Estimating the Conditional Tail Expectation in the Case of Heavy-Tailed Losses

Abstract

The conditional tail expectation (CTE) is an important actuarial risk measure and a useful tool in financial risk assessment. Under the classical assumption that the second moment of the loss variable is finite, the asymptotic normality of the nonparametric CTE estimator has already been established in the literature. The noted result, however, is not applicable when the loss variable follows any distribution with infinite second moment, which is a frequent situation in practice. With a help of extreme-value methodology, in this paper, we offer a solution to the problem by suggesting a new CTE estimator, which is applicable when losses have finite means but infinite variances.

1. Introduction

One of the most important actuarial risk measures is the conditional tail expectation (CTE) (see, e.g., [1]), which is the average amount of loss given that the loss exceeds a specified quantile. Hence, the CTE provides a measure of the capital needed due to the exposure to the loss, and thus serves as a risk measure. Not surprisingly, therefore, the CTE continues to receive increased attention in the actuarial and financial literature, where we also find its numerous extensions and generalizations (see, e.g., [2–8], and references therein). We next present basic notation and definitions.

Let 𝑋 be a loss random variable with cumulative distribution function (cdf) 𝐹. Usually, the cdf 𝐹 is assumed to be continuous and defined on the entire real line, with negative loss interpreted as gain. We also assume the continuity of 𝐹 throughout the present paper. The CTE of the risk or loss 𝑋 is then defined, for every 𝑡∈(0,1), by

CTE_𝐹(𝑡)=𝐄[𝑋∣𝑋>𝑄(𝑡)], (1.1) where 𝑄(𝑡)=inf{𝑥∶𝐹(𝑥)≥𝑡} is the quantile function corresponding to the cdf 𝐹. Since the cdf 𝐹 is continuous, we easily check that

CTE_𝐹1(𝑡)=1−𝑡¹_𝑡𝑄(𝑠)𝑑𝑠. (1.2)

Naturally, the CTE is unknown since the cdf 𝐹 is unknown. Hence, it is desirable to establish statistical inferential results such as confidence intervals for CTE_𝐹(𝑡) with specified confidence levels and margins of error. We shall next show how to accomplish this task, initially assuming the classical moment assumption 𝐄[𝑋²]<∞. Namely, suppose that we have independent random variables 𝑋₁, 𝑋₂, … , each with the cdf 𝐹, and let 𝑋_1∶𝑛<⋯<𝑋_𝑛∶𝑛 denote the order statistics of 𝑋₁,…,𝑋_𝑛. It is natural to define an empirical estimator of CTE_𝐹(𝑡) by the formula

CTE_𝑛1(𝑡)=1−𝑡¹_𝑡𝑄_𝑛(𝑠)𝑑𝑠, (1.3) where 𝑄_𝑛(𝑠) is the empirical quantile function, which is equal to the 𝑖th order statistic 𝑋_𝑖∶𝑛 for all 𝑠∈((𝑖−1)/𝑛,𝑖/𝑛], and for all 𝑖=1,…,𝑛. The asymptotic behavior of the estimator CTE_𝑛(𝑡) has been studied by Brazauskas et al. [9], and we next formulate their most relevant result for our paper as a theorem.

Theorem 1.1. Assume that 𝐄[𝑋²]<∞. Then for every 𝑡∈(0,1), we have the asymptotic normality statement √𝑛(CTE_𝑛(𝑡)−CTE_𝐹(𝑡))(1−𝑡)→_𝑑𝒩(0,𝜎²(𝑡)), (1.4) when 𝑛→∞, where the asymptotic variance 𝜎²(𝑡) is given by the formula 𝜎²(𝑡)=¹_t¹_t(min(𝑥,𝑦)−𝑥𝑦)𝑑𝑄(𝑥)𝑑𝑄(𝑦). (1.5)

The assumption 𝐄[𝑋²]<∞ is, however, quite restrictive as the following example shows. Suppose that 𝐹 is the Pareto cdf with index 𝛾>0, that is, 1−𝐹(𝑥)=𝑥^−1/𝛾 for all 𝑥≥1. Let us focus on the case 𝛾<1, because when 𝛾≥1, then CTE_𝐹(𝑡)=+∞ for every 𝑡∈(0,1). Theorem 1.1 covers only the values 𝛾∈(0,1/2) in view of the assumption 𝐄[𝑋²]<∞. When 𝛾∈[1/2,1), we have 𝐄[𝑋²]=∞ but, nevertheless, CTE_𝐹(𝑡) is well defined and finite since 𝐄[𝑋]<∞. Analogous remarks hold for other distributions with Pareto-like tails, an we shall indeed work with such general distributions in this paper.

Namely, recall that the cdf 𝐹 is regularly varying at infinity with index (−1/𝛾)<0 if

lim_𝑡→∞1−𝐹(𝑡𝑥)1−𝐹(𝑡)=𝑥^−1/𝛾 (1.6) for every 𝑥>0. This class includes a number of popular distributions such as Pareto, generalized Pareto, Burr, Fréchet, Student, and so forth, which are known to be appropriate models for fitting large insurance claims, fluctuations of prices, log-returns, and so forth (see, e.g., [10]). In the remainder of this paper, therefore, we restrict ourselves to this class of distributions. For more information on the topic and, generally, on extreme value models and their manifold applications, we refer to the monographs by Beirlant et al. [11], Castillo et al. [12], de Haan and Ferreira [13], Resnick [14].

The rest of the paper is organized as follows. In Section 2 we construct an alternative, called “new”, CTE estimator by utilizing an extreme value approach. In Section 3 we establish the asymptotic normality of the new CTE estimator and illustrate its performance with a little simulation study. The main result, which is Theorem 3.1 stated in Section 3, is proved in Section 4.

2. Construction of a New CTE Estimator

We have already noted that the “old” estimator CTE_𝑛(𝑡) does not yield the asymptotic normality (in the classical sense) beyond the condition 𝐄[𝑋²]<∞. Indeed, this follows by setting 𝑡=0, in which case CTE_𝑛(𝑡) becomes the sample mean of 𝑋₁,…,𝑋_𝑛, and thus the asymptotic normality of CTE_𝑛(0) is equivalent to the classical Central Limit Theorem (CLT). Similar arguments show that the finite second moment is necessary for having the asymptotic normality (in the classical sense) of CTE_𝑛(𝑡) at any fixed “level” 𝑡∈(0,1). Indeed, note that the asymptotic variance 𝜎²(𝑡) in Theorem 1.1 is finite only if 𝐄[𝑋²]<∞.

For this reason, we next construct an alternative CTE estimator, which takes into account different asymptotic properties of moderate and high quantiles in the case of heavy-tailed distributions. Hence, from now on we assume that 𝛾∈(1/2,1). Before indulging ourselves into construction details, we first formulate the new CTE estimator:

CTE_𝑛1(𝑡)=1−𝑡_𝑡^{1−𝑘/𝑛}𝑄_𝑛(𝑠)𝑑𝑠+𝑘𝑋_𝑛−𝑘,_𝑛,𝑛(1−𝑡)(1−𝛾) (2.1) where we use the simplest yet useful and powerful Hill's [15] estimator

𝛾_𝑛=1𝑘^𝑘_𝑖=1log𝑋_{𝑛−𝑖+1∶𝑛}−log𝑋_{𝑛−𝑘∶𝑛} (2.2) of the tail index 𝛾∈(1/2,1). Integers 𝑘=𝑘_𝑛∈{1,…,𝑛} are such that 𝑘→∞ and 𝑘/𝑛→0 when 𝑛→∞, and we note at the outset that their choices present a challenging task. In Figures 1 and 2, we illustrate the performance of the new estimator CTE_𝑛(𝑡) with respect to the sample size 𝑛≥1, with the integers 𝑘=𝑘_𝑛 chosen according to the method proposed by Cheng and Peng [16]. Note that when 𝑡 increases through the values 0.25, 0.50, 0.75, and 0.90 (panels (a)–(d), resp.), the vertical axes of the panels also increase, which reflects the fact that the larger the 𝑡 gets, the more erratic the “new” and “old” estimators become. Note also that the empirical (i.e., “old”) estimator underestimates the theoretical CTE_𝐹(𝑡), which is a well known phenomenon (see [17]).

(a)

(b)

(c)

(d)

Figure 1 

Values of the CTE estimator ${\widetilde{\mathrm{CTE}\hspace{0.17em}}}_n(t)$ (vertical axis) versus sample sizes $n$ (horizontal axis) evaluated at the levels $t=0.25\text{,}$$t=0.50\text{,}$$t=0.75$, and $t=0.90$ (panels (a)–(d), resp.) in the Pareto case with the tail index $\gamma =2/3$.

(a)

(b)

(c)

(d)

Figure 2 

Values of the CTE estimator ${\widetilde{\mathrm{CTE}\hspace{0.17em}}}_n(t)$ (vertical axis) versus sample sizes $n$ (horizontal axis) evaluated at the levels $t=0.25\text{,}$$t=0.50\text{,}$$t=0.75$, and $t=0.90$ (panels (a)–(d), resp.) in the Pareto case with the tail index $\gamma =3/4$.

We have based the construction of CTE_𝑛(𝑡) on the recognition that one should estimate moderate and high quantiles differently when the underlying distribution is heavy-tailed. For this, we first recall that the high quantile 𝑞_𝑠 is, by definition, equal to 𝑄(1−𝑠) for sufficiently small 𝑠. For an estimation theory of high quantiles in the case of heavy-tailed distributions we refer to, for example, Weissman [18], Dekkers and de Haan [19], Matthys and Beirlant [20], Gomes et al. [21], and references therein. We shall use the Weissman estimator

𝑞_𝑠𝑘=(𝑛)𝛾𝑋_{𝑛−𝑘∶𝑛}𝑠⁻𝛾𝑘,0<𝑠<𝑛, (2.3) of the high quantile 𝑞_𝑠. Then we write CTE_𝐹(𝑡) as the sum CTE_1,𝑛(𝑡)+CTE_2,𝑛(𝑡) with the two summands defined together with their respective empirical estimators CTE_1,𝑛(𝑡) and CTE_2,𝑛(𝑡) as follows:

CTE_1,𝑛1(𝑡)=1−𝑡_𝑡^{1−𝑘/𝑛}1𝑄(𝑠)𝑑𝑠≈1−𝑡_𝑡^{1−𝑘/𝑛}𝑄_𝑛(𝑠)𝑑𝑠=CTE_1,𝑛(𝑡),CTE_2,𝑛1(𝑡)=1−𝑡¹_{1−𝑘/𝑛}1𝑄(𝑠)𝑑𝑠≈1−𝑡¹_{1−𝑘/𝑛}𝑞_1−𝑠𝑑𝑠=CTE_2,𝑛(𝑡). (2.4) Simple integration gives the formula

CTE_2,𝑛(𝑡)=𝑘𝑋_𝑛−𝑘,_𝑛.𝑛(1−𝛾)(1−𝑡) (2.5) Consequently, the sum CTE_1,𝑛(𝑡)+CTE_2,𝑛(𝑡) is an estimator of CTE_𝐹(𝑡), and this is exactly the estimator CTE_𝑛(𝑡) introduced above. We shall investigate asymptotic normality of the new estimator in the next section, accompanied with an illustrative simulation study.

3. Main Theorem and Its Practical Implementation

We start this section by noting that Hill's estimator 𝛾_𝑛 has been thoroughly studied, improved, and generalized in the literature. For example, weak consistency of 𝛾_𝑛 has been established by Mason [22] assuming only that the underlying distribution is regularly varying at infinity. Asymptotic normality of 𝛾 has been investigated under various conditions by a number of researchers, including Csörgő and Mason [23], Beirlant and Teugels [24], Dekkers et al. [25], see also references therein.

The main theoretical result of this paper, which is Theorem 3.1 below, establishes asymptotic normality of the new CTE estimator CTE_𝑛(𝑡). To formulate the theorem, we need to introduce an assumption that ensures the asymptotic normality of Hill's estimator 𝛾_𝑛. Namely, the cdf 𝐹 satisfies the generalized second-order regular variation condition with second-order parameter 𝜌≤0 (see [26, 27]) if there exists a function 𝑎(𝑡) which does not change its sign in a neighbourhood of infinity and is such that, for every 𝑥>0,

lim_𝑡→∞1(𝑎(𝑡)1−𝐹(𝑡𝑥)1−𝐹(𝑡)−𝑥^−1/𝛾)=𝑥^−1/𝛾𝑥^𝜌/𝛾−1.𝜌/𝛾 (3.1) When 𝜌=0, then the ratio on the right-hand side of (3.1) is interpreted as log𝑥. For statistical inference concerning the second-order parameter 𝜌, we refer, for example, to Peng and Qi [28], Gomes et al. [21], Gomes and Pestana [29]. Furthermore, in the formulation of Theorem 3.1, we shall also use the function 𝐴(𝑧)=𝛾²𝑎(𝕌(𝑧)), where 𝕌(𝑧)=𝑄(1−1/𝑧).

Theorem 3.1. Assume that the cdf 𝐹 satisfies condition (3.1) with 𝛾∈(1/2,1). Then for any sequence of integers 𝑘=𝑘_𝑛→∞ such that 𝑘/𝑛→0 and 𝑘^1/2𝐴(𝑛/𝑘)→0 when 𝑛→∞, we have that for any fixed 𝑡∈(0,1), √𝑛(CTE_𝑛(𝑡)−CTE_𝐹(𝑡))(1−𝑡)(𝑘/𝑛)^1/2𝑋_{𝑛−𝑘∶𝑛}→_𝑑𝒩(0,𝜎²_𝛾), (3.2) where the asymptotic variance 𝜎²_𝛾 is given by the formula 𝜎²_𝛾=𝛾⁴(1−𝛾)⁴.(2𝛾−1) (3.3)

The asymptotic variance 𝜎²_𝛾 does not depend on 𝑡, unlike the variance in Theorem 1.1. This is not surprising because the heaviness of the right-most tail of 𝐹 makes the asymptotic behaviour of ∫¹_𝑡(𝑄_𝑛(𝑠)−𝑄(𝑠))𝑑𝑠 “heavier” than the classical CLT-type behaviour of ∫^𝑡₀(𝑄_𝑛(𝑠)−𝑄(𝑠))𝑑𝑠, for any fixed 𝑡. This in turn implies that under the conditions of Theorem 3.1, statement (3.2) is equivalent to the same statement in the case 𝑡=0. The latter statement concerns estimating the mean 𝐄[𝑋] of a heavy-tailed distribution. Therefore, we can view Theorem 3.1 as a consequence of Peng [30], and at the same time we can view results of Peng [30] as a consequence of Theorem 3.1 by setting 𝑡=0 in it. Despite this equivalence, in Section 4 we give a proof of Theorem 3.1 for the sake of completeness. Our proof, however, is crucially based on a powerful technique called the Vervaat process (see [31–33], for details and references).

To discuss practical implementation of Theorem 3.1, we first fix a significance level 𝜍∈(0,1) and use the classical notation 𝑧_𝜍/2 for the (1−𝜍/2)-quantile of the standard normal distribution 𝒩(0,1). Given a realization of the random variables 𝑋₁,…,𝑋_𝑛 (e.g., claim amounts), which follow a cdf 𝐹 satisfying the conditions of Theorem 3.1, we construct a level 1−𝜍 confidence interval for CTE_𝐹(𝑡) as follows. First, we choose an appropriate number 𝑘 of extreme values. Since Hill's estimator has in general a substantial variance for small 𝑘 and a considerable bias for large 𝑘, we search for a 𝑘 that balances between the two shortcomings, which is indeed a well-known hurdle when estimating the tail index. To resolve this issue, several procedures have been suggested in the literature, and we refer to, for example, Dekkers and de Haan [34], Drees and Kaufmann [35], Danielsson et al. [36], Cheng and Peng [16], Neves and Fraga Alves [37], Gomes et al. [38], and references therein. In our current study, we employ the method of Cheng and Peng [16] for an appropriate value 𝑘^∗ of the “parameter” 𝑘. Having computed Hill's estimator and consequently determined 𝑋_𝑛−𝑘^∗_∶𝑛, we then compute the corresponding values of CTE_𝑛(𝑡) and 𝜎𝛾²_𝑛, and denote them by CTE^∗_𝑛(𝑡) and 𝜎𝛾^2∗_𝑛, respectively. Finally, using Theorem 3.1 we arrive at the following (1−𝜍)-confidence interval for CTE_𝐹(𝑡):

CTE^∗_𝑛(𝑡)±𝑧_𝜍/2(𝑘^∗/𝑛)^1/2𝑋_𝑛−𝑘^∗_∶𝑛𝜎𝛾^2∗_𝑛(1−𝑡)√𝑛. (3.4)

To illustrate the performance of this confidence interval, we have carried out a small-scale simulation study based on the Pareto cdf 𝐹(𝑥)=1−𝑥^−1/𝛾, 𝑥≥1, with the tail index 𝛾 set to 2/3 and 3/4, and the level 𝑡 set to 0.75 and 0.90. We have generated 200 independent replicates of three samples of sizes 𝑛=1000, 2000, and 5000. For every simulated sample, we have obtained estimates CTE_𝑛(𝑡). Then we have calculated the arithmetic averages over the values from the 200 repetitions, with the absolute error (error) and root mean squared error (rmse) of the new estimator 𝐶_𝑛(𝑡) reported in Table 1 (𝛾=2/3) and Table 2 (𝛾=3/4). In the tables, we have also reported 95%-confidence intervals (3.4) with their lower and upper bounds, coverage probabilities, and lengths.

Table 1 

Point estimates CTE_𝑛(𝑡) and 95% confidence intervals for CTE_𝐹(𝑡) when 𝛾=2/3.

Table 2 

Point estimates CTE_𝑛(𝑡) and 95% confidence intervals for CTE_𝐹(𝑡) when 𝛾=3/4.

We note emphatically that the above coverage probabilities and lengths of confidence intervals can be improved by employing more precise but, naturally, considerably more complex estimators of the tail index. Such estimators are described in the monographs by Beirlant et al. [11], Castillo et al. [12], de Haan and Ferreira [13], and Resnick [14]. Since the publication of these monographs, numerous journal articles have appeared on the topic. Our aim in this paper, however, is to present a simple yet useful result that highlights how much Actuarial Science and developments in Mathematical Statistics, Probability, and Stochastic Processes are interrelated, and thus benefit from each other.

4. Proof of Theorem 3.1

We start the proof of Theorem 3.1 with the decomposition

(CTE_𝑛(𝑡)−CTE_𝐹(𝑡))(1−𝑡)=𝐴_𝑛,1(𝑡)+𝐴_𝑛,2, (4.1) where

𝐴_𝑛,1(𝑡)=_𝑡^{1−𝑘/𝑛}(𝑄_𝑛𝐴(𝑠)−𝑄(𝑠))𝑑𝑠,_𝑛,2=𝑘/𝑛1−𝛾_𝑛𝑋_{𝑛−𝑘∶𝑛}−¹_{1−𝑘/𝑛}𝑄(𝑠)𝑑𝑠. (4.2) We shall show below that there are Brownian bridges 𝐵_𝑛 such that

√𝑛𝐴_𝑛,1(𝑡)(𝑘/𝑛)^1/2∫𝑄(1−𝑘/𝑛)=−^{01−𝑘/𝑛}𝐵_𝑛(𝑠)𝑑𝑄(𝑠)(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)+𝑜_𝐏√(1),𝑛𝐴_𝑛,2(𝑘/𝑛)^1/2=𝛾𝑄(1−𝑘/𝑛)(1−²𝛾)²𝑛𝑘𝐵_𝑛𝑘(1−𝑛)−𝛾(1−𝛾)²𝑛𝑘¹_{1−𝑘/𝑛}𝐵_𝑛(𝑠)1−𝑠𝑑𝑠+𝑜_𝐏(1). (4.3)(4.4) Assuming for the time being that statements (4.3) and (4.4) hold, we next complete the proof of Theorem 3.1. To simplify the presentation, we use the following notation:

𝑊_1,𝑛∫=−^{01−𝑘/𝑛}𝐵_𝑛(𝑠)𝑑𝑄(𝑠)(𝑘/𝑛)^1/2,𝑊𝑄(1−𝑘/𝑛)_2,𝑛=𝛾²(1−𝛾)²𝑛𝑘𝐵_𝑛𝑘(1−𝑛𝑊),_3,𝑛𝛾=−(1−𝛾)²𝑛𝑘¹_{1−𝑘/𝑛}𝐵_𝑛(𝑠)1−𝑠𝑑𝑠. (4.5) Hence, we have the asymptotic representation

√𝑛(CTE_𝑛(𝑡)−CTE_𝐹(𝑡))(1−𝑡)(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)=𝑊_1,𝑛+𝑊_2,𝑛+𝑊_3,𝑛+𝑜_𝐏(1). (4.6) The sum 𝑊_1,𝑛+𝑊_2,𝑛+𝑊_3,𝑛 is a centered Gaussian random variable. To calculate its asymptotic variance, we establish the following limits:

𝐄[𝑊²_1,𝑛]→2𝛾2𝛾−1,𝐄[𝑊²_2,𝑛𝛾]→⁴(1−𝛾)⁴,𝐄[𝑊²_3,𝑛]→2𝛾²(1−𝛾)⁴,2𝐄[𝑊_1,𝑛𝑊_2,𝑛]→−2𝛾²(1−𝛾)²,2𝐄[𝑊_1,𝑛𝑊_3,𝑛]→2𝛾(1−𝛾)²,2𝐄[𝑊_2,𝑛𝑊_3,𝑛]→−2𝛾³(1−𝛾)⁴. (4.7) Summing up the right-hand sides of the above six limits, we obtain 𝜎²_𝛾, whose expression in terms of the parameter 𝛾 is given in Theorem 3.1. Finally, since 𝑋_{𝑛−𝑘∶𝑛}/𝑄(1−𝑘/𝑛) converges in probability to 1 (see, e.g., the proof of Corollary in [39]), the classical Sultsky's lemma completes the proof of Theorem 3.1. Of course, we are still left to verify statements (4.3) and (4.4), which make the contents of the following two subsections.

4.1. Proof of Statement (4.3)

If 𝑄 were continuously differentiable, then statement (4.3) would follow easily from the proof of Theorem 2 in [39]. We do not assume differentiability of 𝑄 and thus a new proof is required, which is crucially based on the Vervaat process (see [31–33], and references therein)

𝑉_𝑛(𝑡)=^𝑡₀(𝑄_𝑛(𝑠)−𝑄(𝑠))𝑑𝑠+^𝑄(𝑡)_−∞(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥. (4.8) Hence, for every 𝑡 such that 0<𝑡<1−𝑘/𝑛, which is satisfied for all sufficiently large 𝑛 since 𝑡 is fixed, we have that

𝐴_𝑛,1(𝑡)=₀^{1−𝑘/𝑛}(𝑄_𝑛(𝑠)−𝑄(𝑠))𝑑𝑠−^𝑡₀(𝑄_𝑛(𝑠)−𝑄(𝑠))𝑑𝑠=−^{𝑄(1−𝑘/𝑛)}_𝑄(𝑡)(𝐹_𝑛(𝑥)−𝐹(𝑥))𝑑𝑥+𝑉_𝑛𝑘(1−𝑛)−𝑉_𝑛(𝑡). (4.9) It is well known (see [31–33]) that 𝑉_𝑛(𝑡) is nonnegative and does not exceed −(𝐹_𝑛(𝑄(𝑡))−𝑡)(𝑄_𝑛(𝑡)−𝑄(𝑡)). Since the cdf 𝐹 is continuous by assumption, we therefore have that

√𝑛𝑉_𝑛(𝑡)≤|𝑒_𝑛(𝑡)||𝑄_𝑛(𝑡)−𝑄(𝑡)|, (4.10) where 𝑒_𝑛(𝑡) is the uniform empirical process √𝑛(𝐹_𝑛(𝑄(𝑡))−𝐹(𝑄(𝑡)), which for large 𝑛 looks like the Brownian bridge 𝐵_𝑛(𝑡). Note also that with the just introduced notation 𝑒_𝑛, the integral on the right-hand side of (4.9) is equal to ∫^{𝑄(1−𝑘/𝑛)}_𝑄(𝑡)𝑒_𝑛(𝐹(𝑥))𝑑𝑥. Hence,

√𝑛𝐴_𝑛,1(𝑡)(𝑘/𝑛)^1/2∫𝑄(1−𝑘/𝑛)=−^{𝑄(1−𝑘/𝑛)𝑄(𝑡)}𝑒_𝑛(𝐹(𝑥))𝑑𝑥(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)+𝑂_𝐏(1)|𝑒_𝑛(1−𝑘/𝑛)||𝑄_𝑛(1−𝑘/𝑛)−𝑄(1−𝑘/𝑛)|(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)+𝑂_𝐏(1)|𝑒_𝑛(𝑡)||𝑄_𝑛(𝑡)−𝑄(𝑡)|(𝑘/𝑛)^1/2.𝑄(1−𝑘/𝑛) (4.11) We shall next replace the empirical process 𝑒_𝑛 by an appropriate Brownian bridge 𝐵_𝑛 in the first integral on the right-hand side of (4.11) with an error term of magnitude 𝑜_𝐏(1), and we shall also show that the second and third summands on the right-hand side of (4.11) are of the order 𝑜_𝐏(1). The replacement of 𝑒_𝑛 by 𝐵_𝑛 can be accomplished using, for example, Corollary 2.1 on page 48 of Csörgő et al. [40], which states that on an appropriately constructed probability space and for any 0≤𝜈<1/4, we have that

sup_{1/𝑛≤𝑠≤1−1/𝑛}|𝑒_𝑛(𝑠)−𝐵_𝑛(𝑠)|𝑠^1/2−𝜈(1−𝑠)^1/2−𝜈=𝑂_𝐏(𝑛^−𝜈). (4.12) This result is applicable in the current situation since we can always place our original problem into the required probability space, because our main results are “in probability”. Furthermore, since 𝑄(𝑡)≤𝑥≤𝑄(1−𝑘/𝑛), we have that 𝑡≤𝐹(𝑥)≤1−𝑘/𝑛. Hence, statement (4.12) implies that

−∫^{𝑄(1−𝑘/𝑛)}_𝑄(𝑡)𝑒_𝑛(𝐹(𝑥))𝑑𝑥(𝑘/𝑛)^1/2∫𝑄(1−𝑘/𝑛)=−^{𝑄(1−𝑘/𝑛)𝑄(𝑡)}𝐵_𝑛(𝐹(𝑥))𝑑𝑥(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)+𝑂_𝐏∫(1)^{𝑄(1−𝑘/𝑛)𝑄(𝑡)}(1−𝐹(𝑥))^1/2−𝜈𝑑𝑥𝑛^𝜈(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛). (4.13) Changing the variables of integration and using the property (𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)→∞ when 𝑛→∞, we obtain that

−∫^{𝑄(1−𝑘/𝑛)}_𝑄(𝑡)𝐵_𝑛(𝐹(𝑥))𝑑𝑥(𝑘/𝑛)^1/2∫𝑄(1−𝑘/𝑛)=−^{01−𝑘/𝑛}𝐵_𝑛(𝑠)𝑑𝑄(𝑠)(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)+𝑜_𝐏(1). (4.14) The main term on the right-hand side of (4.14) is 𝑊_1,𝑛. We shall next show that the right-most summand of (4.13) converges to 0 when 𝑛→∞.

Changing the variable of integration and then integrating by parts, we obtain the bound

∫^{𝑄(1−𝑘/𝑛)}_𝑄(𝑡)(1−𝐹(𝑥))^1/2−𝜈𝑑𝑥𝑛^𝜈(𝑘/𝑛)^1/2≤𝑄(1−𝑘/𝑛)(1−𝑠)^1/2−𝜈𝑄(𝑠)|_𝑡^{1−𝑘/𝑛}𝑛^𝜈(𝑘/𝑛)^1/2∫𝑄(1−𝑘/𝑛)+𝑂(1)^{𝑡1−𝑘/𝑛}(1−𝑠)^{−1/2−𝜈}𝑄(𝑠)𝑑𝑠𝑛^𝜈(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛). (4.15) We want to show that the right-hand side of bound (4.15) converges to 0 when 𝑛→∞. For this, we first note that

(1−𝑠)^1/2−𝜈𝑄(𝑠)|_𝑡^{1−𝑘/𝑛}𝑛^𝜈(𝑘/𝑛)^1/2=1𝑄(1−𝑘/𝑛)𝑘^𝜈−(1−𝑡)^1/2−𝜈𝑄(𝑡)𝑛^𝜈(𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)→0. (4.16) Next, with the notation 𝜙(𝑢)=𝑄(1−𝑢)/𝑢^1/2+𝜈, we have that

∫_𝑡^{1−𝑘/𝑛}(1−𝑠)^{−1/2−𝜈}𝑄(𝑠)𝑑𝑠𝑛^𝜈(𝑘/𝑛)^1/2=1𝑄(1−𝑘/𝑛)𝑘^𝜈∫^1−𝑡_𝑘/𝑛𝜙(𝑠)𝑑𝑠(𝑘/𝑛)𝜙(𝑘/𝑛)→0 (4.17) when 𝑛→∞, where the convergence to 0 follows from Result 1 in the Appendix of Necir and Meraghni [39]. Taking statements (4.15)–(4.17) together, we have that the right-most summand of (4.13) converges to 0 when 𝑛→∞.

Consequently, in order to complete the proof of statement (4.3), we are left to show that the second and third summands on the right-hand side of (4.11) are of the order 𝑜_𝐏(1). The third summand is of the order 𝑜_𝐏(1) because |𝑒_𝑛(𝑡)‖𝑄_𝑛(𝑡)−𝑄(𝑡)|=𝑂_𝐏(1) and (𝑘/𝑛)^1/2𝑄(1−𝑘/𝑛)→∞. Hence, we are only left to show that the second summand on the right-hand side of equation (4.11) is of the order 𝑜_𝐏(1), for which we shall show that

|𝑒_𝑛(1−𝑘/𝑛)|(𝑘/𝑛)^1/2|𝑄_𝑛(1−𝑘/𝑛)𝑄(1−𝑘/𝑛)−1|=𝑜_𝐏(1). (4.18) To prove statement (4.18), we first note that

|𝑒_𝑛(1−𝑘/𝑛)|(𝑘/𝑛)^1/2≤|𝑒_𝑛(1−𝑘/𝑛)−𝐵_𝑛(1−𝑘/𝑛)|(𝑘/𝑛)^1/2+|𝐵_𝑛(1−𝑘/𝑛)|(𝑘/𝑛)^1/2. (4.19) The first summand on the right-hand side of bound (4.19) is of the order 𝑂_𝐏(1) due to statement (4.12) with 𝜈=0. The second summand on the right-hand side of bound (4.19) is of the order 𝑂_𝐏(1) due to a statement on page 49 of Csörgő et al. [40] (see the displayed bound just below statement (2.39) therein). Hence, to complete the proof of statement (4.18), we need to check that

𝑄_𝑛(1−𝑘/𝑛)𝑄(1−𝑘/𝑛)=1+𝑜_𝐏(1). (4.20) Observe that, for each 𝑛, the distribution of 𝑄_𝑛(1−𝑘/𝑛) is the same as that of 𝑄(𝐸_𝑛⁻¹(1−𝑘/𝑛)), where 𝐸_𝑛⁻¹ is the uniform empirical quantile function. Furthermore, the processes {1−𝐸_𝑛⁻¹(1−𝑠),0≤𝑠≤1} and {𝐸_𝑛⁻¹(𝑠),0≤𝑠≤1} are equal in distribution. Hence, statement (4.20) is equivalent to

𝑄(1−𝐸_𝑛⁻¹(𝑘/𝑛))𝑄(1−𝑘/𝑛)=1+𝑜_𝐏(1). (4.21) From the Glivenko-Cantelli theorem we have that 𝐸_𝑛⁻¹(𝑘/𝑛)−𝑘/𝑛→0 almost surely, which also implies that 𝐸_𝑛⁻¹(𝑘/𝑛)→0 since 𝑘/𝑛→0 by our choice of 𝑘. Moreover, we know from Theorem 0 and Remark 1 of Wellner [41] that

sup_{1/𝑛≤𝑠≤1}𝑠⁻¹|𝐸_𝑛⁻¹(𝑠)−𝑠|=𝑜_𝐏(1), (4.22) from which we conclude that

𝑛𝐸_𝑛⁻¹(𝑘/𝑛)𝑘=1+𝑜_𝐏(1). (4.23) Since the function 𝑠↦𝑄(1−𝑠) is slowly varying at zero, using Potter's inequality (see the 5th assertion of Proposition B.1.9 on page 367 of de Haan and Ferreira [13],we obtain that

𝑄(1−𝐸_𝑛⁻¹(𝑘/𝑛))𝑄(1−𝑘/𝑛)=(1+𝑜_𝐏(1))(𝑛𝐸_𝑛⁻¹(𝑘/𝑛)𝑘)^{−𝛾±𝜃} (4.24) for any 𝜃∈(0,𝛾). In view of (4.23), the right-hand side of (4.24) is equal to 1+𝑜_𝐏(1), which implies statement (4.21) and thus finishes the proof of statement (4.3).

4.2. Proof of Statement (4.4)

The proof of statement (4.4) is similar to that of Theorem 2 in Necir et al. [42], though some adjustments are needed since we are now concerned with the CTE risk measure. We therefore present main blocks of the proof together with pinpointed references to Necir et al. [42] for specific technical details.

We start the proof with the function 𝕌(𝑧)=𝑄(1−1/𝑧) that was already used in the formulation of Theorem 3.1. Hence, if 𝑌 is a random variable with the distribution function 𝐺(𝑧)=1−1/𝑧, 𝑧≥1, then 𝕌(𝑌)=𝑄(𝐺(𝑌))=_𝑑𝑋 because 𝐺(𝑌) is a uniform on the interval [0,1] random variable. Hence,

𝐴_𝑛,2=𝑘/𝑛1−𝛾_𝑛𝕌(𝑌_{𝑛−𝑘∶𝑛})−₀^𝑘/𝑛1𝕌(𝑠)𝑑𝑠, (4.25) and so we have

√𝑛𝐴_𝑛,2(𝑘/𝑛)^1/2=√𝑄(1−𝑘/𝑛)1𝑘(1−𝛾_𝑛𝕌(𝑌_{𝑛−𝑘∶𝑛})−𝕌(𝑛/𝑘)(𝑛/𝑘)∫₀^𝑘/𝑛𝕌(1/𝑠)𝑑𝑠)=√𝕌(𝑛/𝑘)1𝑘(1−𝛾_𝑛𝕌(𝑌_{𝑛−𝑘∶𝑛})−1𝕌(𝑛/𝑘))+√1−𝛾1𝑘(−∫1−𝛾^∞1𝑠⁻²𝕌(𝑛𝑠/𝑘)𝑑𝑠𝕌(𝑛/𝑘)). (4.26) We next show that the right-most term in (4.26) converges to 0 when 𝑛→∞. For this reason, we first rewrite the term as follows:

√𝑘(1−∫1−𝛾^∞1𝑠⁻²𝕌(𝑛𝑠/𝑘)𝑑𝑠√𝕌(𝑛/𝑘))=−𝑘^∞₁1𝑠²(𝕌(𝑛𝑠/𝑘)𝕌(𝑛/𝑘)−𝑠^𝛾)𝑑𝑠. (4.27) The right-hand side of (4.27) converges to 0 (see notes on page 149 of Necir et al. [42]) due to the second-order condition (3.1), which can equivalently be rewritten as

lim_𝑧→∞1(𝐴(𝑧)𝕌(𝑧𝑠)𝕌(𝑧)−𝑠^𝛾)=𝑠^𝛾𝑠^𝜌−1𝜌 (4.28) for every 𝑠>0, where 𝐴(𝑧)=𝛾²𝑎(𝕌(𝑧)). Note that √𝑘𝐴(𝑛/𝑘)→0 when 𝑛→∞. Hence, in order to complete the proof of statement (4.4), we need to check that

√𝑘(11−𝛾_𝑛𝕌(𝑌_{𝑛−𝑘∶𝑛})−1𝕌(𝑛/𝑘)𝛾1−𝛾)=²(1−𝛾)²𝑛𝑘𝐵_𝑛𝑘(1−𝑛)−𝛾(1−𝛾)²𝑛𝑘¹_{1−𝑘/𝑛}𝐵_𝑛(𝑠)1−𝑠𝑑𝑠+𝑜_𝐏(1). (4.29) With Hill's estimator written in the form

𝛾_𝑛=1𝑘^𝑘_𝑖=1log(𝕌(𝑌_{𝑛−𝑖+1∶𝑛})𝕌(𝑌_{𝑛−𝑘∶𝑛})), (4.30) we proceed with the proof of statement (4.29) as follows: √𝑘(1−𝛾1−𝛾_𝑛𝕌(𝑌_{𝑛−𝑘∶𝑛})=√𝕌(𝑛/𝑘)−1)𝑘1−𝛾1−𝛾_𝑛(𝕌(𝑌_{𝑛−𝑘∶𝑛})𝑌𝕌(𝑛/𝑘)−(^{𝑛−𝑘∶𝑛})𝑛/𝑘^𝛾)+√𝑘1−𝛾1−𝛾_𝑛𝑌((^{𝑛−𝑘∶𝑛})𝑛/𝑘^𝛾−1)+√𝑘𝛾_𝑛−𝛾1−𝛾_𝑛. (4.31) Furthermore, we have that

√𝑘𝛾_𝑛−𝛾1−𝛾_𝑛=11−𝛾_𝑛1√𝑘^𝑘_𝑖=1(log(𝕌(𝑌_{𝑛−𝑖+1∶𝑛})𝕌(𝑌_{𝑛−𝑘∶𝑛})𝑌)−𝛾log(^{𝑛−𝑖+1∶𝑛}𝑌_{𝑛−𝑘∶𝑛}+𝛾))1−𝛾_𝑛1√𝑘^𝑘_𝑖=1𝑌(log(^{𝑛−𝑖+1∶𝑛}𝑌_{𝑛−𝑘∶𝑛})−1). (4.32) Arguments on page 156 of Necir et al. [42] imply that the first term on the right-hand side of (4.32) is of the order 𝑂_𝐏(√𝑘𝐴(𝑌_{𝑛−𝑘∶𝑛})), and a note on page 157 of Necir et al. [42] says that √𝑘𝐴(𝑌_{𝑛−𝑘∶𝑛})=𝑜_𝐏(1). Hence, the first term on the right-hand side of (4.32) is of the order 𝑜_𝐏(1). Analogous considerations using bound (2.5) instead of (2.4) on page 156 of Necir et al. [42] imply that the first term on the right-hand side of (4.31) is of the order 𝑜_𝐏(1). Hence, in summary, we have that

√𝑘(1−𝛾1−𝛾_𝑛𝕌(𝑌_{𝑛−𝑘∶𝑛})𝕌(𝑛/𝑘)−1)=1−𝛾1−𝛾_𝑛√𝑘((𝑌_{𝑛−𝑘∶𝑛})𝑛/𝑘^𝛾+𝛾−1)1−𝛾_𝑛1√𝑘^𝑘_𝑖=1𝑌(log(^{𝑛−𝑖+1∶𝑛}𝑌_{𝑛−𝑘∶𝑛})−1)+𝑜_𝐏(1). (4.33) We now need to connect the right-hand side of (4.33) with Brownian bridges 𝐵_𝑛. To this end, we first convert the 𝑌-based order statistics into 𝑈-based (i.e., uniform on [0,1]) order statistics. For this we recall that the cdf of 𝑌 is 𝐺, and thus 𝑌 is equal in distribution to 𝐺⁻¹(𝑈), which is 1/(1−𝑈). Consequently,

√𝑘(1−𝛾1−𝛾_𝑛𝕌(𝑌_{𝑛−𝑘∶𝑛})𝕌(𝑛/𝑘)−1)=1−𝛾1−𝛾_𝑛√𝑘((1(𝑛/𝑘)(1−𝑈_{𝑛−𝑘∶𝑛}))^𝛾+𝛾−1)1−𝛾_𝑛1√𝑘^𝑘_𝑖=1(log((1−𝑈_{𝑛−𝑘∶𝑛})(1−𝑈_{𝑛−𝑖+1∶𝑛}))−1)+𝑜_𝐏(1). (4.34) Next we choose a sequence of Brownian bridges 𝐵_𝑛 (see pages 158-159 in [42] and references therein) such that the following two asymptotic representations hold:

√𝑘((1(𝑛/𝑘)(1−𝑈_{𝑛−𝑘∶𝑛}))^𝛾−1)=−𝛾𝑛𝑘𝐵_𝑛𝑘(1−𝑛)+𝑜_𝐏1(1),√𝑘^𝑘_𝑖=1(log((1−𝑈_{𝑛−𝑘∶𝑛})(1−𝑈_{𝑛−𝑖+1∶𝑛}))−1)=𝑛𝑘𝐵_𝑛𝑘(1−𝑛)−𝑛𝑘¹_{1−𝑘/𝑛}𝐵_𝑛(𝑠)1−𝑠𝑑𝑠+𝑜_𝐏(1). (4.35) Using these two statements on the right-hand side of (4.34) and also keeping in mind that 𝛾_𝑛 is a consistent estimator of 𝛾 (see [22]), we have that

√𝑘(1−𝛾1−𝛾_𝑛𝕌(𝑌_{𝑛−𝑘∶𝑛})𝛾𝕌(𝑛/𝑘)−1)=²1−𝛾𝑛𝑘𝐵_𝑛𝑘(1−𝑛)−𝛾1−𝛾𝑛𝑘¹_{1−𝑘/𝑛}𝐵_𝑛(𝑠)1−𝑠𝑑𝑠+𝑜_𝐏(1). (4.36) Dividing both sides of equation (4.36) by 1−𝛾, we arrive at (4.29). This completes the proof of statement (4.4) and of Theorem 3.1 as well.

Acknowledgments

Our work on the revision of this paper has been considerably influenced by constructive criticism and suggestions by three anonymous referees and the editor in charge of the manuscript, Edward Furman, and we are indebted to all of them. Results of the paper were first announced at the 44th Actuarial Research Conference at the University of Wisconsin, Madison, Wisconsin, July 30–August 1, 2009. The authors are grateful to participants of this most stimulating conference, organized by the Society of Actuaries, for generous feedback. The research has been partially supported by grants from the Society of Actuaries (SOA) and the Natural Sciences and Engineering Research Council (NSERC) of Canada.