A class of unbiased location invariant Hill-type estimators for heavy tailed distributions

Based on the methods provided in Caeiro and Gomes (2002) and Fraga Alves (2001), a new class of location invariant Hill-type estimators is derived in this paper. Its asymptotic distributional representation and asymptotic normality are presented, and the optimal choice of sample fraction by Mean Squared Error is also discussed for some special cases. Finally comparison studies are provided for some familiar models by Monte Carlo simulations.


Introduction
In this paper we study the asymptotic properties of a class of unbiased location invariant Hill-type heavy tailed index estimators. Let {X n , n ≥ 1} be an independent and identically distributed (i.i.d) sequence with distribution function (d.f) F (x) belonging to the domain of attraction of an extreme value distribution, G γ (x) = exp −(1 + γx) −1/γ , γ ∈ R and 1 + γx > 0, i.e. there exist normalizing constants a n > 0 and b n ∈ R such that F n (a n x + b n ) −→ G γ (x) (1.1) as n → ∞ for all x ∈ R. We write F ∈ D(G γ ) if (1.1) holds and call γ the tail index.
The applications of the heavy tailed distributions may be found in many fields such as insurance, finance, climatology and environmental science (cf. Embrechets et al. (1997)), and the problem of estimating the heavy tailed index has been studied extensively. For positive γ, Hill (1975) introduced the well known Hill estimator, which is where X 1,n ≤ X 2,n ≤ · · · ≤ X n,n are the order statistics of X 1 , X 2 , · · · , X n . As an extension of Hill-type estimator for general γ ∈ R, Dekkers et al. (1989) proposed the following moment estimator (ln X n−i,n − ln X n−k,n ) j , j = 1, 2.
In applications both Hill and moment estimators are sensitive to the threshold, say X n−k,n , and these two estimators are not invariant to the affine transformation. Fraga Alves (2001)  (1. 2) The asymptotic properties of γ n (k 0 , k) have been considered when holds as t → ∞, where U (t) = F ← (1 − 1/t), t ≥ 1, where F ← (x) denotes the inverse function of F (x). Based on the methods provided in Fraga Alves (2001) and Dekkers et al. (1989), Ling et al. (2007aLing et al. ( , 2007b proposed a kind of location invariant moment-type tail index estimator and considered its asymptotic properties under some second order regular varying conditions. Meanwhile the problem of searching for the unbiased estimators has been considered by many authors, see, e.g., Peng (1998), Beirlant et al. (2002), Caeiro and Gomes (2002), Martins (2002, 2004), Peng andQi (2006a, 2006b), , , and Qi (2008aQi ( , 2008b. Here we are interested in the class of semi-parametric heavy tail estimator introduced by Caeiro and Gomes (2002): where Γ(·) is the gamma function and They considered the asymptotic distributional representation of M (α) n (k) and the choice of tuning parameter α such that γ (α) n (k) is asymptotically normal with asymptotic null bias under the assumptions that and √ kA(n/k) → λ, finite. Motivated by the works of Fraga Alves (2001) and Caeiro and Gomes (2002), we propose a new class of location invariant estimators for a heavy tailed distribution based on the asymptotic distributional representation of the following statistic: The asymptotic distributional representation of M (α) n (k 0 , k) will be derived under the following second order regular variation condition: and ρ < 0 is the second order parameter and |A(t)| ∈ RV ρ (cf. Corollary 2.3.5 of de Haan and Ferreira (2006)). Here, f ∈ RV β means lim t→∞ f (tx)/f (t) = x β for all x > 0. Based on the convergence M (α) , a location invariant Hill-type estimator for the heavy tailed index may be defined by which converges to γ in probability. Finite sample simulation shows that the positive tuning parameter α may be chosen appropriately to improve the performance of tail index estimation in applications. The paper is organized as follows. In section 2, we provide the main results, i.e. the asymptotic distributional representations of (1.3) and (1.5), and the optimal choice of the sample fraction k 0 by mean squared error (MSE) for some special distributions. Related proofs are deferred to section 4. Simulation studies are performed in section 3.

The main results
Throughout this paper, we assume that X 1 , X 2 , · · · , X n are i.i.d random variables (r.v.s) with d.f F (x), and denote by X 1,n ≤ X 2,n ≤ · · · ≤ X n,n the order statistics of X 1 , · · · , X n . For the heavy tail distribution, i.e. γ > 0, we know that (2.1) We need the following notations to simplify the statements of our main results: 3) The first result is about the asymptotic distributional representation of M n (k 0 , k) converges in probability to Γ(α + 1)γ α . Furthermore, if the second order framework in (1.4) holds, we may obtain the following asymptotic distributional representation where P (α) n is an asymptotically standard normal r.v., and Based on Theorem 2.1, we may derive the asymptotic distributional representation of the proposed estimator γ (α) n (k 0 , k) in (1.5), which is the following result.
Theorem 2.2. Under the conditions of Theorem 2.1, the following asymptotic distributional representation is asymptotically standard normal, and Furthermore, if there exist finite λ 1 and λ 2 such that where V α and b α (γ) are defined in (2.2) and (2.3), respectively. Consequently, for every γ > 0, there exists α 0 given by (k 0 , k) has asymptotic null bias, even when For special A(t), we will consider the optimal choice of the sample fraction k 0 as a function of k, γ, ρ and α following the criterion of Fraga Alves (2001) n (k 0 , k)) is minimal. Note that an asymptotic MSE is just the usual MSE based on some asymptotic relationship. For example, suppose θ n is an estimator of θ based on a random sample of size n and that it satisfies √ a n ( θ n − θ − b n ) → N (µ, σ 2 ) as n → ∞ for some a n , b n , µ and σ 2 . Then the asymptotic MSE of θ n denoted by M SE ∞ ( θ n ) is simply {b n +µ/ √ a n } 2 +σ 2 /a n .
where V α and b α (γ) are defined as before. Then we may obtain the sequence k opt and V α defined as before.

Simulation study
Firstly, we present some α 0 (γ) for heavy index γ such that b α0 (γ) = 0. Table 1 shows that α 0 (γ) is a decreasing function of γ. Firstly we consider the effect of the tuning parameter α on the heavy tail index estimator proposed in this paper. We randomly select a sample from Fréchet d.f F (x) = exp(−x −1/γ ) with γ = 1. For γ = 1, choose α 0 = 1.90 such that b α0 (γ) = 0 (cf. n with sample size n = 3000. Figure 1 shows that γ (1.90) n has a much smaller bias than others. For the optimal sequence k 0 , we denote k 0 := arg min k0 M SE( γ H n (k 0 , k)) and k 0 := arg min k0 M SE( γ (α) n (k 0 , k)). Consequently, we get the following possible measure of asymptotic relative efficiency (AREFF) of the proposed location invariant estimator γ  (1) distribution with sample size n = 3000.  for γ = 0.5, 1 and 2. Simulation shows that for every γ, we may find some α on the left region of α 0 such that the AREFF in (3.1) is greater than one.
Next we compare the relative efficiency of the proposed location invariant Hill-type estimator and that of Fraga Alves (2001) in terms of average mean and MSE for finite sample size. We consider the following two models: - n (k 0 , k) and γ (α0) n (k 0 , k) for the distributions just mentioned. For both distributions, γ (α0) n (k 0 , k) has a small bias and is closer to the real extreme value index value (see Figures 3 and 4). Lastly we compare the coverage probability and the confidence length based on the asymptotic normality of the two location invariant Hill estimators. Note that Corollary 2.2 of Fraga Alves (2001) provides us the 1 − θ asymptotic confidence interval of γ, i.e.
We can also construct a new asymptotic confidence interval of γ based on Corollary 2.1 by choosing α 0 such that b α0 (γ) = 0 for given γ. The asymptotic confidence interval of γ is defined by where k 0 in (3.3) is the same as the one in (3.2) and z θ/2 is the critical value of the standard normal distribution at level θ/2; that is, 1 − Φ(z θ/2 ) = θ/2. We drew 2000 random samples from the Fréchet (1), Burr (2,1) and Pareto (2) distributions. The simulation was repeated 2000 times, we computed the coverage probabilities of I N (0.95) and the interval lengths for n = 200, 300, · · · , 2000. These coverage probabilities and interval lengths are reported in Tables 2 and 3. We may conclude that the two estimators have comparable coverage probabilities only for Pareto distribution. Generally, the proposed estimator γ (α0) n (k 0 , k) Table 3 Average length comparison for 95% confidence levels based on (3.2) and (3.3).
Fréchet (1)  has better coverage probability than that of γ H n (k 0 , k). But it has a wider confidence interval length than that of γ H n (k 0 , k) especially when the sample size n is small. Meanwhile, Figure 2 tells us that there exists α 0 such that γ (α0) n (k 0 , k) has asymptotic null bias and better asymptotic efficiency than γ H n (k 0 , k). This new location invariant Hill type estimator may be useful in empirical analysis.

Proofs
Before proving the main results, we need some lemmas.
Proof of Theorem 2.1. For the asymptotic distributional representation of M (α) n (k 0 , k), we only consider the case of γ + ρ = 0. By Lemma 4.2, we have Let Y 1 , Y 2 , · · · , Y n be i.i.d Pareto r.v.s with F Y (y) = 1−1/y, y ≥ 1 and let Y 1,n ≤ Y 2,n ≤ · · · ≤ Y n,n denote the order statistics of Y 1 , Y 2 , · · · , Y n . Now replace t by Y n−k,n , x by Y n−i,n /Y n−k,n and y by Y n−k0,n /Y n−k,n , respectively, and note By Taylor's expansion, we may get So, By (4.2) and the law of large numbers, the asymptotic distributional representation of M (α) which is the desired result.
Proof of Theorem 2.2. Note that both For γ + ρ = 0, by using the conditions of Theorem 2.2, following joint expansion in distribution,  Note that (4.4) may be deduced from Wold device and Delta-method, as for arbitrary a, b ∈ R, we have a M (2α) n (k 0 , k)/Γ(2α + 1) The last step follows since Z n .