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There are several ways to establish the asymptotic normality of $L$-statistics, which depend on the choice of the weights-generating function and the cumulative distribution selection of the underlying model. In this study, we focus on stablishing computational formulas for the asymptotic variance of two robust $L$-estimators: the method of trimmed moments (MTM) and the method of winsorized moments (MWM). We demonstrate that two asymptotic approaches for MTM are equivalent for a specific choice of the weights-generating function. These findings enhance the applicability of these estimators across various underlying distributions, making them effective tools in diverse statistical scenarios. Such scenarios include actuarial contexts, such as payment-per-payment and payment-per-loss data scenarios, as well as in evaluating the asymptotic distributional properties of distortion risk measures. The effectiveness of our methodologies depends on the availability of the cumulative distribution function, ensuring broad usability in various statistical environments.

Numerous robust estimators exist as alternatives to the maximum likelihood estimator (MLE) when a completely observed ground-up loss severity sample dataset is available. However, the options for robust alternatives to MLE become significantly limited when dealing with grouped loss severity data, with only a handful of methods like least squares, minimum Hellinger distance, and optimal bounded influence function available. This paper introduces a novel robust estimation technique, the Method of Truncated Moments (MTuM), specifically designed to estimate the tail index of a Pareto distribution from grouped data. Inferential justification of MTuM is established by employing the central limit theorem and validating them through a comprehensive simulation study., Comment: 18 pages, 1 figure, 6 tables

The classical B\"{u}hlmann credibility model has been widely applied to premium estimation for group insurance contracts and other insurance types. In this paper, we develop a robust B\"{u}hlmann credibility model using the winsorized version of loss data, also known as the winsorized mean (a robust alternative to the traditional individual mean). This approach assumes that the observed sample data come from a contaminated underlying model with a small percentage of contaminated sample data. This framework provides explicit formulas for the structural parameters in credibility estimation for scale-shape distribution families, location-scale distribution families, and their variants, commonly used in insurance risk modeling. Using the theory of \(L\)-estimators (different from the influence function approach), we derive the asymptotic properties of the proposed method and validate them through a comprehensive simulation study, comparing their performance to credibility based on the trimmed mean. By varying the winsorizing/trimming thresholds in several parametric models, we find that all structural parameters derived from the winsorized approach are less volatile than those from the trimmed approach. Using the winsorized mean as a robust risk measure can reduce the influence of parametric loss assumptions on credibility estimation. Additionally, we discuss non-parametric estimations in credibility. Finally, a numerical illustration from the Wisconsin Local Government Property Insurance Fund indicates that the proposed robust credibility approach mitigates the impact of model mis-specification and captures the risk behavior of loss data from a broader perspective.

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

The primary objective of this scholarly work is to develop two estimation procedures - maximum likelihood estimator (MLE) and method of trimmed moments (MTM) - for the mean and variance of lognormal insurance payment severity data sets affected by different loss control mechanism, for example, truncation (due to deductibles), censoring (due to policy limits), and scaling (due to coinsurance proportions), in insurance and financial industries. Maximum likelihood estimating equations for both payment-per-payment and payment-per-loss data sets are derived which can be solved readily by any existing iterative numerical methods. The asymptotic distributions of those estimators are established via Fisher information matrices. Further, with a goal of balancing efficiency and robustness and to remove point masses at certain data points, we develop a dynamic MTM estimation procedures for lognormal claim severity models for the above-mentioned transformed data scenarios. The asymptotic distributional properties and the comparison with the corresponding MLEs of those MTM estimators are established along with extensive simulation studies. Purely for illustrative purpose, numerical examples for 1500 US indemnity losses are provided which illustrate the practical performance of the established results in this paper.

With some regularity conditions maximum likelihood estimators (MLEs) always produce asymptotically optimal (in the sense of consistency, efficiency, sufficiency, and unbiasedness) estimators. But in general, the MLEs lead to non-robust statistical inference, for example, pricing models and risk measures. Actuarial claim severity is continuous, right-skewed, and frequently heavy-tailed. The data sets that such models are usually fitted to contain outliers that are difficult to identify and separate from genuine data. Moreover, due to commonly used actuarial "loss control strategies" in financial and insurance industries, the random variables we observe and wish to model are affected by truncation (due to deductibles), censoring (due to policy limits), scaling (due to coinsurance proportions) and other transformations. To alleviate the lack of robustness of MLE-based inference in risk modeling, here in this paper, we propose and develop a new method of estimation - method of truncated moments (MTuM) and generalize it for different scenarios of loss control mechanism. Various asymptotic properties of those estimates are established by using central limit theory. New connections between different estimators are found. A comparative study of newly-designed methods with the corresponding MLEs is performed. Detail investigation has been done for a single parameter Pareto loss model including a simulation study.