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In this paper, the problem of robust estimation and validation of location-scale families is revisited. The proposed methods exploit the joint asymptotic normality of sample quantiles (of i.i.d random variables) to construct the ordinary and generalized least squares estimators of location and scale parameters. These quantile least squares (QLS) estimators are easy to compute because they have explicit expressions, their robustness is achieved by excluding extreme quantiles from the least-squares estimation, and efficiency is boosted by using as many non-extreme quantiles as practically relevant. The influence functions of the QLS estimators are specified and plotted for several location-scale families. They closely resemble the shapes of some well-known influence functions yet those shapes emerge automatically (i.e., do not need to be specified). The joint asymptotic normality of the proposed estimators is established, and their finite-sample properties are explored using simulations. Also, computational costs of these estimators, as well as those of MLE, are evaluated for sample sizes n = 10^6, 10^7, 10^8, 10^9. For model validation, two goodness-of-fit tests are constructed and their performance is studied using simulations and real data. In particular, for the daily stock returns of Google over the last four years, both tests strongly support the logistic distribution assumption and reject other bell-shaped competitors.

"The rich are getting richer" implies that the population income distributions are getting more right skewed and heavily tailed. For such distributions, the mean is not the best measure of the center, but the classical indices of income inequality, including the celebrated Gini index, are all mean-based. In view of this, Professor Gastwirth sounded an alarm back in 2014 by suggesting to incorporate the median into the definition of the Gini index, although noted a few shortcomings of his proposed index. In the present paper we make a further step in the modification of classical indices and, to acknowledge the possibility of differing viewpoints, arrive at three median-based indices of inequality. They avoid the shortcomings of the previous indices and can be used even when populations are ultra heavily tailed, that is, when their first moments are infinite. The new indices are illustrated both analytically and numerically using parametric families of income distributions, and further illustrated using capital incomes coming from 2001 and 2018 surveys of fifteen European countries. We also discuss the performance of the indices from the perspective of income transfers.

To accommodate numerous practical scenarios, in this paper we extend statistical inference for smoothed quantile estimators from finite domains to infinite domains. We accomplish the task with the help of a newly designed truncation methodology for discrete loss distributions with infinite domains. A simulation study illustrates the methodology in the case of several distributions, such as Poisson, negative binomial, and their zero inflated versions, which are commonly used in insurance industry to model claim frequencies. Additionally, we propose a very flexible bootstrap-based approach for the use in practice. Using automobile accident data and their modifications, we compute what we have termed the conditional five number summary (C5NS) for the tail risk and construct confidence intervals for each of the five quantiles making up C5NS, and then calculate the tail probabilities. The results show that the smoothed quantile approach classifies the tail riskiness of portfolios not only more accurately but also produces lower coefficients of variation in the estimation of tail probabilities than those obtained using the linear interpolation approach., Comment: 22 pages, 1 figure, 7 tables

When constructing parametric models to predict the cost of future claims, several important details have to be taken into account: (i) models should be designed to accommodate deductibles, policy limits, and coinsurance factors, (ii) parameters should be estimated robustly to control the influence of outliers on model predictions, and (iii) all point predictions should be augmented with estimates of their uncertainty. The methodology proposed in this paper provides a framework for addressing all these aspects simultaneously. Using payment-per-payment and payment-per-loss variables, we construct the adaptive version of method of winsorized moments (MWM) estimators for the parameters of truncated and censored lognormal distribution. Further, the asymptotic distributional properties of this approach are derived and compared with those of the maximum likelihood estimator (MLE) and method of trimmed moments (MTM) estimators. The latter being a primary competitor to MWM. Moreover, the theoretical results are validated with extensive simulation studies and risk measure sensitivity analysis. Finally, practical performance of these methods is illustrated using the well-studied data set of 1500 U.S. indemnity losses. With this real data set, it is also demonstrated that the composite models do not provide much improvement in the quality of predictive models compared to a stand-alone fitted distribution specially for truncated and censored sample data., Comment: 35 pages, 4 figures, etc

In this paper, we consider robust estimation of claim severity models in insurance, when data are affected by truncation (due to deductibles), censoring (due to policy limits), and scaling (due to coinsurance). In particular, robust estimators based on the methods of trimmed moments (T-estimators) and winsorized moments (W-estimators) are pursued and fully developed. The general definitions of such estimators are formulated and their asymptotic properties are investigated. For illustrative purposes, specific formulas for T- and W-estimators of the tail parameter of a single-parameter Pareto distribution are derived. The practical performance of these estimators is then explored using the well-known Norwegian fire claims data. Our results demonstrate that T- and W-estimators offer a robust and computationally efficient alternative to the likelihood-based inference for models that are affected by deductibles, policy limits, and coinsurance., Comment: 32 pages, 2 figures