Vytaras Brazauskas

Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution

Abstract Estimation of the tail index parameter of a single-parameter Pareto model has wide application in actuarial and other sciences. Here we examine various estimators from the standpoint of two competing criteria: efficiency and robustness against upper outliers. With the maximum likelihood estimator (MLE) being efficient but nonrobust, we desire alternative estimators that retain a relatively high degree of efficiency while also being adequately robust. A new generalized median type estimator is introduced and compared with the MLE and several well-established estimators associated with the methods of moments, trimming, least squares, quantiles, and percentile matching. The method of moments and least squares estimators are found to be relatively deficient with respect to both criteria and should become disfavored, while the trimmed mean and generalized median estimators tend to dominate the other competitors. The generalized median type performs best overall. These findings provide a basis for revision and updating of prevailing viewpoints. Other topics discussed are applications to robust estimation of upper quantiles, tail probabilities, and actuarial quantities, such as stop-loss and excess-of-loss reinsurance premiums that arise concerning solvency of portfolios. Robust parametric methods are compared with empirical nonparametric methods, which are typically nonrobust.

Robust Estimation of Tail Parameters for Two-Parameter Pareto and Exponential Models via Generalized Quantile Statistics

Robust estimation of tail index parameters is treated for (equivalent) two-parameter Pareto and exponential models. These distributions arise as parametric models in actuarial science, economics, telecommunications, and reliability, for example, as well as in semiparametric modeling of upper observations in samples from distributions which are regularly varying or in the domain of attraction of extreme value distributions. New estimators of “generalized quantile” type are introduced and compared with several well-established estimators, for the purpose of identifying which estimators provide favorable trade-offs between efficiency and robustness. Specifically, we examine asymptotic relative efficiency with respect to the (efficient but nonrobust) maximum likelihood estimator, and breakdown point. The new estimators, in particular the generalized median types, are found to dominate well-established and popular estimators corresponding to methods of trimming, least squares, and quantiles. Further, we establish that the least squares estimator is actually deficient with respect to both criteria and should become disfavored. The generalized median estimators manifest a general principle: “smoothing” followed by “medianing” produces a favorable trade-off between efficiency and robustness.

Favorable Estimators for Fitting Pareto Models: A Study Using Goodness-of-fit Measures with Actual Data

Several recent papers treated robust and efficient estimation of tail index parameters for (equivalent) Pareto and truncated exponential models, for large and small samples. New robust estimators of “generalized median” (GM) and “trimmed mean” (T) type were introduced and shown to provide more favorable trade-offs between efficiency and robustness than several well-established estimators, including those corresponding to methods of maximum likelihood, quantiles, and percentile matching. Here we investigate performance of the above mentioned estimators on real data and establish — via the use of goodness-of-fit measures — that favorable theoretical properties of the GM and T type estimators translate into an excellent practical performance. Further, we arrive at guidelines for Pareto model diagnostics, testing, and selection of particular robust estimators in practice. Model fits provided by the estimators are ranked and compared on the basis of Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling statistics.

Folded- and Log-Folded-t Distributions as Models for Insurance Loss Data

A rich variety of probability distributions has been proposed in the actuarial literature for fitting of insurance loss data. Examples include: lognormal, log-t, various versions of Pareto, loglogistic, Weibull, gamma and its variants, and generalized beta of the second kind distributions, among others. In this paper, we supplement the literature by adding the log-folded-normal and log-folded-t families. Shapes of the density function and key distributional properties of the ‘folded’ distributions are presented along with three methods for the estimation of parameters: method of maximum likelihood; method of moments; and method of trimmed moments. Further, large and small-sample properties of these estimators are studied in detail. Finally, we fit the newly proposed distributions to data which represent the total damage done by 827 fires in Norway for the year 1988. The fitted models are then employed in a few quantitative risk management examples, where point and interval estimates for several value-at-risk measures are calculated.

Small sample performance of robust estimators of tail parameters for pareto and exponential models

Robust estimation of tail index parameters is treated for (equivalent) two-parameter Pareto and exponential models. These distributions arise as parametric models in actuarial science, economics, telecommunications, and reliability, for example, as well as in semiparametric modeling of upper observations in samples from distributions which are regularly varying or in the domain of attraction of extreme value distributions. In a recent previous paper, new estimators of generalized median (GM) type were introduced and shown to provide more favorable trade-offs between efficiency and robustness than several well-established estimators, including those corresponding to methods of maximum likelihood, trimming, and quantiles. Here we establish-via simulation-that the superiority of the GM type estimators remains valid even for small sample sizes n=10 and 25. To bridge between small and large sample sizes, we also include the cases n=50 and 100. Further, we arrive at guidelines for selection of a particlar GM estimator in practice, depending upon the sample size, upon whether protection is desired against upper outliers onlu, or against both upper and lower outliers, and upon whether the level of possible contamination by outliers is high or low. Comparisons of estimators are made on the basis of relative efficiency with respect to the maximum likelihood estimator, breakdown points, and premium-protection plots.

Interval Estimation of Actuarial Risk Measures

Abstract This article investigates performance of interval estimators of various actuarial risk measures. We consider the following risk measures: proportional hazards transform (PHT), Wang transform (WT), value-at-risk (VaR), and conditional tail expectation (CTE). Confidence intervals for these measures are constructed by applying nonparametric approaches (empirical and bootstrap), the strict parametric approach (based on the maximum likelihood estimators), and robust parametric procedures (based on trimmed means). Using Monte Carlo simulations, we compare the average lengths and proportions of coverage (of the true measure) of the intervals under two data-generating scenarios: “clean” data and “contaminated” data. In the “clean” case, data sets are generated by the following (similar shape) parametric families: exponential, Pareto, and lognormal. Parameters of these distributions are selected so that all three families are equally risky with respect to a fixed risk measure. In the “contaminated” case, the “clean” data sets from these distributions are mixed with a small fraction of unusual observations (outliers). It is found that approximate knowledge of the underlying distribution combined with a sufficiently robust estimator (designed for that distribution) yields intervals with satisfactory performance under both scenarios.

Fisher information matrix for the Feller-Pareto distribution

In this paper, the exact form of Fisher information matrix for the Feller-Pareto (FP) distribution is determined. The FP family is a very general unimodal distribution which includes a variety of distributions as special cases. For example: - A hierarchy of Pareto models: Pareto (I), Pareto (II), Pareto (III), and Pareto (IV) (see Arnold (Pareto Distributions, International Cooperative Publishing House, Fairland, MD, 1983)); and - Transformed beta family which in turn includes such general families as Burr, Generalized Pareto, and Inverse Burr (see Klugman et al. (Loss Models: From Data to Decisions, Wiley, New York, 1998)). Application of these distributions covers a wide spectrum of areas ranging from actuarial science, economics, finance to biosciences, telecommunications, and extreme value theory.

Nested L-statistics and their use in comparing the riskiness of portfolios

Inspired by the problem of testing hypotheses about the equality of several risk measure values, we find that the ‘nested L-statistic’—a notion introduced herein—is natural and particularly convenient. Indeed, the test statistic that we explore in this paper is a nested L-statistic. We discuss large-sample properties of the statistic, investigate its performance using a simulation study, and consider an example involving the comparison of risk measure values where the risks of interest are those associated with tornado damage in different time periods and different regions.

Information Matrix for Pareto(IV), Burr, and Related Distributions

Abstract In this paper the exact form of information matrix for Pareto(IV) and related distributions is determined. The Pareto(IV) family being very general includes more specialized families of Pareto(I), Pareto(II), and Pareto(III), and the Burr family of distributions, as special cases. These distributions, for example, arise as tractable parametric models in actuarial science, economics, finance, and telecommunications. Additionally, a useful mathematical result with its own domain of importance is obtained. In particular, explicit formula for the improper integral , with b > 0 and non-negative integer m, is derived.

Modeling Severity and Measuring Tail Risk of Norwegian Fire Claims

The probabilistic behavior of the claim severity variable plays a fundamental role in calculation of deductibles, layers, loss elimination ratios, effects of inflation, and other quantities arising in insurance. Among several alternatives for modeling severity, the parametric approach continues to maintain the leading position, which is primarily due to its parsimony and flexibility. In this article, several parametric families are employed to model severity of Norwegian fire claims for the years 1981 through 1992. The probability distributions we consider include generalized Pareto, lognormal-Pareto (two versions), Weibull-Pareto (two versions), and folded-t. Except for the generalized Pareto distribution, the other five models are fairly new proposals that recently appeared in the actuarial literature. We use the maximum likelihood procedure to fit the models and assess the quality of their fits using basic graphical tools (quantile-quantile plots), two goodness-of-fit statistics (Kolmogorov-Smirnov and Anderson-Darling), and two information criteria (AIC and BIC). In addition, we estimate the tail risk of “ground up” Norwegian fire claims using the value-at-risk and tail-conditional median measures. We monitor the tail risk levels over time, for the period 1981 to 1992, and analyze predictive performances of the six probability models. In particular, we compute the next-year probability for a few upper tail events using the fitted models and compare them with the actual probabilities.