it easy to interpret the parameters of the prior distribution.

Instead of determining for a fire insurance portfolio the loss distribution purely based on the claims experience, we try to determine it based on the sums insured.

Semi-Pareto processes, of which Pareto processes form a proper sub-class, are discussed here. A semi-Pareto process has semi-Pareto inputs. Asymptotic properties of the maximum and minimum of the first n observations are examined as well as the geometric maximum and geometric minimum. A characterization of the semi-Pareto distribution is given. A canonical representation of a special class of Pareto process is also given.

In reliability studies, it is well known that the mean residual life function determines the distribution function uniquely. In this paper we show how closely we can determine a distribution when its median residual life function M[S | t] is known. This amounts to solving the functional equation , where R is the reliability function. We actually study a more general functional equation f(φ(t)) = sf(t) called Schroder's equation. It is shown that, under mild assumptions on φ, the solution is of the form f(t) = f0(t)k(log f0 (t)), where f0 is a well-behaved particular solution which can be constructed and k is a periodic function; thus the solution is not unique. Two examples are solved to illustrate the method. Finally, these examples are used to solve the problem of linear M[S | t] studied by Schmittlein and Morrison. As an extra benefit, all of our results hold equally well for the more general sth percentile residual life function.

It is shown that the gamma distribution is the unique mixing distribution of exponentials that leads to a linearly increasing mean residual lifetime function. A corollary of this result is that regardless of the mixing process the Pareto distribution is the unique aggregate distribution mixture with this property.