On the Generalized Hill Process for Small Parameters and Applications

Let $X_{1},X_{2},...$ be a sequence of independent copies (s.i.c) of a real random variable (r.v.) $X\geq 1$, with distribution function $df$ $F(x)=\mathbb{P}% (X\leq x)$ and let $X_{1,n}\leq X_{2,n} \leq ... \leq X_{n,n}$ be the order statistics based on the $n\geq 1$ first of these observations. The following continuous generalized Hill process {equation*} T_{n}(\tau)=k^{-\tau}\sum_{j=1}^{j=k}j^{\tau}(\log X_{n-j+1,n}-\log X_{n-j,n}), \label{dl02} {equation*} $\tau>0$, $1\leq k \leq n$, has been introduced as a continuous family of estimators of the extreme value index, and largely studied for statistical purposes with asymptotic normality results restricted to $\tau>1/2$. We extend those results to $0<\tau \leq 1/2$ and show that asymptotic normality is still valid for $\tau=1/2$. For $0<\tau<1/2$, we get non Gaussian asymptotic laws which are closely related to the Riemann function $% \zeta(s)=\sum_{n=1}^{\infty} n^{-s},s>1$


Introduction
The aim of this note is to settle an asymptotic theory for some functional forms of Hill's estimators. Precisely, let X 1 , X 2 , ... be a sequence of independent copies (s.i.c) of a real random variable (r.v.) X with distribution function (df ) F (x) = P(X ≤ x), x ∈ R. Since we are only concerned with the upper tail of F in this paper, we assume without loss of generality that X ≥ 1 and define a s.i.c. of the r.v. Y = log X denoted Y 1 , Y 2 , ... with df G(x) = P (Y ≤ x) = F (e x ), x ≥ 0. Finally Y 1,n = log X 1,n ≤ ... ≤ Y n,n = log X n,n are their respective order statistics. The following generalized continuous Hill process (1.1) T n (τ ) = k −τ j=k j=1 j τ (log X n−j+1,n − log X n−j,n ) , has been introduced and studied in [6] and [5], for a continuous parameter τ > 0, and as throughout this paper, 1 ≤ k = k(n) ≤ n, k → +∞, k/n → 0 as n → +∞.
But some margins of (1.1) appeared before that, since T n (1) is the Hill estimator introduced in [8] in 1975. Also, De Haan and Resnick [4] proposed DR n = T n (0)/(log k) as a simple estimator of the extreme value index. It has been shown in [10] that for F ∈ D(Λ), DR n -when appropriately centered and normalized -converges in law to a multiple of a Gumbel law.
However, we are only concerned in this paper with values τ > 0. In this case, the process (1.1) provides a family of estimators of the Univariate Extreme Value Index as shown in [5] in a sense to be defined shortly, after a brief recall on Extreme Value Theory.
Since this work arises in the just mentioned theory, it would be appropriate to make some reminders before going any further. The reader is referred to de Haan [3], de Haan and Ferreira [2], Resnick [12] and Galambos [7] for a modern and large account of the Extreme Value Theory. However, the least to say is that the df F is said to be attracted to a non degenerated df M iff the maximum X n,n = max (X 1 , ...X n ), when appropriately centered and normalized by two sequences of real numbers (a n > 0) n≥0 and (b n ) n≥0 , converges to M , in the sense that (1.2) lim n→+∞ P (X n,n ≤ a n x + b n ) = lim n→+∞ F n (a n x + b n ) = M (x), for continuity points x of M . If (1.2) holds, it is said that F is attracted to M or F belongs to the domain of attraction of M , written F ∈ D(M ). It is well-kwown that the three nondegenerate possible limits in (1.2), called extremal df 's, are the following.
The Gumbel df the Fréchet df with parameter γ > 0 and the Weiblull df with parameter β > 0 where I A denotes the indicator function of the set A.
Actually the limiting df M is defined by an equivalence class of the binary relation R on the set D of df s on R, defined as follows Theses facts allow to parameterize the class of extremal df 's. For this purpose, suppose that (1.2) holds for the three df 's given in (1.3), (1.4) and (1.5). If we take sequences (c n > 0) n≥1 and (d n ) n≥1 such that the limits in (1.6) are a = γ = 1/α and b = 1 (in the case of Fréchet extremal domain), and a = −β = −1/α and b = −1 (in the case of Weibull extremal domain), and finally, if we interpret (1 + γx) −1/γ as exp(−x) for γ = 0 (in the case of Gumbel extremal domain), we are entitled to write the three extremal df 's in the parameterized shape The parameter γ is called extreme value index. Originally, Hill(1975) [8] introduced the so-called Hill estimator T n (1) of the parameter γ of the Pareto df 1 − Cx −1/γ I (x≥0) . So did Pickands [11] as well. In the same way, the Generalized Hill process T n (τ ) provides an infinite family of estimators for the extremal index value in the sense that for any τ > 0, for F∈ D(G γ ), 0 < γ ≤ +∞, τ T n (τ ) → γ a.s as n → +∞.
This motivated Diop and Lo [6] to find out asymptotic normality results for this family in order to construct statistical tests. Although their results seem to be valuable, they restricted themselves to values τ > 1/2. Actually, this restriction is imposed by the so-called Hungarian approximation methods based on M. Csőrgö et al. [1] results that they used. It is then necessary to call for another approach to get around this obstacle.
In this paper, we will complete the work of [6] by giving the asymptotic law of T n (τ ) for 0 < τ ≤ 1/2 under the hypothesis F ∈ D(G γ ), 0 ≤ γ < +∞. Our results are also complements to the related ones for T n (0) in connection with those in [11], [4] and [10]. We point out for once that D(G 0 ) = D(Γ) for γ = +∞.
We restrict ourselves ourselves here to the cases 0 ≤ γ < +∞, since the case γ < 0, may be studied through the transform F (x 0 (F ) − 1/˙)) ∈ D(G −γ ) for estimating γ. This leads to replace X n−j+1,n by x 0 (F ) − 1/1/X j,n in (1.1). However, a direct investigation of (1.1) for γ < 0 is possible. This requires the theory of sums of dependent random variables while this paper uses results on sums of independant random variabes, as it will be seen shortly. We consequently consider a special handling of this case in a dinstinct paper.
Our achievement is the full description of the asymptotic behavior of T n (τ ), under the hypothesis F ∈ D(G γ ), 0 ≤ γ < +∞, for 0 < τ ≤ 1/2, which is still Gaussian for τ = 1/2 and non Gaussian for τ < 1/2. In the latter case, the limiting laws are described in connection with the Riemann function. Our best results concern τ ∈]0, 1/2[, in which case the usual conditions, (C2) is useless and and (C3) becomes very week. The asymptotic results become very general (see Subsection 4.2).
Before we state our results, we need to introduce some representations used in the theorems and to describe the conditions we shall require, at the appropriate time, in the next section. In Section 3, we will state our results and their proofs. In Section 4, statistical illustrations of the results and simulation study results are provided while Section 5 includes the key tools of this paper, which ends by a general conclusion in Section 6.

Representations Tools
Throughout the paper, we use the usual representation of Y 1 , Y 2 , ...
.. are independent and uniform random variables on (0, 1) in the sense of equality in distribution (denoted by = d ) In connexion with this, we also use the following Malmquist representation (see [13], p. 336) where E 1 , ..., E n are independent standard exponential random variables.
Now we recall the classical representations of df 's attracted to some nondegenerated extremal df . For each df F in the extremal domain, an appropriate representation is given for Theorem 1. We have : , γ > 0, then there exist two measurable functions p(u) and b(u) of u ∈ (0, 1) such that sup(|p(u)| , |b(u)|) → 0 as u → 0 and a positive constant c so that (2.3) 1} < +∞ and there exist two measurable functions p(u) and b(u) for u ∈ (0, 1) and a positive constant c as defined in (2.3) such that (2) Representation of de Haan (Theorem 2.4.1 in [3]) : If G ∈ D(Λ), then there exist two measurable functions p(u) and b(u) of u ∈ (0, 1) and a positive constant c as defined in (2.3) such that for we have for some constant d ∈ R, It is important to remark at once that any df F in the domain of attraction is associated with a couple of functions (p, b) linked to G(x) = F (e x ) used in each appropriate representation. Our conditions will rely on them. In order to state them, we set first for 0 < τ ≤ 1/2 From there, our conditions are the following, where λ is an arbitrary real number greater than one. The first is The second is The third is We finally point out that all the limits in the sequel are meant as n → +∞ unless the contrary is specified. We are now able to state our results in the following section.

The results
for 0 < τ < 1/2, and where is a centered and normed random variable and is the Riemann function.
This concludes the proof since, by Lemma 1 in Section 5, V n (τ ) converges in probability to a N (0, 1) random variable for τ = 1/2 and to L(τ ) for τ < 1/2. Now let F ∈ D(G γ ), 0 < γ < +∞. We will have similar proofs but we do not explicit the convergences in distribution as we did in the previous case. We have by (2.3) and the usual representations, We have, for large values of k, where g 1,n,0 is defined in (3.2), which tends to zero in probability by (C1) and (3.4). Next where g 2,n,0 defined in (3.2). Then S n (3)/σ n (τ ) → 0 by (C2), Lemma 1 and the methods described above. Finally, always by Lemma 1, we get (1) and this converges in distribution to N (0, γ 2 ) random variable for τ = 1/2 and to γL(τ ) for τ < 1/2. The proof is completed with the remark that a n (τ ) → τ −1 .

4.2.
A class of non asymptotically Gaussian estimators. For 0 < τ < 1/2, we also have under (C1) and (C2), Here again, we get a family of estimators of γ. However, the limiting law is not Gaussian. This seems to be new. Remark here that we cannot simplify (4.3) to like for τ = 1/2 because of σ n (τ ) being finite and because of (S3) below. This results make a connexion of Extreme Value Theory and Number Theory since they deeply depend of the Riemann function closely related to the prime numbers since, for s > 1, where the product extends to the prime numbers p ≥ 2 and [x] denotes the integer part of x. The limiting law L(τ ) is characterized by its characteristic function (it) n n ζ(n(1 − τ ))ζ(2(1 − τ )) −n/2 ), calculated and justified in (5.1). This random variable has all its moments finite and is related to the Riemann function. Indeed put ψ ∞ (t) = exp(φ(t)) and let φ (r) denote the r th derivative function of φ.
We easily see that and thus EX = 0, EX 2 = 1.
We get here the interesting thing that (C2) always holds for any df F ∈ D(G γ ), 0 < γ < +∞. And this is a quite general and rare result in EVT. For γ = 0, the condition (C3) is also very weak since g 1,n and g 2,n are only requested to tend to zero faster that log k. We get then very general asymptotic laws that sound as a compensation of the lack of normality! Since these limits seem new in the extreme value index estimation, it may be reasonable that we give some comments on how to use them. Although we cannot give a simple expression of the df of L(τ ), F L(τ ) (u) = P(L(τ ) ≤ u), we may use computer-based methods to compute approximations of its values. We may fix a great value of k and then consider a large sample, of size B, of the random variable where ζ(s, m) = m j=1 j −s , with a sufficiently large value of m. We surely get the empirical distribution function (edf ) of this sample as a fair uniform approximation of F L(τ ) . Fix m = 10000, B = 10000, we may observe in the right graph in Figure 3, that the df 's of F L(τ ) for τ ∈ {0.1, 0.2, 0.25, 0.3, 0.25, 0.3, 0.35, 0.40, 0.48, 0.48} are very close one another. When we consider a τ uniformly drawn from ]0, 1/2[, we obtain a significantly different df as illustrated in the left graph in Figure  3, where we simply add this last df to those already in the first graph.
From there, we are able to compute the probabilities and the quantiles of such limiting laws, as illustrated in the Table 1 in the Appendix. Statistical tests may then be based on these results.

4.3.
Simulations. We wish to show the performance of these class of statistics as estimators of γ and compare them to some other ones available in the literature, such as Hill's one T n (1) (see [8]), the Pickands estimator (see [11]) P n (k) = (log 2) −1 log {(X n−k,n − X n−2k,n )/(X n−2k,n − X n−4k,n )} , and the Lo estimator (see [9]) L n (k) with where g(i, j) = 1/2 for i = j and g(i, j) = 1 otherwise. We may see that the performance of these estimators largely depend on the nature of the tail 1 − F. In the simple case where 1 − F (x) = x −1/γ , all these estimators perform pretty well for n = 300, k ∼ n 0.75 for γ = 0.5 and T n (1/2)/a n (1/2) has particularly good results. They also present good performance for the model The results are less sound for small η (η = 10 for example) for sample sizes around 300. We check the performances in Table 2 in the Appendix, where we report the estimated values for γ for the sample case and the MSE, and in Figure 1 all in the Appendix. The asymptotic normality of (4.1) is illustrated in the graphs of Figure 2 as good.
The simulation results for the asymptotic normality of (4.1) are also acceptable since the empirical p-value is high, of order 90%.

Lemmas and Tools
We begin by this simple lemma where we suppose that we are given a sequence of independent and uniformly distributed random variables U 1 , U 2 , ... as in (2.1).
Proof. By using the Malmquist representation (2.2), we have It follows that E(V n (k)) = a n (k) and Var(V n (k)) = σ 2 n (k). Put V * n (τ ) = σ n (k) −1 (V n (k) − a n (k)). Then For 0 < τ < 1/2, we have by (S1) below, k τ σ n (τ ) → A(2(1 − τ )) −1/2 and Now, we have to prove that L(τ ) is a well-defined random variable with all finite moments. The characteristic function of V * n (τ ) is By using the development of log(1 − ·) and by the Lebesgues Theorem, one readily proves that Recall as in [14], p.506, that for s > 1, where [x] denotes the integer part of x. This leads to By using this, we see that the absolute value of the general term of the series in (5.1) is dominated, for large values of n, as follows This shows that ψ ∞ (t) is well defined and characterizes the df of L(τ ).

Conclusion
The family {T n (τ ), 0 < τ ≤ 1/2} is only studied for F belonging to D(G γ ), γ ≥ 0. This includes the negative case under the appropriate transform. However the remaining case, that is the Weibull domain, presents a radically different approach including sums of dependent random variables and deserves a seperated study. Furthermore, we still have to develop a Bayesian approach by considering a random choice of the parameter τ leading to an optimal choice among a large class of admissible laws. the paper.