On the Maximum Likelihood Estimation of Extreme Value Index Based on -Record Values
In this paper, we study the existence and consistency of the maximum likelihood estimator of the extreme value index based on -record values. Following the method used by Drees et al. (2004) and Zhou (2009), we prove that the likelihood equations, in terms of -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.
Let , be a sequence of independent and identically distributed random variables (i.i.d.) having a continuous distribution function . For each , denote by the order statistics of the -sample . We first recall some basic notions of the univariate extreme value theory. Assume that belongs to the max-domain of attraction of an extreme value distribution, denoted by with , i.e., there exist sequences and such thatfor . The parameter is called the extreme value index. The first-order condition is equivalent to that there exists an auxiliary function such thatfor all , where . For more details on the max-domain of attraction, see De Haan and Ferreira  and references therein.
The estimation of the extreme value index plays an important role in the classical extreme value theory, and many estimators have been proposed in the literature such as the Hill estimator , Pickands estimator , and moment estimator suggested by Dekkers et al. . The books by Beirlant et al.  and De Haan and Ferreira  provide good reviews on this estimation problem.
Alternatively, condition (1) is equivalent tofor all , where is a positive function and is the right endpoint of , i.e., . is the so-called generalized Pareto distribution (GPD) function.
Based on (3), Smith  constructed the maximum likelihood (ML) estimator for by solving two estimation equations, Drees et al.  derived its asymptotic normality for when the threshold is chosen as an upper order statistic, while Zhou  studied in detail its existence and consistency when . 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 Theorem 2.3.3 in ) or the characterization of tail distributions (e.g., ). There are quite few publications which are devoted to the estimation of the extreme value index based on record values, see, for example, Berred , Khaled et al. , and El Arrouchi and Imlahi . 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 -record values.
This paper is organized as follows. In Section 2, we give the likelihood equations based on -record values. Section 3 is devoted to existence and consistency of the solutions of these equations, whose proofs will be given in Section 4.
2. Likelihood Equations Based on -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  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.  and Nevzorov  and the references therein for a review of the general theory of records.
Let be an integer. Define the sequences of -record times and -record values (see ) 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 , the conditional distribution of , given , is the same as the unconditional distribution of the -record values arising from i.i.d. random variables , with the left-truncated distribution
Let be an intermediate sequence of integers satisfying and as , and let
From Lemma 1, the conditional distribution of , given , equals the unconditional distribution of the -record values arising from i.i.d. random variables , with distribution which, in view of (3), can be approximated by the generalized Pareto distribution (see ). Using this information, one can construct an estimation of the unknown parameters and by a maximum likelihood estimation; that is, given the -record values , we maximize the likelihood functionwith , , and .
Remark 1. Observe that if , when , and so, the maximum of does not exist. However, this case will be disregarded since has been taken as a sequence tending to infinity.
The likelihood equations are then given in terms of the partial derivatives:The maximum likelihood estimators for the extreme value index and the scale, and , are obtained by solving the following likelihood equations:The equations for are defined by continuity. If , they can be simplified toIt follows thatPutIn view of (11), any root of (10) satisfies . Conversely, if is a nonzero root of , we obtain as the solution of (10). We can readily check that has a trivial root which must be omitted even if in reality, .
3. 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 , and assume that, as ,Then, there exists a sequence of estimators and a random integer such thatand as ,Moreover, if additionally, as , thenas , where is the auxiliary function in (2).
Theorem 2. Suppose (1) holds for . Assume that, as ,and with probability 1, the following relation does not hold for sufficiently large :Then, there exists a sequence of estimators and a random integer such thatand as ,Moreover, if additionally, as , then
Remark 2. Extra condition (18) ensures the existence of a nonzero solution of the likelihood equations for . Hence, the solution of the likelihood equations for will almost surely not be equal to 0 if, for example, possesses a density.
We first recall the following representation of the -record values. Let be an i.i.d sequence of standard exponential random variables, and denote by , their partial sums. Let be the hazard function of . It is easy to see that for . Since is continuous, the function is strictly increasing, and hence, we have the following representation (see relation (4.7), p. 167 in ):
So, from now on, we shall assume, without loss of generality, that , for and .
Before proving the above theorems, we need the following lemmas.
Lemma 2. For a sequence , , and , we have as , uniformly on , where is the largest integer not exceeding .
Proof. First, we write, for ,where . It follows thatSince, for all , , we haveBy using the Komlós–Major–Tusnády approximation [18, 19], we can define Wiener processes such thatNext, observe thatNote that, for the first terms,For the last term, we use Theorem 3.2B in Hanson and Russo . It implies thatCombining this with (25), (28), (29), and the above conditions on , we getwhich completes the proof of lemma.
Lemma 3. Suppose (1) holds for and as . Let . Then, for any , we have as ,In addition, if is close to 0 and is large enough, we have
Proof. Write , and note that when (1) is satisfied for , it is well known that is regularly varying at infinity with index so that locally uniformly in . Next, from Lemma 2, as ; it follows readily that as , and so, , , and as .
Similarly, we observe thatFrom Lemma 2, for all , and by the fact that is an increasing function, we have, for all and ,Hence, the dominated convergence theorem ensures thatBy the same arguments, we have as .
Next, again by using the dominated convergence theorem and after straightforward calculations, we obtainPut . Since, for all , , then . Hence, for all , . Thus, . Consequently, there exists , for any ; when is large enough,The same arguments show that .
Remark 3. This lemma can be proved for . Indeed, Potter’s inequality (see Proposition 0.8.(ii) in ) implies that, for any , there exists such that, for and , . Then, for all small enough,By Lemma 2, , which leads, when , to . Similarly, .
Lemma 4. Suppose (1) holds for and as . Let . Then, for any , we have as ,In addition, if is close to 0 and is large enough, we have
Proof. This proof is similar to the previous proof with straightforward modifications. When (1) is satisfied for , it is well known that and is regularly varying at infinity with index . WriteAgain from Lemma 2, ; it follows readily that , and so, .
Similarly, we writeSince, from Lemma 2, for all and observe that, for all and ,it implies by the dominated convergence theorem thatBy the same arguments, we have as .
Next, again by using the dominated convergence theorem and after straightforward calculations, we havePut . Since, for all , , then . Hence, for all , . Thus, . Consequently, there exists , for any ; when is large enough,The same arguments show that .
Lemma 5. Suppose (1) holds for and as . Let . Then, for any and as ,Furthermore, for sufficiently large , we have
Proof. For arbitrary , letFirst, we haveSuppose now (2) holds for , i.e., , andSince is monotone, this limit holds locally uniformly in .
Next, observe thatBy using Lemma 2, it follows readily that, for all ,Since, for any ,it follows by using the dominated convergence theorem that, as ,Similarly, for any and all ,and so, as ,Hence, as ,Note that, for , for all , which implies that, for all and , . Thus, and .
Consequently, for sufficiently large , we have almost surely
Proof of Theorem 1. Here, we present the proof only for . For , the proof is essentially the same.
By choosing a suitable positive sequence as , there exists, from Lemma 3, a random integer such that, for any , and . This ensures, by the mean value theorem, the existence of a random variable a.s. such that a.s. when .
Since is an increasing function, we have almost surelyFrom Lemma 3, and ; this implies that , i.e., is strongly consistent.
To prove the almost sure convergence of , we use the fact that, as , (see Lemma 1.2.9, p. 22 in ).
So, as ,Since, for sufficiently large , a.s., we have eventuallywhich leads to as . Hence, as ,By applying the law of the iterated logarithm, we have almost surelyIf as , then . Combining this with the fact that the function is regularly varying at infinity with index , the consistency of is proved for the positive case.
Proof of Theorem 2. First, we choose a suitable positive sequence as . It follows from Lemma 4 that there exists a random integer such that, for any , and .. Since, after straightforward calculations, we have almost surelyThis ensures that when is large enough, changes the sign in the neighborhood of 0. Combining this with the fact that and have the same sign, it is proved that almost surely, for sufficiently large , there exists a nonzero root of on .
Recall that is an increasing function. This implies almost surelySince as , , and the consistency is proved.
Now, we prove the almost sure convergence of . For this, we writeSince, for sufficiently large , a.s., we have eventuallywhich leads to as . Hence, as ,Under (2), Lemma 2 ensures thatTherefore,Finally, if as , we have, by applying the law of the iterated logarithm, . Combining this with the fact that the function is slowly varying at infinity, i.e., for all , (see Lemma 1.2.9, p. 22 in ), the consistency of is then proved for .
Proof of Lemma 1. Recalling the following representation,where is the continuous hazard function of the distribution function , and , be independent random variables having the standard exponential distribution.
It follows, without loss of generality, thatWe know that , and by continuity, . Then,which gives by independence