Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications

In this paper, we introduce a new robust estimator for the extreme value index of Pareto-type distributions under randomly right-truncated data and establish its consistency and asymptotic normality. Our considerations are based on the Lynden-Bell integral and a useful huberized M-functional and M-estimators of the tail index. A simulation study is carried out to evaluate the robustness and the finite sample behavior of the proposed estimator. Moreover, an extreme quantiles estimation was also derived and applied to real data-set of lifetimes of automobile brake pads.


Introduction
Let (X j ,Y j ), 1 ≤ j ≤ N, denote a sample of bivariate positive and independent random variables (rv's) defined over some probability space (Ω, A , P) , with continuous marginal cumulative distribution functions (cdf's) F and G respectively. Suppose that X is right-truncated by Y , in the sense that the rv of interest X j is only observed when X j ≤ Y j . We assume that both survival functions F := 1 − F and G := 1 − G are regularly varying at infinity, with respective indices (−1/γ 1 ) and (−1/γ 2 ) , i.e, F ∈ RV −1/γ 1 and G ∈ RV −1/γ 2 . That is, for any t > 0, lim x→∞ F (tx) /F (x) = t −1/γ 1 and lim where γ j > 0 ( j = 1, 2) is the so-called extreme value index (e.v.i) is a well-known parameter to measure the tail heaviness of a distribution. Distributions satisfying (1) play a very crucial role in extreme value analysis. They include many commonly used models such as Pareto, Burr, Fréchet and Lévy-stable distributions, known to be suitable models for adjusting large insurance claims, log-returns, large fluctuations, etc. (see for instance, Resnick, 2006). Recently, Benatmane et al. (2020) have proposed a new so-called composite Rayleigh-Pareto distribution, and they showed that such a model will be a better fit for some heavy tailed insurance claims data (actual data on Algerian fire insurance losses and Danish fire loss data). In many real applications, in case of presence of random right truncation (RRT), the rv of interest X may not be fully available. This truncation can occur in many areas, for example, it is usual that the insurer's claim data do not correspond to the underlying losses, because they are truncated from above. We refer to Escudero and Ortega (2008) for a recent paper on insurance claims under RRT. In what follows, the star notation ( * ) relates to any characteristic of the observed subsequence denoted by (X * i ,Y * i ) ; 1 ≤ i ≤ n, (n ≤ N) subject to X * i ≤ Y * i from the N-sample. As a consequence of truncation, the size of actually observed sample, n, is a binomial rv with parameters N and p := P (X ≤ Y ) . We shall assume that p > 0, otherwise, nothing will be observed. Consequently, the marginal cdf's of X * and Y * , respectively denoted by F * and G * , becomes the corresponding tails are We can easily show that (see Proposition B.1.10 in de Haan and Ferreira, 2007) F * ∈ RV −1/γ * 1 and G * ∈ RV −1/γ * 2 with respective indices Since F and G are heavy-tailed. Therefore, the Woodroofe's nonparametric estimator (see, Woodroofe, 1985) of F, is defined by in which C n is the empirical estimator of Another most popular estimator for F, is the well known Lynden-Bell nonparametric maximum likelihood estimator, originally proposed in Lynden-Bell (1971), defined by Recently, Gardes and Stupfler (2015) was briefly exploited the above relations (2) to define an estimator for the parameter of interest γ 1 by considering the classical Hill (see, Hill, 1975) estimators of γ * 1 and γ * 2 as functions of two distinct numbers of upper observations: . They proved the asymptotic normality of this estimator under the tail dependence and the second-order regular variation conditions. Recently, Worms and Worms (2016) proposed an asymptotically normal estimator for γ 1 by considering a Lynden-Bell integrals with a deterministic threshold t n > 0 given bŷ The case of a random threshold, is addressed by Benchaira et al. (2016) who propose a Hill-type estimator under RRT Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications based on a Woodroofe integration as follows: All of these e.v.i estimators, as well as the classical Hill-type (in complete data case) are non-robust, in the sense that they are very sensitive to extreme observations, data contamination or model deviation and tend to be highly volatile for small samples (this is illustrated in our simulation study). Also, the rate of convergence of these estimators are based on the optimal value of the numbers of top statistics k or the threshold t n , but this rate are slower than the parametric rate √ n. Moreover, estimating the optimal value of k is virtually impossible when the sample size n is small and this leads to unstable estimates for small samples and large confidence intervals (see, Resnick, 1997, for a detailed discussion). The alternative approach is inspired by the theory of robust inference (see for instance, Huber (1981) and Hampel et al. (1986)) instead of exact consistency, this theory aim at stability for small samples, possibly at the cost of a small asymptotic bias. However, as observed by Beran and Schell (2012), in some practical cases, such as natural disasters, operational risk assessment or reinsurance data are sparse (with n often somewhere between 20 and 50) and distributions are expected to be heavy tailed with an unknown e.v.i. Robust estimation of e.v.i. focuses primarily on complete data case, see Brazauskas and Serfling (2000), Beran and Schell (2012) and references therein. The incomplete data case has first been considered by Sayah et al. (2014), who dealt with heavy-tailed and right censored data. The aim of the current paper is to provide a robust e.v.i. estimator for heavy tailed data under RLT.
The paper is organized as follows. In Section 2, we introduce our new e.v.i. estimator under RRT, and establish its consistency and asymptotic normality. The proposed estimator is compared with those already existed and its finite sample behavior is checked by simulation in Section 3. As an application, we introduce, in Section 4, an estimator for very high quantiles, which we apply to a real data-set of lifetimes of automobile brake pads. Section 5 contain some concluding notes.

Framework and statement of the results
Recall that the condition (1) can be rephrased asF (x) = x −1/γ 1 L F (x) and G (x) = x −1/γ 2 L G (x) , where L F and L G are slowly varying functions at infinity. Assuming that lim x→∞ L F (x) = c > 0 leads to the class of so-called Pareto-like (or heavy-tailed) distributions, i.e. distribution satisfying 1 − F (x) ∼ cx −1/γ 1 as x tends to infinity. This, the tail of such distribution behaves asymptotically like the tail of Pareto distribution. Thus, we suggests to robustify the Pareto maximum likelihood estimator of γ 1 in order to obtain sensible estimates for the class of Pareto-type distributions despite possible deviations from the single-parameter Pareto model (see, Beran and Schell, 2012, for a detailed discussion). A natural estimate of γ 1 can therefore be based on a Huberized Pareto score function : where The reason for huberization is that the Pareto distribution is only an approximation of the true cdf. By huberizing, the estimate becomes robust against a large class of deviations from this approximation. Since deviations are mainly relevant in the center of the distribution, the lower truncation parameter v is more important. As an alternative to Hill-type estimation, Beran (1997) proposed to use all data but huberize the Pareto score function at lower quantiles. This method has been investigated in the complete data case in Beran and Schell (2012). Moreover, ψ v,u (x, γ) is defined for any choice of γ > 0 and −1 ≤ v < u ≤ ∞. Thus, as shown by Beran and Schell (2012), robustness needs to be achieved for lower quantiles whereas extreme observations on the right are those we are interested in. In particular, ψ −1,∞ (x, γ) = γ −1 log (x) − 1 for x ≥ 1. Consequently, a natural choice is u = ∞ and robustification on the left is obtained only if v > −1.
Under the assumptions above, and denote by F a set of distributions with support in R + . Then the functional T (F) defined on F as the solution t = t 0 of the equation Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications is called huberized tail index M-functional. Consequently, by using relations (1.9) and (1.10) in Stute and Wang (2008) in the left truncation case, a natural adaptation of this integral β F (t) in the framework of RRT, leads to the corresponding Huberized Lynden-Bell integral estimator of the e.v.i. γ 1 as any solution sequence T n of the empirical equationβ Remark 2.1. It is worth mentioning that for complete data case (no truncation), we have n = N, X * = X and C n = and consequently T n reduce to the Beran and Schell estimator (see e.g. Beran and Schell, 2012).
Next, we investigate the asymptotic properties of the estimator of the tail index γ 1 under the large class of Pareto-type distributions assumptions. To formulate our main result, the following conditions are required: Remark 2.2. In comparison with the optimal value of the numbers of top statistics k in the Hill-type estimators, the parameter v play a less crucial role, since the rate of convergence does not depend on v. In contrast to Hill-type estimators under truncation (see, equations 3 and 5), all data are used. The role of v is only to determine a threshold below which data have a bounded influence on the estimator. Note also that, the equation (7) defining our estimator has a solution for n ≥ 2 almost surely.
is a consistent estimator of γ 1 . Assume further that ∂ ∂t ψ v,u (x,t) dF (x) = 0 hold in a neighborhood of γ 1 . Then √ n γ (Z) where d → stands for convergence in distribution and in which Remark 2.3. Condition (A1) is standard in heavy-tailed and RRT context. The condition γ 1 < γ 2 ensures that the tail of the truncated rv of interest X is note too contaminated by the truncation rv Y. In addition, (A1) implies that, the right endpoints of X and Y are infinite and thus they are equal. Assumption (A2) already appeared in Stute and Wang (2008), who showed that, σ 2 < ∞ under (A2), therefore, σ 2 v,u < ∞ too. Since G ≤ 1, it implies ψ 2 v,u (x,t) dF (x) < ∞, which is the assumption when no truncation occurs (see, Theorem 2 in Beran and Schell, 2012). In our case, (A2) is satisfied when 0 < γ 1 < γ 2 .
Proof. The proof is essentially based on Theorem 4.3 in He and Yang (1998) and Corollary 1.1. in Stute and Wang (2008). Note that ψ v,u (x,t) is monotone and continuous in t and β F (t) possesses an isolated root at t = γ 1 . Let ε > 0, then under (A1) and (A2) by strong low of large numbers under truncation (see, Theorem 4.3 in He and Yang, 1998), we haveβ Hence, there exists some n ∈ N such that According to the monotonicity of ψ v,u (x,t) in t, together with the assumption of the existence of a solution sequencê we then get P γ 1 + ε <γ The existence of such a solution sequence for a continuous function in a neighborhood of γ 1 follows from (9) for n large enough. Thus,γ 1 is a consistent estimator of γ 1 . Let us now focus on the asymptotic normality ofγ The assumed differentiability of ψ v,u (x,t) in t allows a Taylor expansion around γ 1 which yields It was shown in Theorem 4.3 of He and Yang (1998) that for any non-negative measurable real function ϕ := ∂ ∂t ψ, under the condition that ϕ v,u (x,t) dF (x) = 0 hold in a neighborhood of γ 1 , we get Under assumptions (A1) and (A2), we can apply the central limit theorem under right truncation (see, Corollary 1.1 in Stute and Wang, 2008) for the Lynden-Bell integral, obtaining where σ 2 is given by (8). Consequently, the limit variance follows from (11) and (12). This concludes the proof of Theorem 2.1.
Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications

Simulation study
This following section examines the performance of our estimatorγ (Z) 1 given by solving the empirical equation (7), in which, the huberizing constants are v = 0 and u = ∞, and compare it with estimators proposed by Gardes  1 + x 1/θ −θ /γ 1 andḠ (x) = 1 + x 1/θ −θ /γ 2 x ≥ 0. We fix θ = 1/4 and choose the values 0.6 and 0.8 for γ 1 and p = 0.7 (resp. 0.9), that means the percentage of truncation is 30% (resp. 10%). The pertaining γ 2 -value is obtained by solving the equation p = γ 2 / (γ 1 + γ 2 ), for each couple (γ 1 , p) . We obtained 1000 pseudo-random samples X * and Y * of size n pN. Next, we calculate the estimators values frame the observed data X * and Y * . For choosing the optimal number k n of upper order statistics used in the computation ofγ 1 we adopt the Reiss and Thomas algorithm Reiss and Thomas (2007). In those simulations, we used the random threshold X * (n−k n ) instead of t n in the definition ofγ 1 . Also note that we only consider k n := k 1 = k 2 in the expression (3), in this caseγ is the one considered in Benchaira et al. (2015). Finally, we compute the absolute bias (abias) and root mean squared error (rmse) of these estimators, the results are summarized in Table 1 and Table 2. We see that our new estimator shows good performance compared to existing methods for small sample sizes.  Now, in order to study the sensitivity to outliers of our newly estimator, we consider an ε-contaminated model known by mixture of Pareto distributions Note that, for γ 1 < λ and ε > 0, (13) corresponds to a Pareto distribution contaminated by a longer tailed distribution.
Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications In this context, we proceed our study as follows: We fix λ = 2 and consider four different contamination levels ε = 0.05, 0.15, 0.25, 0.35, and we vary γ 1 among 0.6 and 0.8. For each value of ε, 1000 samples of size N = 200 were generated from the model (13) truncated by a simple Pareto modelḠ (x) = x −1/γ 2 , with p = 0.7 and 0.9. Our illustration, made with respect to the biases and rmse's, are summarized in Table 3. The values of the first line are those of the case where ε = 0 (i.e., uncontaminated case). Both the bias and the rmse of our estimator are note sensitive to outliers. Then we can conclude its robustness, giving us, in fact, an excellent level of protection against contamination data. We conclude from tables 1, 2 and 3 that our newly estimator perform better (with the smallest bias and root mean squared error), compared to existing methods, for small sample sizes and for both uncontaminated and contaminated cases.

Estimation of an upper quantile
Estimation of e.v.i. is very important in the determination of high quantiles, upper tail probabilities, mean excess functions, and excess-of-loss and stop-loss reinsurance premiums. Consequently, small errors in estimation of this quantity can produce substantial impact in applications. Thus, for robust estimation of quantities based on γ 1 robust estimation of γ 1 itself is crucial. In other words, for a heavy tailed distributions, robust estimation of the high quantile Q ε corresponding to upper tail probability ε, becomes of interest, and this may be carried out by robust estimation of γ 1 . We refer to Brazauskas and Serfling (2000) for a detailed account of this issue. Let (α n ) be some sequence of quantiles orders tending to 0, such that α n = o F (s n ) , where (s n ) is a sequence of positive deterministic thresholds growing to infinity with n. Consequently, the quantile of F of order (1 − α n ) is such that F (Q α n ) = α n . We can then define an estimatorQ α n ,s n of Q α n : A similar estimator is proposed in Worms and Worms (2016), but instead ofγ  (t n ) given by (4). Before we state the asymptotic normality ofQ α n ,s n , we set d n := F (s n ) /α n and assume that d n → ∞ and √ n/ log (d n ) → ∞, as n → ∞.
Moreover, from the classical second order condition (see, Bingham et al. 1989) for L F , it follows that where L F is slowly varying function at infinity and h is a positive measurable function, slowly varying with index ρ < 0. SetH :=FḠ, where H is the distribution function of min(X,Y ). The asymptotic normality result will then require the following conditions on s n : nH (s n ) → ∞, as n → ∞ and nH (s n )h (s n ) → λ , as n → ∞ (for some λ > 0).
Theorem 4.1. Under (14), (15), (16) and the assumptions of Theorem 2.1, we have √ n log (d n ) Proof. The result follows by analogous arguments as in the proof of Theorem 2 in Worms and Worms (2016). Recall that the high quantile Q α n corresponding to order (1 − α n ) is such that F (Q α n ) = α n , and its estimator is defined bŷ For convenience, we set Λ n :=F n (s n ) /F (s n ) . Indeed, we havê We will show that √ n log(d n ) I n1 is asymptotically centred Gaussian rv with variance σ 2 v,u and √ n log(d n ) I n j P → 0, j = 2, 3. Concerning the term I n1 , by using the mean value theorem, it follows that √ n log (d n ) where δ n ≤ γ (Z) 1 − γ 1 log (d n ). Assumption (14) and Theorem 2.1, allows us to conclude that δ n tends to 0. We use then Theorem 2.1 to get.
Let us now focus on the negligible terms I n2 and I n3 . By using the mean value theorem, we get with M n tending to 1. In view of assumptions (A1) and (15), the sequence (Λ n ) converge to 1 in probability (see, Lemma 2 in Worms and Worms, 2016), we have then √ n log (d n ) (Λ n − 1) = o p (1) .
For I n3 , in view of the regular variation ofF, (1) can be rephrased asF (x) = x −1/γ 1 L F (x) , where L F is slowly varying Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications function at infinity and by definition of Q α n , we get Therefore, we use the following representation of L F (see, Smith 1987, page 1195) where h is a positive measurable function, slowly varying with index ρ < 0. We have, Q α n /s n tends to infinity, then h (Q α n ) /h (s n ) tends to 0 and It follows that and then the desired negligibility of I n3 follows from assumption (16), which ends the proof of the Theorem.

Real data example : automobile brake pad lifetime
In reliability, a real data-set of lifetimes of automobile brake pads already considered in Lawless (2002), was recently analyzed in Gardes and Stupfler (2015) and Benchaira et al. (2016) as an application of heavy-tailed and RRT data. We follow the same steps as those in Gardes and Stupfler (2015) who transformed this sample into a right-truncated data, which originally is left-truncated. We are dealing with a data-set of small size (n = 98), consequently, robust estimation of γ 1 can produce substantial robust estimation of the high quantile. Then, our procedure should be preferred to that based on no robust estimation of γ 1 . In these situation, we used the random threshold X * (n−k n ) instead of s n in the definition ofQ α n ,s n . We select the optimal number of top statistics, via the numerical procedure of (Reiss and Thomas, 2007, page 137) and we get k = 10 and we estimate the tail index γ 1 given in (5) and (7) we getγ  Table 4. For instance, we conclude that the brake pad lifetime is estimated to be less than 17063 km for 1 % of the cars while only be out of a thousand brake pads lasts less than 10200 km.

Concluding notes
The main objective of this paper was to propose a robust estimator for the extreme value index of Pareto-type distributions under randomly right-truncated data by using a Lynden-Bell integral and a useful huberized M-functional and M-estimators of the tail index. It has been shown that our newly estimator is more robust and perform better than the estimators proposed in Gardes and Stupfler (2015), Worms and Worms (2016), Benchaira et al. (2016), for small sample sizes and for both uncontaminated and contaminated cases. In our further research we will study this robust estimator in more detail. We will investigate its influence function and its relative asymptotic efficiency. Note also that, The degree of robustness is determined by the tuning parameters v and u. This paper does not treat the choice of these parameters, it remains a likely topic for future investigations. Finally, we emphasize that our approach may also be employed to derive several robust estimators of upper tail probabilities, mean excess functions, and excess-of-loss and stop-loss reinsurance premiums, in case of presence of random right truncation.