On the Maximum Likelihood Estimation of Extreme Value Index Based on k-Record Values

In this paper, we study the existence and consistency of the maximum likelihood estimator of the extreme value index based on k-record values. Following the method used by Drees et al. (2004) and Zhou (2009), we prove that the likelihood equations, in terms of k-record values, eventually admit a strongly consistent solution without any restriction on the extreme value index, which is not the case in the aforementioned studies.


Introduction
Let X 1 , X 2 , . . ., be a sequence of independent and identically distributed random variables (i.i.d.) having a continuous distribution function F. For each n ≥ 1, denote by X 1,n ≤ · · · ≤ X n,n the order statistics of the n-sample (X 1 , . . . , X n ). We first recall some basic notions of the univariate extreme value theory. Assume that F belongs to the max-domain of attraction of an extreme value distribution, denoted by F ∈ D(G c ) with c ∈ R, i.e., there exist sequences a n > 0 and b n ∈ R such that for 1 + cx > 0. e parameter c is called the extreme value index. e first-order condition F ∈ D(G c ) is equivalent to that there exists an auxiliary function a > 0 such that for all x > 0, where U(t) � inf x: F(x) ≥ 1 − 1/t { }, t ≥ 1. For more details on the max-domain of attraction, see De Haan and Ferreira [1] and references therein. e estimation of the extreme value index c plays an important role in the classical extreme value theory, and many estimators have been proposed in the literature such as the Hill estimator [2], Pickands estimator [3], and moment estimator suggested by Dekkers et al. [4]. e books by Beirlant et al. [5] and De Haan and Ferreira [1] provide good reviews on this estimation problem.
Alternatively, condition (1) is equivalent to for all 1 + cx > 0, where σ(t) > 0 is a positive function and x * is the right endpoint of F, i.e., x * � sup x: Based on (3), Smith [6] constructed the maximum likelihood (ML) estimator for (c, σ) by solving two estimation equations, Drees et al. [7] derived its asymptotic normality for c > − 1/2 when the threshold is chosen as an upper order statistic, while Zhou [8] studied in detail its existence and consistency when c > − 1. On the contrary, the theory of record values is connected very closely to the extreme value theory through, like, for example, Resnick's duality theorem (see eorem 2.3.3 in [9]) or the characterization of tail distributions (e.g., [10]). ere are quite few publications which are devoted to the estimation of the extreme value index based on record values, see, for example, Berred [11], Khaled et al. [12], and El Arrouchi and Imlahi [13]. We intend to investigate this problem in this paper, so we are interested here to propose an alternative of the above ML estimation based on the k-record values.
is paper is organized as follows. In Section 2, we give the likelihood equations based on k-record values. Section 3 is devoted to existence and consistency of the solutions of these equations, whose proofs will be given in Section 4.

Likelihood Equations Based on k-Record Values
Record values are of importance in many situations of real life as well as in many statistical applications involving data relating to natural phenomena, sports, economics, reliability, and life tests. Chandler [14] was the first to introduce the concept of record values, record times, and inter-record times in order to analyze weather data. We refer to Arnold et al. [9] and Nevzorov [15] and the references therein for a review of the general theory of records. Let k ≥ 1 be an integer. Define the sequences of k-record times ] (k) i , i ≥ 1 and k-record values R (k) i , i ≥ 1 (see [16]) by Similar to the conditional approach used for order statistics, our equations may be found by using the following lemma which will be proved at the end of Section 4.

Lemma 1.
For all integers 1 ≤ k < n, the conditional distribution of (R (k) n− k+1 , . . . , R (k) n ), given R (k) n− k � y, is the same as the unconditional distribution of the k-record values (S (k) 1 , . . . , S (k) k ) arising from i.i.d. random variables Z 1 , Z 2 , . . ., with the left-truncated distribution Let k � k n be an intermediate sequence of integers satisfying k n ⟶ ∞ and k n /n ⟶ 0 as n ⟶ ∞, and let From Lemma 1, the conditional distribution of (Y 1 , . . . , Y k ), given Y 0 � y 0 , equals the unconditional distribution of the k-record values (S (k) 1 , . . . , S (k) k ) arising from i.i.d. random variables Z 1 , Z 2 , . . ., with distribution F y 0 (· + y 0 ) which, in view of (3), can be approximated by the generalized Pareto distribution H c (·/σ) (see [7]). Using this information, one can construct an estimation of the unknown parameters c and σ by a maximum likelihood estimation; that is, given the k-record values with and so, the maximum of L does not exist. However, this case will be disregarded since k has been taken as a sequence k n tending to infinity. e likelihood equations are then given in terms of the partial derivatives: e maximum likelihood estimators for the extreme value index and the scale, c and σ, are obtained by solving the following likelihood equations: e equations for c � 0 are defined by continuity. If c ≠ 0, they can be simplified to It follows that Put In view of (11), any root (c, σ) of (10) satisfies as the solution of (10). We can readily check that h n (t) � 0 has a trivial root t * � 0 which must be omitted even if in reality, c � 0.

Existence and Consistency
Our main results are the following theorems, stating the existence and the consistency of ML estimators. Theorem 1. Suppose (1) holds for c ≠ 0, and assume that, as n ⟶ ∞, en, there exists a sequence of estimators (c n , σ n ) and a random integer N > 1 such that and as n ⟶ ∞, Moreover, if additionally, (n log logn) 1/2 /k ⟶ 0 as n ⟶ ∞, then σ n a e n/k− 1 ⟶ a.s σ. (16) as n ⟶ ∞, where a(t) is the auxiliary function in (2).
and with probability 1, the following relation does not hold for sufficiently large n: en, there exists a sequence of estimators (c n , σ n ) and a random integer N > 1 such that P c n , σ n is a ML estimator for all n ≥ N � 1, and as n ⟶ ∞, Moreover, if additionally, (n log log n) 1/2 /k ⟶ 0 as n ⟶ ∞, then σ n a e n/k− 1 ⟶ a.s σ, n ⟶ ∞. (21) Remark 2. Extra condition (18) ensures the existence of a nonzero solution of the likelihood equations for c � 0. Hence, the solution of the likelihood equations for c � 0 will almost surely not be equal to 0 if, for example, F possesses a density.

Proofs
We first recall the following representation of the k-record values. Let e n , n ≥ 1 be an i.i.d sequence of standard exponential random variables, and denote by S n � e 1 + · · · + e n , n ≥ 1, their partial sums.
Let F(x)) be the hazard function of F. It is easy to see that U(x) � H ⟵ (log(x)) for x ≥ 1. Since F is continuous, the function H ⟵ is strictly increasing, and hence, we have the following representation (see relation (4.7), p. 167 in [17]): So, from now on, we shall assume, without loss of generality, that R (k) n− i � U(e S n− i /k ), for n ≥ 1 and 0 ≤ i ≤ k. Before proving the above theorems, we need the following lemmas. Proof. First, we write, for 0 ≤ s ≤ 1, Since, for all s By using the Komlós-Major-Tusnády approximation [18,19], we can define Wiener processes

Journal of Probability and Statistics
Next, observe that Note that, for the first terms, For the last term, we use eorem 3.2B in Hanson and Russo [20]. It implies that Combining this with (25), (28), (29), and the above conditions on k, we get which completes the proof of lemma.

Lemma 3. Suppose (1) holds for c > 0 and
en, for any 0 < δ < 1, we have as n ⟶ +∞, In addition, if δ is close to 0 and n is large enough, we have n < 0 a.s..
Next, observe that n− k � U e S n−[sk] /k − U e S n− k /k /a e S n− k /k U e S n /k − U e S n− k /k /a e S n− k /k .
is, by using the above representation, proves the desired conclusion.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.