# On the Maximum Likelihood Estimation of Extreme Value Index Based onk-Record Values

Authors: Abderrahim Louzaoui; Mohamed El Arrouchi
Journal: Journal of Probability and Statistics
 (2020)

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## Abstract

In this paper, we study the existence and consistency of the maximum likelihood estimator of the extreme value index based onk-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.


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## Body

## 1. Introduction

LetX1,X2,…, 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 X1,n≤⋯≤Xn,n the order statistics of the n-sample X1,…,Xn. 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∈DGγ with γ∈ℝ, i.e., there exist sequences an>0 and bn∈ℝ such that(1)limn⟶∞ℙan−1Xn,n−bn≤x=Gγx=exp−1+γx−1/γ,for 1+γx>0. The parameter γ is called the extreme value index. The first-order condition F∈DGγ is equivalent to that there exists an auxiliary function a>0 such that(2)limt⟶∞Utx−Utat=xγ−1γ,for all x>0, where Ut=infx:Fx≥1−1/t,t≥1. For more details on the max-domain of attraction, see De Haan and Ferreira [1] and references therein.The estimation of the extreme value indexγ 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]. The 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(3)limt⟶x∗ℙX1−t≤xσtX1>t=Hγx=1−1+γx−1/γ,for all 1+γx>0, where σt>0 is a positive function and x∗ is the right endpoint of F, i.e., x∗=supx:Fx<1≤∞. Hγ is the so-called generalized Pareto distribution (GPD) function.Based on (3), Smith [6] constructed the maximum likelihood (ML) estimator for γ,σ by solving two estimation equations, Drees et al. [7] derived its asymptotic normality for γ>−1/2 when the threshold is chosen as an upper order statistic, while Zhou [8] studied in detail its existence and consistency when γ>−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 Theorem 2.3.3 in [9]) or the characterization of tail distributions (e.g., [10]). There 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.This paper is organized as follows. In Section2, 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.


## 2. Likelihood Equations Based onk-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.Letk≥1 be an integer. Define the sequences of k-record times νik,i≥1 and k-record values Rik,i≥1 (see [16]) by(4)ν1k=k,νi+1k=minj>νik:Xj−k+1,j>Xνik−k+1,νik,Rik=Xνik−k+1,νik,i≥1.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 Section4.Lemma 1.
For all integers1≤k<n, the conditional distribution of Rn−k+1k,…,Rnk, given Rn−kk=y, is the same as the unconditional distribution of the k-record values S1k,…,Skk arising from i.i.d. random variables Z1,Z2,…, with the left-truncated distribution(5)Fyz=Fz−Fy1−Fy,z>y.Letk=kn be an intermediate sequence of integers satisfying kn⟶∞ and kn/n⟶0 as n⟶∞, and let(6)Y0,Y1,…,Yk≔Rn−kk,Rn−k+1k−Rn−kk,…,Rnk−Rn−kk.From Lemma1, the conditional distribution of Y1,…,Yk, given Y0=y0, equals the unconditional distribution of the k-record values S1k,…,Skk arising from i.i.d. random variables Z1,Z2,…, with distribution Fy0⋅+y0 which, in view of (3), can be approximated by the generalized Pareto distribution Hγ⋅/σ (see [7]). Using this information, one can construct an estimation of the unknown parameters γ and σ by a maximum likelihood estimation; that is, given the k-record values Rn−kk,Rn−k+1k,…,Rnk, we maximize the likelihood function(7)L=kk1−HγY1σk∏i=1khγ,σYi1−HγYi/σ,with Yi=Rn−i+1k−Rn−kk, 1≤i≤k, and hγ,σy=∂Hγy/σ/∂y.Remark 1.
Observe that ifγ<−k, L⟶∞ when γ/σ↓−1/Y1, and so, the maximum of L does not exist. However, this case will be disregarded since k has been taken as a sequence kn tending to infinity.
The likelihood equations are then given in terms of the partial derivatives:(8)∂logL∂γ=−∑i=1kYi/σ1+γ/σYi+kγ2log1+γσY1−kγY1/σ1+γ/σY1=0,∂logL∂σ=−kσ+∑i=1kγ/σ2Yi1+γ/σYi+kσγ/σ2Y11+γ/σY1=0.
The maximum likelihood estimators for the extreme value index and the scale,γ^ and σ^, are obtained by solving the following likelihood equations:(9)1γlog1+γσY1=1,1k∑i=1k11+γ/σYi=1γγ/σY11+γ/σY1.
The equations forγ=0 are defined by continuity. If γ≠0, they can be simplified to(10)log1+γσY1=γ,1+1γ/σY1⋅1k∑i=1k11+γ/σYi=1γ.
It follows that(11)1+1γ/σY1log1+γσY1⋅1k∑i=1k11+γ/σYi=1.
Put(12)fnt≔log1+tY1,qnt≔1+1tY1fnt,gnt≔1k∑i=1k11+tYi,hnt=qnt⋅gnt−1.
In view of (11), any root γ^,σ^ of (10) satisfies hnγ^/σ^=0. Conversely, if t∗ is a nonzero root of hnt=0, we obtain γ^,σ^=fnt∗,fnt∗/t∗ as the solution of (10). We can readily check that hnt=0 has a trivial root t∗=0 which must be omitted even if in reality, γ=0.


## 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 γ≠0, and assume that, as n⟶∞,(13)k=kn⟶∞,kn⟶0,klogn⟶∞.
Then, there exists a sequence of estimatorsγ^n,σ^n and a random integer N>1 such that(14)ℙγ^n,σ^nis a ML estimator for alln≥N=1,and as n⟶∞,(15)γ^n⟶a.s.γ.
Moreover, if additionally,nloglogn1/2/k⟶0 as n⟶∞, then(16)σ^naen/k−1⟶a.s.σ.as n⟶∞, where at is the auxiliary function in (2).Theorem 2.
Suppose (1) holds for γ=0. Assume that, as n⟶∞,(17)k=kn⟶∞,kn⟶0,klogn⟶∞,and with probability 1, the following relation does not hold for sufficiently large n:(18)Rnk+Rn−kk=2k∑i=1kRn−i+1k.
Then, there exists a sequence of estimatorsγ^n,σ^n and a random integer N>1 such that(19)ℙγ^n,σ^nis a ML estimator for alln≥N=1,and as n⟶∞,(20)γ^n⟶a.s.0.
Moreover, if additionally,nloglogn1/2/k⟶0 as n⟶∞, then(21)σ^naen/k−1⟶a.s.σ,n⟶∞.Remark 2.
Extra condition (18) ensures the existence of a nonzero solution of the likelihood equations for γ=0. Hence, the solution of the likelihood equations for γ=0 will almost surely not be equal to 0 if, for example, F possesses a density.


## 4. Proofs

We first recall the following representation of thek-record values. Let en,n≥1 be an i.i.d sequence of standard exponential random variables, and denote by Sn=e1+⋯+en,n≥1, their partial sums. Let Hx=−log1−Fx be the hazard function of F. It is easy to see that Ux=H⟵logx 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]):(22)Rnk,n≥1=dH⟵Snk,n≥1=UeSn/k,n≥1.So, from now on, we shall assume, without loss of generality, thatRn−ik=UeSn−i/k, for n≥1 and 0≤i≤k.Before proving the above theorems, we need the following lemmas.Lemma 2.
For a sequencek=kn⟶∞, k/n⟶0, and k/logn⟶∞, we have as n⟶∞, Sn−sk−Sn−k/k⟶a.s.1−s uniformly on s∈0,1, where u is the largest integer not exceeding u.Proof.
First, we write, for0≤s≤1,(23)Sn−sk−Sn−k=Sn−sk−n−sk−Sn−k−n−k+1−sk=Sn−sk′−Sn−k′+1−sk,where Si′=Si−i. It follows that(24)sup0≤s≤1Sn−sk−Sn−kk−1−s≤sup0≤s≤1Sn−sk′−Sn−k′k+sup0≤s≤11−skk−1−s.
Since, for alls∈0,1, 1−sk−1<1−sk≤1−sk, we have(25)sup0≤s≤11−skk−1−s=o1,n⟶∞.
By using the Komlós–Major–Tusnády approximation [18, 19], we can define Wiener processes Wt,t≥0 such that(26)max1≤i≤NSi′−Wi=a.s.OlogN,N⟶∞.
Next, observe that(27)sup0≤s≤1Snsk′−Sn−k′≤Sn−k′−Wn−k+sup0≤s≤1Sn−sk′−Wn−sk+sup0≤s≤1Wn−sk−Wn−k≤Sn−k′−Wn−k+max0≤j≤kSn−j′−Wn−j+max0≤s≤kWn−k+s−Wn−k.
Note that, for the first terms,(28)Sn−k′−Wn−k=a.s.Ologn−k,n⟶∞,(29)max0≤j≤kSn−j′−Wn−j=a.s.Ologn,n⟶∞.
For the last term, we use Theorem 3.2B in Hanson and Russo [20]. It implies that(30)max0≤s≤kWn−k+s−Wn−k=a.s.Oklognk+loglogk1/2,n⟶∞.
Combining this with (25), (28), (29), and the above conditions on k, we get(31)sup0≤s≤1Sn−sk′−Sn−k′k=a.s.o1,n⟶∞,which completes the proof of lemma.Lemma 3.
Suppose (1) holds for γ>0 and k⟶∞,k/n⟶0,k/logn⟶∞ as n⟶∞. Let Tnδ=1+δ/Rn−kk. Then, for any 0<δ<1, we have as n⟶+∞,(32)fnTnδ⟶a.sfδ≔log1+δeγ−δ,qnTnδ⟶a.sqδ≔1+11+δeγ−1log1+δeγ−δ,gnTnδ⟶a.sgδ≔∫01ds1+δe1−sγ−δ,hnTnδ⟶a.shδ≔qδgδ−1,fnTn−δ⟶a.sf−δ,qnTn−δ⟶a.sq−δ,gnTn−δ⟶a.sg−δ,hnTnδ⟶a.s.h−δ.
In addition, ifδ is close to 0 and n is large enough, we have(33)hnTnδ>0a.s.,hnTn−δ<0a.s..Proof.
WritefnTnδ=log1+1+δUeSn/k/UeSn−k/k−1, and note that when (1) is satisfied for γ>0, it is well known that U is regularly varying at infinity with index γ so that limt⟶∞Utx/Ut=xγ locally uniformly in x>0. Next, from Lemma 2, Sn−Sn−k/k⟶a.s.1 as n⟶+∞; it follows readily that fnTnδ⟶a.s.fδ as n⟶+∞, and so, qnTnδ⟶a.s.qδ, fnTn−δ⟶a.s.f−δ, and qnTn−δ⟶a.s.q−δ as n⟶+∞.
Similarly, we observe that(34)gnTnδ=∫0111+δUeSn−sk/k/UeSn−k/k−δds.
From Lemma2, UeSn−sk/k/UeSn−k/k⟶a.s.e1−sγ for all s∈0,1, and by the fact that U is an increasing function, we have, for all 0<δ<1 and s∈0,1,(35)11+δUeSn−sk/k/UeSn−k/k−δ≤1a.s..
Hence, the dominated convergence theorem ensures that(36)gnTnδ⟶a.s.gδ,n⟶∞.
By the same arguments, we havegnTn−δ⟶a.s.g−δ as n⟶∞.
Next, again by using the dominated convergence theorem and after straightforward calculations, we obtain(37)q0+=γ1−e−γ,q′0+=eγ−γ−1eγ−1,g0+=1−e−γγ,g′0+=−1−e−γ22γ,h0+=0,h′0+=2−γ−2+γe−γ2γ.
PutPγ=2−γ−2+γe−γ. Since, for all γ>0, 1+γ<eγ, then P′γ=1+γ−eγ/eγ<0. Hence, for all γ>0, Pγ<0. Thus, h′0+<0. Consequently, there exists δ0>0, for any 0<δ<δ0; when n is large enough,(38)hnTnδ<h0+=0a.s..
The same arguments show thathnTn−δ>0a.s..Remark 3.
This lemma can be proved forδ=0. Indeed, Potter’s inequality (see Proposition 0.8.(ii) in [17]) implies that, for any ϵ>0, there exists t0 such that, for t≥t0 and x≥1, 1−ϵxγ−ϵ<Utx/Ut<1+ϵxγ+ϵ. Then, for all ϵ>0 small enough,(39)gnTn0−1+ϵ1−e−ϵ−γϵ+γ=∫01UeSn−k/kUeSn−sk/k−1+ϵe−ϵ+γ1−sds≤1+ϵϵ+γsup0≤s≤1Sn−sk−Sn−kk−1−s.
By Lemma2, limsupn⟶∞gnTn0≤1+ε1−e−ε−γ/ε+γa.s., which leads, when ϵ⟶0, to limsupn⟶∞gnTn0≤1−e−γ/γa.s.. Similarly, liminfn⟶∞gnTn0≥1−e−γ/γa.s..Lemma 4.
Suppose (1) holds for γ<0 and k⟶∞,k/n⟶0,k/logn⟶∞ as n⟶∞. Let Tnδ=−1+δ/U∞−Rn−kk. Then, for any −1<δ<0, we have as n⟶+∞,(40)fnTnδ⟶a.sfδ≔log1+δeγ−δ,qnTnδ⟶a.sqδ≔1+11+δeγ−1log1+δeγ−δ,gnTnδ⟶a.sgδ≔∫01ds1+δe1−sγ−δ,hnTnδ⟶a.shδ≔qδgδ−1,qnTn−δ⟶a.sq−δ,gnTn−δ⟶a.sg−δ,hnTnδ⟶a.s.h−δ.
In addition, ifδ is close to 0 and n is large enough, we have(41)hnTnδ<0a.s.,hnTn−δ>0a.s..Proof.
This proof is similar to the previous proof with straightforward modifications. When (1) is satisfied for γ<0, it is well known that U∞<∞ and Vx≔U∞−Ux is regularly varying at infinity with index γ. Write(42)qnTnδ=1−11+δ1−VeSn/k/VeSn−k/klog1−1+δ1−VeSn/kVeSn−k/k.
Again from Lemma2, Sn−Sn−k/k⟶a.s.1; it follows readily that qnTnδ⟶a.s.qδ, and so, qnTn−δ⟶a.s.q−δ.
Similarly, we write(43)gnTnδ=∫0111+δ1−VeSn−sk/k/VeSn−k/k−δds.
Since, from Lemma2, VeSn−sk/k/VeSn−k/k⟶a.s.e1−sγ for all s∈0,1 and observe that, for all −1<δ<0 and s∈a0,1,(44)11+δ1−VeSn−sk/k/VeSn−k/k−δ≤−1δa.s.,it implies by the dominated convergence theorem that(45)gnTnδ⟶a.sgδ,n⟶∞.
By the same arguments, we havegnTn−δ⟶a.s.g−δ as n⟶∞.
Next, again by using the dominated convergence theorem and after straightforward calculations, we have(46)q0−=γ1−e−γ,q′0−=eγ−γ−1eγ−1,g0−=1−e−γγ,g′0−=−1−e−γ22γ,h0−=0,h′0−=2−γ−2+γe−γ2γ.
PutPγ=2−γ−2+γe−γ. Since, for all γ<0, 1+γ>eγ, then P′γ=1+γ−eγ/eγ>0. Hence, for all γ<0, Pγ<0. Thus, h′0−>0. Consequently, there exists δ0<0, for any δ0<δ<0; when n is large enough,(47)hnTnδ<h0−=0a.s..
The same arguments show thathnTn−δ>0a.s..Lemma 5.
Suppose (1) holds for γ=0 and k⟶∞,k/n⟶0,k/logn⟶∞ as n⟶∞. Let Tnδ=δ/Rnk−Rn−kk. Then, for any 0<δ<1/2 and as n⟶+∞,(48)gnTnδ⟶a.sgδ≔log1+δδ,gnTn−δ⟶a.sg−δ≔−log1−δδ,hnTnδ⟶a.shδ≔1+δlog1+δδ2−1,hnTn−δ⟶a.sh−δ≔1−δlog1−δδ2−1.
Furthermore, for sufficiently largen, we have(49)hnTnδ<0a.s.,hnTn−δ<0a.s..Proof.
For arbitrary0<δ<1/2, let(50)Tnδ=δRnk−Rn−kk,Tn−δ=−δRnk−Rn−kk.
First, we have(51)qnTnδ=1+δlog1+δδ,qnTn−δ=−1−δlog1−δδ,gnTnδ=∫01ds1+δRn−skk−Rn−kk/Rnk−Rn−kk,gnTnδ=∫01ds1−δRn−skk−Rn−kk/Rnk−Rn−kk.
Suppose now (2) holds for γ=0, i.e., ∀x>0, and(52)limt⟶∞Utx−Utat=logx.
SinceU is monotone, this limit holds locally uniformly in x>0.
Next, observe that(53)Rn−skk−Rn−kkRnk−Rn−kk=UeSn−sk/k−UeSn−k/k/aeSn−k/kUeSn/k−UeSn−k/k/aeSn−k/k.
By using Lemma2, it follows readily that, for all s∈0,1,(54)Rn−shk−Rn−kkRnk−Rn−kk⟶a.s.1−s,n⟶∞.
Since, for any0<δ<1/2,(55)0<11+δRn−skk−Rn−kk/Rnk−Rn−kk<1a.s.,it follows by using the dominated convergence theorem that, as n⟶∞,(56)gnTnδ⟶a.s.∫01ds1+δ1−s=log1+δδ.
Similarly, for any0<δ<1/2 and all s∈0,1,(57)0<11−δRn−skk−Rn−kk/Rnk−Rn−kk<2,a.s.,and so, as n⟶∞,(58)gnTn−δ⟶a.s.∫01ds1−δ1−s=−log1−δδ.
Hence, asn⟶∞,(59)hnTnδ⟶a.shδ=1+δlog1+δδ2−1,hnTn−δ⟶a.sh−δ=1−δln1−δδ2−1.
Note that, forφx=x/x+1−logx+1, φ′x=x2+2x+2/2x+1x+1>0 for all x>−1, which implies that, for all x>−1 and x≠0, 1+xlog1+x/x2<1. Thus, hδ<0 and h−δ<0.
Consequently, for sufficiently largen, we have almost surely(60)hnTnδ<0,hnTn−δ<0.Proof of Theorem1.
Here, we present the proof only forγ>0. For γ<0, the proof is essentially the same.
By choosing a suitable positive sequenceδn⟶0 as n⟶∞, there exists, from Lemma 3, a random integer N>1 such that, for any n≥N, hnTnδn<0a.s.. and hnTn−δn>0a.s.. This ensures, by the mean value theorem, the existence of a random variable Tn∗∈Tn−δn,Tnδn a.s. such that hnTn∗=0 a.s. when n≥N.
Sincefn is an increasing function, we have almost surely(61)fnTn−δn≤γ^n=fnTn∗≤fnTnδn.
From Lemma3, fnTnδn⟶a.s.f0=γ and fnTn−δn⟶a.s.f0=γ; this implies that γ^n⟶a.s.γ, i.e., γ^n is strongly consistent.
To prove the almost sure convergence ofσ^n, we use the fact that, as t⟶∞, at∼γUt (see Lemma 1.2.9, p. 22 in [1]).
So, asn⟶∞,(62)σ^naeSn−k/k=fnTn∗Tn∗aeSn−k/k∼fnTn∗γTn∗UeSn−k/ka.s..
Since, for sufficiently largen, Tn−δn≤Tn∗≤Tnδn a.s., we have eventually(63)1−δn≤Tn∗UeSn−k/k≤1+δna.s.,which leads to Tn∗UeSn−k/k⟶a.s.1 as n⟶∞. Hence, as n⟶∞,(64)σ^naeSn−k/k∼a.s.1.
By applying the law of the iterated logarithm, we have almost surely(65)Sn−kk−n−kk=Onloglogn1/2k,n⟶∞.
Ifnloglogn1/2/k⟶0 as n⟶∞, then Sn−k/k−n−k/k⟶a.s.0. Combining this with the fact that the function a is regularly varying at infinity with index γ, the consistency of σ^n is proved for the positive case.Proof of Theorem2.
First, we choose a suitable positive sequenceδn⟶0 as n⟶∞. It follows from Lemma 4 that there exists a random integer N>1 such that, for any n≥N, hnTnδn<0a.s.. and hnTn−δn<0a.s.. Since, after straightforward calculations, we have almost surely(66)hn0=0,hn′0=Rnk−Rn−kk2−1k∑i=1kRn−i+1k−Rn−kk≠0.
This ensures that whenn is large enough, hnt changes the sign in the neighborhood of 0. Combining this with the fact that hnTnδ and hnTn−δ have the same sign, it is proved that almost surely, for sufficiently large n, there exists a nonzero root Tn∗ of hnt=0 on Tn−δ,Tnδ.
Recall thatfn is an increasing function. This implies almost surely(67)log1−δn=fnTn−δn≤γ^n=fnTn∗≤fnTnδn=log1−δn.
Sinceδn⟶0 as n⟶∞, γ^n⟶a.s.0, and the consistency is proved.
Now, we prove the almost sure convergence ofσ^n. For this, we write(68)σ^naeSn−k/k=fnTn∗Tn∗aeSn−k/k=log1+Tn∗Y1Tn∗Y1Y1aeSn−k/ka.s..
Since, for sufficiently largen, Tn−δn≤Tn∗≤Tnδn a.s., we have eventually(69)−δn≤Tn∗Y1≤δna.s.,which leads to Tn∗Y1⟶a.s.0 as n⟶∞. Hence, as n⟶∞,(70)log1+Tn∗Y1Tn∗Y1⟶a.s.1.
Under (2), Lemma 2 ensures that(71)Y1aeSn−k/k=UeSn/k−UeSn−k/kaeSn−k/k⟶a.s.1,n⟶∞.
Therefore,(72)σ^naeSn−k/k⟶a.s.1,n⟶∞.
Finally, ifnloglogn1/2/k⟶0 as n⟶∞, we have, by applying the law of the iterated logarithm, Sn−k/k−n−k/k⟶a.s.0. Combining this with the fact that the function at is slowly varying at infinity, i.e., for all x>0, limt⟶∞atx/at=1 (see Lemma 1.2.9, p. 22 in [1]), the consistency of σ^n is then proved for γ=0.Proof of Lemma1.
Recalling the following representation,(73)Rn−kk,Rn−k+1k,…,Rnk=dH⟵Sn−kk,H⟵Sn−k+1k,…,H⟵Snk,where Hx=−log1−Fx is the continuous hazard function of the distribution function F, Sn=e1+⋯+en and e1,e2,…, be independent random variables having the standard exponential distribution.
It follows, without loss of generality, that(74)ℙRn−k+1k>x1,…,Rnk>xk|Rn−kk=y=ℙH⟵Sn−k+1k>x1,…,H⟵Snk>xk|H⟵Sn−kk=y.
We know thatH⟵t>x⟺t>Hx, and by continuity, H⟵t=x⟹Hx=t. Then,(75)ℙRn−k+1k>x1,…,Rnk>xk|Rn−kk=y=ℙen−k+1k>Hx1−Hy,…,en−k+1+⋯+enk>Hxn−Hy|H⟵Sn−kk=y,which gives by independence(76)ℙRn−k+1k>x1,…,Rnk>xk|Rn−kk=y=ℙen−k+1>kHx1−Hy,…,en−k+1+⋯+en>kHxn−Hy=ℙS1k>Hx1−Hy,…,Sk>Hxn−Hy.
Observe thatHz−Hy=−log1−Fz/1−Fy is the hazard function of the distribution function Fy. This, by using the above representation, proves the desired conclusion.



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