Improving the bias of a pseudo-maximum likelihood estimate of the extreme value index by k-records

The paper focusses on the estimation of the extreme value index in terms of k-records based on a maximum likelihood approach, which is suggested recently by Louzaoui and El Arrouchi (J Probab Stat, 2020). Its asymptotic normality is well investigated in order to propose a bias correction while ensuring that the new estimator becomes asymptotically unbiased and still normal. Some numerical studies are also provided in order to show how the proposed estimators behave in practice.


Introduction
For an n-sample X 1 , X 2 , … , X n from a continuous distribution function F, let X 1,n ≤ ⋯ ≤ X n,n be the corresponding order statistics. Recall that k-record process is defined in terms of the kth largest observations, see Dziubdziela and Kopocinski [10]. For any integer k, let and then, the k-record values are defined by R (k) Next, suppose that F belongs to the max-domain of attraction of an extreme value distribution G ( F ∈ D(G ) ) where ∈ ℝ is the extreme value index. That is, there exist sequences a n > 0 and b n ∈ ℝ such that (k) where 1 + x > 0 . Let U(y) = inf{z ∶ 1 − F(z) ≤ 1∕y} , for y ≥ 1 . The first order condition (1), in term of U, is equivalent, for all x > 0 , to where a(.) is a some positive auxiliary function. It can be proved that (1) or (2) is equivalent to where 1 + x > 0 and (.) > 0 is a function with t * = sup{y ∶ F(y) < 1} ≤ ∞ is the right endpoint of F. The function D is known as the Generalized Pareto Distribution. See [6] for more theoretical discussion on the max-domain of attraction. The problem of estimating the extreme value index on the basis of the largest observations of the sample (X 1 , … , X n ) has received very special attention in the classical extreme value theory. Many statistics based on higher order statistics have been proposed to estimate such as Hill's estimator [14], Pickands's estimator [16], Moment estimator derived by Dekkers et al. [8], Maximum likelihood (ML) estimator suggested by Drees et al. [9]. For further informations, [6] and [3] gives a good introduction which are rich in application, but gives even more theoretical and practical details on the estimation problem of the extreme value index. In the other hand, the example of the Resnick's duality theorem [2,Theorem 2.3.3] or the caracterization of tail distributions [11] show that the extreme value theory is very linked to the theory of record values. A recent development in record theory can be found in [13] and [1]. Observing k-records only prevents the possibility of applying conventional estimators in extreme value statistics and therefore, the construction of estimators based on record values is essential [12]. This problem has not been sufficiently studied in the literature which has been revisited recently by Louzaoui and El Arrouchi [15] by using a maximum likelihood (ML) approach based on the top k + 1 highest k-records. More precisely, for k = k m an intermediate sequence of integers satisfying k m → ∞ , k m ∕m → 0 as m → ∞ with m is the number of k-records observed, they are showed that the conditional joint distribution of k from independent and identically distributed random variables . This result can be used to construct a pseudo maximum likelihood estimation (̂,̂) of the unknown parameters ( , ) ; that is, based on the sample of , we can maximize the likelihood function , Consequently, ̂∶≡̂m(k) and ̂∶≡̂m(k) are obtained by solving the likelihood equations f m (t) ∶= log 1 + tY 1 . Any solution (̂,̂) of (4) satisfies h m (̂∕̂) = 0 . Conversely, (̂,̂) = (f m (t * ), f m (t * )∕t * ) is solution of (4) for any non-zero solution t * of h m (t) = 0 . It can be easily seen that h m (t) = 0 has a zero solution which must be dropped even if really = 0.
Under the first order condition (2), Louzaoui and El Arrouchi [15] have shown the existence of a random N such that the likelihood equations have a consistent solution (̂m,̂m) for all m ≥ N . Here, we study their asymptotic normality under the so-called second order condition and derive another estimation of the extreme value index which is asymptotically unbiased and normal. The remainder of this paper is organized as follows. In Sect. 2, we establish the asymptotic normality of the ML estimators for ≠ 0 and then we propose a bias correction. Section 3 will devoted to some numerical studies which lend further support to our theoretical results with discussion. Finally, in Sect. 4, a real data set is analyzed by using the suggested methods.

Main results
The study of the asymptotic normality of ML estimators requires a second order condition which is a refinement of (2), see [6]. For some positive, there exists a auxiliary function A(t) (with constant sign and A(t) → 0 as t → ∞ ) and a real index ≤ 0 , such that, ∀x > 0, The parameter governs the rate of convergence in (2). It can be shown that necessarily |A| ∈ RV . The parameter is of primordial importance in the adaptive choice of the threshold to be considered in the estimation of the extreme value index [6,12]. For < 0 this condition becomes, ∀x > 0 We now state our main result, stating asymptotic normality of ML estimators. (4) be an independent and identically distributed sequence of standard exponential random variable and ) . It can be seen easily that, for x ≥ 1 , U(x) = H ← (log(x)) and H ← is a strictly increasing function, since F is continuous. From this and the Relation (4.7) in [17], we get the following representation Without loss of generality, we can assume that For the case > 0 , Louzaoui and El Arrouchi [15] have given bounds for the solution t * of h m (t) = 0 . More Precisely, under the first order condition (2) and From (5) and by Theorem B.2.18 in [6], there exists, for each > 0 , a t 0 = t 0 ( ) such that for x ≥ 1 and t > t 0 , and observe that as m → ∞ , t → ∞ , x → e and x −1 ± x + → e ( ) ± e + almost surely, see Lemma 1 in [15]. We get, for each > 0 , almost surely and so ≤ e ( ) almost surely. Thus, as m → ∞ , almost surely Hence, as m → ∞ Notice that the central limit theorem implies, as m → ∞ where N 1 is a random variable having a standard normal distribution.
On the other hand, by (m log log m) 1∕2 ∕k → 0 and using the law of the iterated logarithm, we have as m → ∞ .
and by the fact that A ∈ RV , we get as m → ∞ Choosing m such that √ k m → 0 and combining (7), (8), (10) with � is asymptotically normal with mean ( ) and variance � is asymptotically normal with mean ( ) and variance 2 . Since f m is an increasing function, we have for sufficiently large m which gives the result (i).
To prove the asymptotic normality of ̂m , we use the following expansion First consider T 1 . For sufficiently large m, we have almost surely Hence, as m → ∞ Next consider T 2 . We have by Theorem 2.3.3 in [6] and consequently, for each > 0 , there exists a t 0 = t 0 ( ) such that for t > t 0 , x ≥ 1, Take t = e (m−k)∕k and x = e (S m−k −(m−k))∕k and observe again that as m → ∞ , t → ∞ , x → 1 and x x −1 ± x + + → ± almost surely. We get and so, by the central limit theorem, as m → ∞ Notice that, without loss of generality, we can take N 1 and N 2 are independent random variables. Thirdly, consider T 3 . From (11), we have for sufficiently large m, Next, adapting the Lemma 4.5.4 in [6] to the case where is positive, we get that, if

Combining this with
√ k m → 0 , we have as m → ∞ Thus, as m → ∞ Finally, the combination of the three parts proves (ii). The proof for < 0 is the same as before with slight modifications. It proved by Louzaoui and El Arrouchi [15] that if k → ∞, k∕m → 0 and k∕ log m → ∞ as m → ∞ , the first order condition (2) ensures the existence of a solution t * of h m (t) = 0 such that, almost surely, T The rest is similar except that the relation (12) becomes for < 0 as .
provided the second order condition. Finally, the statement (2) follows directly from (7) and (10). In order to obtain an unbiased estimator for , it can be seen from the asymptotic expansion (7) that is necessary to eliminate the term A(e m∕k ) and to replace by any consistent estimator. Define, for integers n ≥ k ≥ 1, and where N (k) (n) denote the number of k-record values in the sequence X 1 , … , X n and [x] is the largest integer less than or equal to x. Then we have the following theorem.

Theorem 2.3 Assume (5) holds for
< 0 . Assume k = k n → ∞ , k∕n → 0 , log(n∕k) log log n = o(k) and k∕ log n → ∞ as n → ∞.  Furthermore, since log(k log(n∕k)) < log log n , we have as n → ∞ , almost surely Combining this with the fact A ∈ RV , we get from (13) and so, for 0 < s ≤ 1 which, by the Donsker's invariance principle, gives for 0 < s ≤ 1 and n → ∞, This gives the first part of (i) and (ii). Next, observe that if k ∈ {n∕ log n, n∕(2 log n), n∕(4 log n)} , then all conditions on the sequence k are fulfilled with lim n→∞ √ k�A(n∕k)� = ∞ . Since, as n → ∞ , A(2 log n) ∼ 2 A(log n) and A(4 log n) ∼ 4 A(log n) , we have from the statement (ii) 1. In the same way, we can obtain a similar results to those of Theorem 2.3 and Corollary 2.5 in the case where < 0. 2. The method used here cannot work for = 0 because the bounds found in [15] are almost surely constant, and therefore they are not asymptotically normal. 3. This method is not applicable on bounds proposed by Zhou [18] since they are not symmetrical.

Simulation results
We now present some numerical results for the proposed bias correction. We consider here the Generalized Pareto distribution with F(x) = 1 − (1 + x) −1∕ , for x ≥ 0 and , > 0 , the Burr IV distribution with F(x) = {( ∕x − 1) 1∕ + 1} − , 0 < x < , > 0 and the standard Cauchy distribution with F(x) = 1 2 + 1 arctan(x) , ̂n = (log 2) −1 log (A(log n)) −1 � n (n∕(2 log n)) − − (� n (n∕(4 log n)) − ) for x ∈ ℝ . For each of these distributions, we generate a random sample of size n. Moreover, for each of these random samples, the record values are picked up and then the corresponding estimates are computed. We report the simulation results in Tables 1, 2, 3 and 4. ̃ and ̄ are the averages of 10,000 estimates of ̃ and ̄ with MSE(̃) and MSE(̄) denoted respectively their mean square errors. The simulated values are calculated for three sizes n against k with a reasonable number of the record N (k) (n) (by using the approximation N (k) (n) ∼ k log(n∕k) ). We remark that when the mean squared error values were rounded to the fourth decimal place, some values were repeated. We observe that the simulated values of ̃ and ̄ are close to the theoretical value of , and frequently, we have ̄ is closer to theoretical than ̃ . Unfortunately, the balancing of the MSE's did not allow us to

Real data
In this section, we apply our estimation method on rainfall data, collected monthly from 1975 until 2007 at Melk Zhar Station in the Souss Massa region of Morocco (Fig. 1). This estimation are compared with the most used methods: the block maxima (GEV from (1)) and the POT (GPD from (3)). Table 5 shows the estimated parameters of the GEV distribution. The estimated shape parameter is positive, but the 95% confidence interval extends also below zero which  stability of parameter estimates. The linearity of the mean residual life and the stability of the GPD parameters are both reached when t = 20 which the excesses are composed by 89 observations. From Table 6, we have ̂= 2.5 × 10 −8 which is very close to 0. Next, using equations in (4), Table 7 summarizes our estimates for some selected values for k which ensures again the closeness of to 0. Consequently, the Gumbel model (GEV with = 0 ) is a suitable model for our data. This is supported by diagnostic plots in Fig. 2. By adopting the Gumbel model, the associated return level z at return period 1∕ is z = − log(− log(1 − )) , where and are estimated in Table 8. It means that on average, z is exceeded   Table 9. Hence, the return level estimates indicate that the maximum value 144.6 (maximum total monthly rainfall recorded in Melk Zhar, see Fig. 1) will not be exceeded in the next 20 years, but it will be exceeded in the next 50 years.