Robust Estimation of the Tail Index of a Single Parameter Pareto Distribution from Grouped Data

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.


Introduction
To protect the privacy of policyholders (e.g., individuals, small businesses, privately owned companies, local government funds), data vendors and publicly available databases provide summarized data, that is, in a grouped format.For statistical inference, we view such data as i.i.d.realizations of a random variable that was subjected to interval censoring by multiple, say m, contiguous intervals.The existing parametric statistical estimating tool for such grouped sample data is mainly dominated by maximum likelihood.But maximum likelihood estimator (MLE) typically results in sensitive fitted predictive models if the sample data is coming from a contaminated distribution, Tukey (1960), or if there are heavier point masses assigned at some particular values, Poudyal et al. (2023), specially for actuarial payment-per-payment and payment-perloss data scnarios.To overcome the sensitivity of MLE, several robust estimation techniques are established in the literature for different sample data scenarios except for grouped data.Therefore, the motivation of this scholarly work is to fulfil this gap by proposing a robust estimation method and for grouped data.
A general class of L-statistics, Chernoff et al. (1967), provides a broad selection of robust estimators including methods of trimmed moments (MTM) and winsorized moments (MWM).Brazauskas et al. (2009) and Zhao et al. (2018) have implemented MTM and MWM, respectively, in actuarial framework but for completely observed ground-up loss severity data.For incomplete loss data scenarios, in a series of papers, Poudyal (2021a), Poudyal andBrazauskas (2022, 2023), Poudyal et al. (2023) consider both MTM and MWM with comprehensive simulation and sensitivity analysis.In these papers, it has been shown that trimming and winsorizing are useful methods of robustifying moment estimation under extreme claims, Gatti and Wüthrich (2023).Poudyal (2021b) introduced a novel method, called the Method of Truncated Moments (MTuM), which employs fixed lower and upper truncation thresholds.MTuM is designed to work for completely observed ground-up loss dataset.In this approach, tail sample observations can be random.
For the grouped data, Aigner and Goldberger (1970) studied the problem of estimating the scale parameter of single parameter Pareto distribution via MLE and four variants of least squares.As a robust alternative to MLE, Lin and He (2006) considered approximate minimum Hellinger distance estimator (Beran, 1977a,b) for grouped data which can be asymptotically as efficient as the MLE.Under small model contaminations, Victoria-Feser and Ronchetti (1997) established that the optimal bounded influence function estimators are more robust than the MLEs for grouped data.The concept of optimal grouping, in the sense of minimizing the loss of information, has been introduced by Schader and Schmid (1986), but this approach is still under likelihood framework, Kleiber and Kotz (2003, p. 82).Therefore, the goal of this manuscript is to explore the robustness of the MTuM estimator specifically for the tail index of grouped single parameter Pareto distributions and to assess its performance against the corresponding MLE.Asymptotic distributional properties, such as normality, consistency, and the asymptotic relative efficiency in relation to the MLE, are established for the purpose of inferential justification.In addition, the paper strengthens its theoretical concepts with extensive simulation studies.It is noteworthy that the moments, when subject to threshold truncation and/or censorship, are consistently finite, irrespective of the underlying true distribution.
The rest of the paper is structured as follows.In Section 2, we briefly summarize the grouped data scenarios including different probablility functions.Section 3 is focused on the development of MTuM procedures for grouped data along with the inferential justification.In Section 4, we conduct an extensive simulation study to complement the theoretical results for different scenarios.Finally, concluding remarks and further directions are summarized in Section 5.

Pareto Grouped Data
Due to the complexity of the involved theory, we only investigate single parameter Pareto distribution in this scholarly work.As considered by Poudyal (2021b, §3), let Y ∼ Pareto I(α, x 0 ) with the distribution function F Y (y) = 1 − (x 0 /y) α , y > x 0 , zero elsewhere.Here, α > 0 represents the shape parameter, often referred to as the tail index, and x 0 > 0 is the known lower bound threshold.Consequently, if we define with its distribution function given by F X (x) = 1 − e −x/θ .Hence, estimating α is equivalent to estimating the exponential parameter θ.Thus, for the purpose of analytic simplicity, we investigate θ, rather than α.The development and asymptotic behavior of MTuM estimators will be explored for a grouped sample drawn from an exponential distribution.
∞ be the group boundaries for the grouped data where we define c 0 := 0, and Computation of the empirical distribution function at the group boundaries is clear, but inside the intervals, the linearly interpolated empirical cdf as defined in Klugman et al. (2019, §14.2), is the most common one.The linearly interpolated empirical cdf, called "ogive" and denoted by F n , is defined as (2.1) In the complete data case, we observe the following empirical frequencies of X: where n j is the sample size.
Clearly, the empirical distribution as it is impossible to draw a straight line joining two points (c m , F n (c m )) and (∞, 1) The corresponding linearized population cdf F G is defined by (2.2) The corresponding density function f n , called the histogram, is defined as The empirical quantile function (the inverse of F n ) is then computed as (2.4) Similarly, (2.5) If the loss variable X observed in a grouped format is affected by additional transformations: truncation, interval censoring, coverage modifications, then in those cases, the underlying distribution function would have to be modified accordingly.For example, if m groups (n observations in total) are provided and it is known that only data above deductible d appeared, then the distributional assumption is that we observe 3 MTuM for Grouped Data For both MTM and MWM, if the right trimming/winsorzing proportion b is such that does not exist as the linearized empirical distribution F n is not defined in the interval Thus, in order to apply the MTM/MWM approach for grouped sample, we always need to make sure that is problematic for different samples with the fixed right trimming/winsorzing b.With this fact in consideration, the asymptotic distributional properties of MTM and MWM estimators and from grouped data are very complicated and not easy to analytically derive if not intractable.But with MTuM, we can always choose the right truncated threshold T such that T ≤ c m .Therefore, we proceed with MTuM approach for grouped data in the rest of this section.Let 0 ≤ t and T ≤ c m , with t < T , be the left and right truncation points, respectively.
By using the empirical cdf, Eq. (2.1) and pdf Eq. (2.3), the sample truncated moments for a grouped data as defined by Poudyal (2021b) is given by Let us introduce the following notations: The following corollary is an immediate consequence of Proposition 3.1.
Proposition 3.2.The function g tT (θ) has the following limiting values Proof.These limits can be established by using L'Hôpital's rule.
Now, assuming the Conjecture 3.1 is true then with Proposition 3.2, we have Theorem 3.1.The equation µ = g tT (θ) has a unique solution θ MTuM provided that Solve the equation µ = µ for θ MTuM , say θ =: g θ ( µ).Then, again by the delta method, we conclude that θ ∼ AN g θ (µ), n −1 (g ′ θ (µ)) 2 Σ µ .Note that if both the left-and right-truncation points lie on the same interval, then µ Then, we get a fixed point function as θ = G(θ), where However, we need to consider the condition µ(A 2 + Q) > P .Therefore, we need to be careful about the initialization of θ as the right truncation point T cannot be a boundary point.Because if it was, then A 2 = 0 and we would not be able to divide by A 2 in the fixed point function θ = G(θ).
Now, let us compute the derivative of g θ with respect to µ, using implicit differentiation.
Case 1: Assume that the two truncation points are in two consecutive intervals, i.e., assume that l = r.Then θ ′ = g ′ 1 P j (θ) .
The asymptotic performance of MTuM estimator is measured through the asymp- The primary justification for employing MLE as a standard/benchmark for comparison lies in its optimal asymptotic behavior in terms of variance, though this comes with the typical proviso of being subject to ' "under certain regularity conditions".Therefore, the desired ARE as given by Eq. (3.6) is computed as The numerical values of ARE θ MTuM , θ MLE from Exp(θ = 10) with group boundaries vector G := (0 : 4 : 30, ∞) is summarized in

Simulation Study
This section augments the theoretical findings established in Section 3 with simulations.
The primary objective is to determine the sample size required for the estimators to be unbiased (acknowledging that they are asymptotically unbiased), to validate the asymptotic normality, and to ensure that their finite sample relative efficiencies (RE) are converging towards the respective AREs.For calculating RE, MLE is utilized as the reference point.Consequently, the concept of asymptotic relative efficiency outlined in equation (3.6) is adapted for finite sample analysis as follows: RE(MT uM, MLE) = asymptotic variance of MLE estimator variance of a competing estimator MTuM .
As we move in sequence from Table 4.1 to Table 4.5, it becomes noticeable that the convergence of the ratio of the estimated θ with the true θ, i.e., θ/θ, approaches the true asymptotic value of 1 at a more gradual pace/slower.Additionally, both our intuition and the data presented in the tables suggest that when there is a wider gap between the thresholds (namely, t and T ), the estimators tend to approach the true values at a slower rate.

Concluding Remarks
In this scholarly work, we have purposed a novel Method of Truncated Moments Regarding future directions, this paper focused mainly on estimating the mean parameter of an exponential distribution (equivalently estimating tail index of a single parameter Pareto distribution) given a grouped sample data, so the purposed methodology could be extended to more general situations and models.However, and specially for multi-parameter distributions, it is very complicated to investigate the nature of the function g tT (Parameters) given in Eq. (3.2) and Conjecture 3.1, if not intractable.Asymptotic inferential justification of the designed MTuM methodology for multi-parameter distributions is equally difficult.In this regard, a potential future direction is to think algorithmically (i.e., designing simulation based estimators for complex models, Guerrier et al., 2019) rather than establishing inferential justification, Efron and Hastie (2016, p. xvi).Furthermore, it remains to be assessed how this novel MTuM estimator performs across various practical risk analysis scenarios.

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MTuM) estimator to estimate the tail index from grouped Pareto loss severity data, as a robust alternative to MLE.Theoretical justifications regarding the designed estimators' existence and asymptotic normality are established.The finite sample performance, for various sample sizes and different group boundaries vectors has been investigated in detail via simulation study.
repeat this process 10 times getting θ1 , . . ., θ10 .Then we compute mean θ and standard deviation se( θ) of θ1 , . . ., θ10 and finally θ θ and se( θ) θ are reported on the table.Similarly, finite-sample relative efficiency (RE) of MTuM w.r.t.grouped MLE are computed as RE 1 , . . ., RE 10 and their mean, standard deviations are reported for different vectors of group boundaries.