# Asymptotic Prediction for Future Observations of a Random Sample of Unknown Continuous Distribution

Authors: Magdy El-Adll; H. M. Barakat; Amany Aly
Journal: Complexity
 (2022)

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## Abstract

When the firstr lower extreme order statistics (failure times) of a sample of large size n,1<r<s<n, are observed, asymptotic predictive intervals of the future extreme order statistic with a rank s are constructed. The only assumption that we adopt is that the first failure time is attracted to the Weibull distribution. In addition, we suggest an efficient point estimator of its shape parameter and then a confidence interval is constructed for it. Moreover, new interesting asymptotic properties of the distributions that belong to the minimum domain of attraction of the Weibull distribution are revealed. Furthermore, extensive simulation studies are conducted to demonstrate the efficiency of the proposed methods. Two real datasets are analyzed to illustrate and corroborate the obtained results. The main results concern large samples.


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## Body

## 1. Introduction

Past experiences of the distribution theory of order statistics and their functions (such as quasi-ranges) for finite samples have repeatedly demonstrated that the exact distribution functions (DFs) of such statistics are computationally messy, except in rare cases. Thus, one might be interested in a large sample in the hopes of finding a simple DF that approximates the intractable DF of the statistic that we have. When the laws of extremes govern the random quantities of interest (equipment breakdown, the strength of materials, air pollution, droughts, floods, effects of food additives, etc.), the extreme-value theory (EVT) provides, in some cases, exact probabilistic models but, in most cases, approximate probabilistic ones. A complicated situation can therefore be replaced by a simple asymptotic model if the fundamental conditions of the actual situation are compatible with the model assumptions. Interested readers of the EVT and its applications are referred to Barakat et al., [1], David and Ngaraja [2], Galambos [3], Gnedenko [4], Haan [5], Haan and Ferreira [6], and Resnick [7].Since the papers of Hill [8] and Pickands [9], the estimation problem of the tail index of the asymptotic limits of extremes has dominated several authors. Among them are Cabrala et al. [10], Fedotenkov [11], Haan [12], Haan and Ferreira [6], Smith and Weissman [13], and the references included therein. In many applications of the EVT, inference based on censored observations is of special interest. A predictive interval is commonly used in reliability theory, lifetime problems, and industrial applications to predict the number of defective units that will be produced on a specific production line. Censoring occurs when survival times are known only for a portion of the items under consideration. The complete survival time may not have been observed by the experimenters, either intentionally or unintentionally.Order statistics are useful in lifetime experiments because ifn items are tested, the weakest component will fail first, followed by the second weakest, and so on until all components fail. In industry, for example, we are interested in the time to failure after undergoing a life test. In such cases, the lifetimes are already arranged in ascending order of magnitude. The practical importance of such experiments is evident. Moreover, it is now possible to terminate the experiment before its conclusion, by stopping after a given time (type I censoring) or after a given number of failures (type II censoring). Predicting the time at which future observations will fail or the mean failure time for unobserved lifetimes is an indispensable procedure in several situations. Predictive intervals or point predictors are of particular interest in these situations. In the late 1990s, an excellent review of the development of prediction problems can be found in Kaminsky and Nelson [14]. Many authors have studied point and interval predictions in the statistical literature for both frequentist and Bayesian approaches. Among them are Ahsanullah [15], Aitcheson and Dunsmore [16], AL-Hussaini [17], AL-Hussaini and Ahmad [18], AL-Hussaini and Al-Awadi [19], Aly [20], Barakat et al. [21–24], El-Adll [25, 26], Geisser [27], Guilbaud [28], Kaminsky and Rhodin [29], Lawless [30], Lingappaiah [31], Nagaraja [32, 33], Raqab [34], and Raqab and Barakat [35].Barakat et al. [36] proposed a general finite-sample method for predicting future observations (future order statistics) from any arbitrary continuous distribution. Later, this result was extended to the fractional record-values by Aly et al. [37]. The main barrier to these results being useful in practical applications appears to be that the underlying lifetime distribution should be known. So, while the method is exact, it assumes knowledge of something which cannot be known with certainty. How can interested researchers overcome these obstacles? This is the problem that we will tackle in this paper. Namely, the main goal of this paper is to diminish the required knowledge about the distribution of the underlying lifetimes. The EVT enables us to make the necessary inference by obtaining the asymptotic DF of the proposed pivotal quantities from a DF F in the Weibull minimum domain of attraction. In other words, we can estimate the shape parameter of the Weibull limit and construct predictive intervals of unobserved lower extremes.The practical importance of the Weibull distribution and its extensions in reliability analysis is evidence. Recent contributions in this context include, Ahmad and Ghazal [38], Alkarni et al. [39], Hashem and Alyami [40], El-Adll [25], and Smith and Weissman [13], among others.The rest of this paper is organized as follows. In Section2, some asymptotic distributional properties concerning lower extremes are revealed. As a result, an estimator and asymptotic confidence interval of the shape parameter of the Weibull limit are proposed and an analytical comparison with the Smith and Weissman [13] estimator is performed using Pitman’s measure of closeness. In Section 3, different asymptotic predictive intervals of future unobserved lower extreme order statistics are constructed, whenever the left endpoint and shape parameter are assumed to be known or unknown. Section 4 is devoted to a comprehensive simulation study to assess the suggested estimates and predictive intervals. Finally, in Section 5, two real-lifetime datasets are analyzed. The simulation study which is carried out in Section 4 shows that the proposed estimates are comparable to those estimates obtained by de Haan [12] and Smith and Weissman [13].Throughout this paper, we use the symbols “n→,” “→nw,” “→np,” “→na.s,” and “Xn=nwYn” which stand for convergence, weak convergence, convergence in probability, and almost surely convergence, and the random variables (RVs) Xn and Yn have the same limit DF, as n⟶∞, respectively. Finally, “X=dY” means that the RVs X and Y have the same DF.


## 2. Estimation of the Positive Lower Extreme Value Index

LetX1,X2,…,Xn be a random sample of size n from a continuous DF F, and let X1:n<X2:n<⋯<Xn:n be their corresponding order statistics. We make no further assumptions on F except that it belongs to the minimum domain of attraction of the Weibull distribution with a shape parameter α>0, written F∈DWα, where Wαx=1−exp−xα,x≥0. In the literature, the shape parameter α is known as the lower extreme value index (L-EVI). The L-EVI is the primary parameter of extreme events, and it is mathematically related to the asymptotic behaviour of the left tail of the DF F. By the EVT, the assumption F∈DWα implies that F should have a finite left endpoint (guarantee life) x0, where Fx0+x=0, for x<0, Fx0+x>0, for x>0, and(1)limt↓0Fx0+xtFx0+t=xα.In view of the EVT, the above condition is necessary and sufficient for the existence of a normalizing constantan>0 such that(2)Fr:nanx+x0=PXr:n≤anx+x0=IFanx+x0r,n−r+1⟶nwΓrxα,for any fixed integer r<n, where Ixa,b denotes the beta distribution with shape parameters a,b, written as Beta a,b, i.e.,(3)Ixa,b=1βa,b∫0xta−11−tb−1dt,0≤x≤1,βa,b=ΓaΓbΓa+b,a,b>0,and Γrx is the gamma DF with the shape parameter r, i.e.,(4)Γrx=1Γr∫0xtr−1e−tdt.Among the distributions which are often used in life-testing and satisfy relation (2) are the exponential, Weibull, gamma, exponentiated exponential, and beta. For example, for the standard exponential distribution (denoted by EXP(1)), we have x0=0, an=1/n, and α=1. On the contrary, it is well known that (e.g., see [41])(5)X1:n⟶na.sx0.In this section, we propose an asymptotically consistent estimator for the L-EVIα. It is worth mentioning that the most important life DFs which are used frequently in the life-testing and reliability theory belong to the minimum domain of attraction of the Weibull distribution.We begin with a lemma, which is an essential tool for our studies. On the contrary, this lemma individually expresses an interesting fact about the distributions that belong to the minimum domain of attraction of the Weibull distribution.Lemma 1.
LetF∈DWα with a known left endpoint x0. Furthermore, let k,ℓ,r and s be fixed integers such that 1≤k<ℓ≤r<s<n. Then, Xk:n−x0/Xℓ:n−x0 and Xr:n−x0/Xs:n−x0 are asymptotically independent RVs having Beta k,ℓ−k and Beta r,s−r distributions, respectively.Proof.
First, we consider the caseℓ<r<s<n. Since F∈DWα, there exists normalizing constant an>0 for which (2) is satisfied. An application of Rényi’s representation [42] (see also p. 18 in [2]) and Continuous Mapping Theorem to relation (2) thus yields(6)Xk:n−x0/anXℓ:n−x0/anα=∑i=1kZi∑i=1ℓZi:=Λ1and(7)Xr:n−x0/anXs:n−x0/anα=∑i=1rZi∑i=1sZi:=Λ2,where Z1,Z2,…,Zs are iid RVs having EXP(1). The lemma would be proved if we could prove that the RVs Λ1 and Λ2 are independent. For this purpose, we define(8)Y1=∑j=1kZj,Y2=∑j=k+1ℓZj,Y3=∑j=ℓ+1rZj,Y4=∑j=r+1sZj.
Clearly,Y1,Y2,Y3, and Y4 are independent RVs, which are distributed as gamma DFs with the shape parameters k,ℓ−k,r−ℓ, and s−r, respectively. Consequently, the probability density function (PDF) of the random vector Y¯=Y1,…,Y4 is(9)fY¯y¯=Ck,ℓ,r,sy1k−1y2ℓ−k−1y3r−ℓ−1y4s−r−1e−y1+y2+y3+y4,0<yi<∞,i=1,…,4,where Ck,ℓ,r,s=ΓkΓℓ−kΓr−ℓΓs−r−1. Now, consider the 4-dimensional transformation:(10)Vi=∑j=1iYj∑j=14Yj,i=1,2,3,V4=∑j=14Yj,which is a one-to-one and onto transformation and maps 0,∞4 to A⊂ℝ4, where(11)A=v1,v2,v3,v4∈ℝ4:0<v1<v2<v3<1and0<v4<∞.
The inverse transformation isy1=v1v4,y2=v2−v1v4,y3=v3−v2v4,y4=1−v3v4 and the Jacobian of the transformation is J=v43. Therefore, in view of (9), the joint PDF of the random vector V¯=V1,V2,V3,V4 is(12)fV¯v¯=JfY¯v1v4,v2−v1v4,v3−v2v4,1−v3v4=Ck,ℓ,r,sv1k−1v2−v1ℓ−k−1v3−v2r−ℓ−11−v3s−r−1v4s−1e−v4,0<v1<v2<v3<1,v4>0.
The joint PDF ofV1,V2, and V3 is obtained by integrating (12) over v4 as(13)fV1,V2,V3v1,v2,v3=Ck,ℓ,r,s∗v1k−1v2−v1ℓ−k−1v3−v2r−ℓ−11−v3s−r−1,0<v1<v2<v3<1,where Ck,ℓ,r,s∗=ΓsCk,ℓ,r,s. Similarly, the joint PDF of V1,V=V1/V2, and V3 is given by(14)fV1,V,V3v1,v,v3=Ck,ℓ,r,s∗v1ℓ−1vk−r1−vℓ−k−1vv3−v1r−ℓ−11−v3s−r−1,0<v1<v3<1,0<v<1.
After some algebraic calculations, the joint PDF ofV and V3 can be expressed as(15)fV,V3v,v3=1βℓ−k,kvk−11−vℓ−k−1×1βs−r,rv3r−11−v3s−r−1,0<v<1,0<v3<1.
Therefore, by the factorization theorem,V and V3 are independent RVs distributed as Beta k,ℓ−k and Beta r,s−r, respectively. Since, V=Λ1 and V3=Λ2, the proof of this case is completed. The proof of the case ℓ=r is similar to the case ℓ<r except for some evident modifications. Namely, in the case ℓ=r, we have Λ1:=∑i=1kZi/∑i=1rZi and Λ2:=∑i=1rZi/∑i=1sZi. Moreover, the RVs defined in (8) become Y1=∑j=1kZj, Y2=∑j=k+1rZj, and Y3=∑j=r+1sZj, where Y1,Y2, and Y3 are independent RVs, which are distributed as gamma DFs with the shape parameters k,r−k, and s−r, respectively. In addition, the 4-dimensional transformation defined in (10) becomes the 3-dimensional transformation:(16)Vi=∑j=1iYj∑j=13Yj,i=1,2,V3=∑j=13Yj.
Hence, we can write the joint PDF of the random vectorV¯=V1,V2,V3 as(17)fV¯v¯=v32fY¯v1v3,v2−v1v3,1−v2v3=1ΓkΓr−kΓs−rv1k−1v2−v1r−k−11−v2s−r−1v3s−1e−v3,0<v1<v2<1,v3>0.
By integrating overv3, the joint PDF of V1 and V2 is(18)fV1,V2v1,v2=ΓsΓkΓr−kΓs−rv1k−1v2−v1k−r−11−v2s−r−1,0<v1<v2<1.
Now, setU1=V1/V2 and U2=V2. Thus, after simple algebra, it can be shown that U1 and U2 are independent RVs distributed as Beta k,r−k and Beta r,s−r, respectively. Since, U1=Λ1 and U2=Λ2, the second case is proved. Consequently, the proof of the lemma is completed.Corollary 1.
Under the conditions of Lemma1, the RVs(19)ηi=−αilogXi:n−x0Xi+1:n−x0,i=1,2,…,r−1,r⩾2,are asymptotically iid according to EXP(1). Moreover,(20)Bj=ηj∑i=1r−1ηi,j=1,2,…,r−1,are asymptotically iid RVs having Beta 1,r−2 distribution. Furthermore, the RVs(21)Ui∗:=∑j=1iBj,i=1,2,…,r−2,are asymptotically distributed as an ordered sample of size r−2 from the uniform distribution over (0,1), denoted by U0,1.Proof.
According to Lemma1,(22)X1:n−x0X2:n−x0α,X2:n−x0X3:n−x02α,…,Xr−1:n−x0Xr:n−x0r−1αare asymptotically independent RVs having U0,1 distribution. Consequently, η1,η2,…,ηr−1 are asymptotically independent RVs having the common EXP(1) distribution. Thus, η⋆=η1+η2+⋯+ηr−1 is asymptotically distributed as a gamma distribution with the shape parameter r−1. Therefore, Bj=ηj/η⋆,j=1,2,…,r−1, are asymptotically independent RVs having the common Beta 1,r−2 distribution. It is not difficult to verify that RV Ui∗ has the asymptotic Beta i,r−2−i+1 distribution for i=1,2,…,r−2. This completes the proof of the corollary.


### 2.1. Estimation ofα When x0 Is Known

Theorem 1.
Assume that the assumptions of Lemma1 are satisfied. Furthermore, define the statistic:(23)α⋆=r−1∑i=1r−1ilogXi+1:n−x0Xi:n−x0−1,r⩾2.
Then, a1001−δ% asymptotic confidence interval for α is(24)α⋆νχδ/22ν,α⋆νχ1−δ/22ν,where ν=2r−1 and χδ2ν, 0<δ<1, is the δ th quantile of the chi-square distribution with ν degrees of freedom. Moreover, when r is large enough (see Remark 1), statistic (23) is a consistent estimator for α.Proof.
Clearly,(25)α⋆α=r−1η1+η2+⋯+ηr−1=1η¯.
Thus, in view of Lemma1 and its Corollary 1, the DF of the pivotal quantity P=να/α⋆ weakly converges, as n⟶∞, to the chi-square distribution with ν=2r−1 degrees of freedom. On the contrary, Lemma 1 implies that, for large n, η¯ is the sample mean of a random sample of size r−1 from the EXP(1) distribution. Thus, from the law of large number, η¯→1. This completes the proof of the theorem.Remark 1.
The stipulation “r is large enough” means that r should be large enough in order that the law of large number is applicable; meanwhile, r should be chosen such that Xr:n is still a lower extreme order statistic. The simulation study was carried out to assess the performance of the estimator α⋆ (see Table 1) which reveals that, at n=1000, the best value of the estimator α⋆ is attained when r=50, while small values of rr≤5 or large values of rr≥200 give unsatisfactory estimates.Table 1
Estimates ofα attached with their MSEs when the left endpoint is known or unknown, based on X1:n,…,Xr:n for different values of r and n, where Xi,i=1,2,…,r, follow Pareto distribution Ft=1−5/t2.5,t⩾5, i.e., F∈DW1.
nrThe left endpointx0 is assumed to be knownThe left endpointx0 is assumed to be unknownα˜ (MSE)α⋆ (MSE)95%CP⋆L⋆,U⋆α^MSEα˜ (MSE)α⋆ (MSE)95%CP⋆(L⋆,U⋆)10061.461 (0.899)1.218 (0.524)95.06(0.395, 2.495)2.789 (143960.)1.960 (2.870)1.307 (0.960)94.97(0.356, 2.864)101.199 (0.236)1.079 (0.165)94.83(0.493, 1.890)1.628 (5111.2)1.375 (0.443)1.100 (0.204)94.90(0.475, 1.983)151.085 (0.099)1.013 (0.081)94.62(0.554, 1.608)1.055 (0.801)1.180 (0.153)1.022 (0.091)94.64(0.544, 1.649)201.025 (0.058)0.974 (0.052)93.90(0.586, 1.458)0.975 (0.158)1.089 (0.077)0.980 (0.056)93.98(0.581, 1.482)250.982 (0.040)0.942 (0.040)92.46(0.604, 1.355)0.938 (0.109)1.030 (0.047)0.947 (0.042)92.63(0.601, 1.372)300.946 (0.032)0.915 (0.035)90.15(0.612, 1.276)0.915 (0.089)0.984 (0.034)0.919 (0.036)90.64(0.610, 1.289)500101.243 (0.264)1.119 (0.180)95.54(0.512, 1.960)2.273 (6629.4)1.428 (0.515)1.142 (0.233)95.25(0.493, 2.060)201.098 (0.077)1.043 (0.062)95.49(0.628, 1.562)1.017 (0.187)1.166 (0.110)1.049 (0.069)94.91(0.622, 1.586)301.049 (0.041)1.014 (0.036)95.32(0.679, 1.415)0.968 (0.105)1.090 (0.053)1.017 (0.039)94.82(0.676, 1.427)401.021 (0.028)0.996 (0.026)95.05(0.708, 1.332)0.949 (0.081)1.051 (0.033)0.998 (0.027)94.54(0.706, 1.339)501.002 (0.021)0.982 (0.020)94.23(0.727, 1.276)0.939 (0.070)1.025 (0.023)0.984 (0.021)94.16(0.725, 1.281)750.971 (0.013)0.958 (0.014)92.58(0.753, 1.189)0.923 (0.058)0.986 (0.013)0.960 (0.014)92.27(0.752, 1.192)1000.944 (0.012)0.935 (0.013)88.07(0.760, 1.128)0.913 (0.051)0.955 (0.011)0.936 (0.013)88.12(0.760, 1.130)1000101.250 (0.289)1.125 (0.199)94.95(0.514, 1.970)2.534 (13135.)1.417 (0.513)1.134 (0.235)94.72(0.490, 2.044)201.103 (0.081)1.048 (0.066)95.20(0.631, 1.569)1.021 (0.205)1.163 (0.108)1.047 (0.068)95.05(0.620, 1.583)301.059 (0.045)1.024 (0.039)95.28(0.685, 1.428)0.973 (0.108)1.095 (0.054)1.022 (0.040)95.02(0.679, 1.434)401.038 (0.031)1.012 (0.028)94.70(0.720, 1.354)0.956 (0.084)1.064 (0.036)1.011 (0.029)94.73(0.716, 1.357)501.024 (0.023)1.003 (0.021)95.07(0.742, 1.303)0.947 (0.072)1.044 (0.025)1.002 (0.022)94.80(0.739, 1.305)601.012 (0.018)0.996 (0.017)94.97(0.758, 1.265)0.942 (0.065)1.029 (0.019)0.995 (0.017)94.86(0.755, 1.267)701.004 (0.015)0.989 (0.014)94.85(0.770, 1.236)0.938 (0.060)1.018 (0.016)0.989 (0.015)94.84(0.768, 1.237)800.996 (0.012)0.983 (0.012)94.71(0.779, 1.212)0.935 (0.056)1.008 (0.013)0.983 (0.012)94.62(0.777, 1.212)900.989 (0.011)0.978 (0.011)94.41(0.786, 1.192)0.932 (0.053)1.000 (0.011)0.978 (0.011)93.87(0.784, 1.193)1000.983 (0.010)0.973 (0.010)93.89(0.791, 1.174)0.930 (0.051)0.993 (0.010)0.973 (0.010)93.81(0.790, 1.175)1500.958 (0.008)0.951 (0.008)89.33(0.805, 1.110)0.923 (0.043)0.964 (0.007)0.951 (0.008)89.62(0.804, 1.111)2000.935 (0.008)0.930 (0.009)81.03(0.806, 1.064)0.917 (0.039)0.940 (0.008)0.930 (0.009)81.21(0.805, 1.064)Remark 2.
Corollary1 and Theorem 1 provide a preliminary check of the assumption F∈DWα. In other words, we can apply any goodness-of-fit test, such as the Kolmogorov–Smirnov (K-S) test, to check whether the U0,1 distribution fits the observed data u1∗,u2∗,…,ur−2∗. If the U0,1 distribution failed to fit this data, we conclude that F∈DWα. Otherwise, we can go ahead with our inference.


### 2.2. Estimation ofα when x0 is Unknown

When the left endpointx0 of F is unknown, we can invoke the strong asymptotic relation (5) combining with Theorem 1 to suggest an approximate 1001−δ% asymptotic confidence interval for α≤1 corresponding to (24) (see Remark 3) as(26)α⋆ν⋆χδ/22ν⋆,α⋆ν⋆χ1−δ/22ν⋆,where ν⋆=2r−2 and(27)α⋆=r−2∑i=2r−1i−1logXi+1:n−X1:nXi:n−X1:n−1,r>2.Remark 3.
Although there is no rigorous proof of the asymptotic properties of the confidence interval (26), its performance is discussed in Section 4 via a simulation study. This simulation study (Tables 1 and 2) demonstrates that this confidence interval performs well for α≤1. Moreover, the simulation study reveals that statistic (27) serves as a good estimator for α≤1, while this is not the case for α>1. The poor performance of estimator (27) and, consequently, the poor performance of the confidence interval (26), for α>1, may be interpreted by noting that, by increasing the value of α, we need a larger sample size to get the same accuracy from relation (5) (i.e., to get the same closeness between X1:n and x0).
It is worth mentioning that, atα=1, the asymptotic behaviour of the point estimator (27) and confidence interval (26) can be proved theoretically as in Theorem 1. This fact can be easily seen by the following lemma.Table 2
Estimates ofα attached with their MSEs when the left endpoint is known or unknown, based on X1:n,…,Xr:n for different values of r and n, where Xi,i=1,2,…,r, follow the exponentiated exponential DF Ft=1−exp−t−10/1002/3,t⩾10, i.e., F∈DW2/3.
nrThe left endpointx0 is assumed to be knownThe left endpointx0 is assumed to be unknownα˜ (MSE)α⋆ (MSE)95%CP⋆L⋆,U⋆α^ (MSE)α˜ (MSE)α⋆ (MSE)95%CP⋆L⋆,U⋆100100.830 (0.120)0.747 (0.082)95.11(0.341, 1.308)1.024 (7.711)1.067 (0.324)0.854 (0.141)95.39(0.368, 1.539)150.760 (0.055)0.709 (0.042)95.09(0.388, 1.127)0.841 (0.202)0.905 (0.123)0.784 (0.064)95.45(0.418, 1.265)200.728 (0.033)0.692 (0.027)95.03(0.416, 1.036)0.787 (0.090)0.835 (0.067)0.752 (0.039)95.44(0.446, 1.137)250.707 (0.024)0.679 (0.021)94.76(0.435, 0.976)0.761 (0.064)0.793 (0.043)0.729 (0.027)95.53(0.462, 1.056)300.692 (0.018)0.668 (0.016)94.71(0.448, 0.933)0.745 (0.050)0.763 (0.029)0.712 (0.020)95.67(0.473, 0.999)500100.838 (0.131)0.754 (0.090)95.41(0.345, 1.321)1.043 (12.095)1.072 (0.339)0.857 (0.149)95.44(0.370, 1.546)200.745 (0.038)0.707 (0.031)95.02(0.426, 1.059)0.799 (0.097)0.852 (0.075)0.767 (0.043)95.05(0.454, 1.159)300.717 (0.021)0.693 (0.018)94.99(0.464, 0.967)0.761 (0.057)0.790 (0.037)0.737 (0.024)94.73(0.490, 1.035)400.702 (0.014)0.685 (0.013)95.05(0.487, 0.916)0.744 (0.043)0.759 (0.023)0.721 (0.016)95.08(0.510, 0.968)500.694 (0.011)0.680 (0.010)94.95(0.503, 0.884)0.735 (0.036)0.741 (0.016)0.711 (0.012)94.88(0.524, 0.926)750.681 (0.007)0.672 (0.006)94.74(0.528, 0.834)0.721 (0.027)0.714 (0.009)0.695 (0.007)95.01(0.545, 0.863)1000.672 (0.005)0.665 (0.004)94.86(0.541, 0.802)0.713 (0.022)0.698 (0.006)0.684 (0.005)95.34(0.555, 0.825)1000100.840 (0.129)0.756 (0.088)95.23(0.346, 1.324)1.042 (85.092)1.077 (0.345)0.861 (0.152)95.22(0.372, 1.553)200.744 (0.038)0.707 (0.031)95.14(0.425, 1.058)0.796 (0.097)0.852 (0.077)0.767 (0.045)94.79(0.454, 1.159)300.717 (0.021)0.693 (0.018)94.92(0.464, 0.967)0.760 (0.057)0.790 (0.037)0.737 (0.025)94.87(0.490, 1.034)400.703 (0.014)0.685 (0.013)95.11(0.487, 0.917)0.743 (0.043)0.760 (0.023)0.722 (0.017)94.76(0.511, 0.969)500.696 (0.011)0.682 (0.010)95.43(0.504, 0.885)0.734 (0.036)0.743 (0.017)0.713 (0.012)95.10(0.526, 0.928)600.691 (0.008)0.679 (0.008)95.42(0.517, 0.863)0.728 (0.031)0.731 (0.013)0.707 (0.010)95.07(0.537, 0.900)700.687 (0.007)0.677 (0.007)95.41(0.527, 0.846)0.724 (0.028)0.723 (0.010)0.702 (0.008)94.80(0.545, 0.878)800.684 (0.006)0.675 (0.006)95.26(0.535, 0.832)0.720 (0.026)0.716 (0.009)0.698 (0.007)94.97(0.552, 0.861)900.681 (0.005)0.674 (0.005)95.34(0.541, 0.821)0.717 (0.024)0.710 (0.007)0.694 (0.006)94.75(0.557, 0.847)1000.679 (0.005)0.672 (0.005)95.28(0.546, 0.811)0.715 (0.023)0.705 (0.006)0.691 (0.005)95.09(0.561, 0.835)1500.671 (0.003)0.666 (0.003)94.99(0.564, 0.777)0.707 (0.018)0.689 (0.004)0.680 (0.003)95.33(0.575, 0.794)2000.664 (0.002)0.661 (0.002)94.34(0.572, 0.756)0.702 (0.016)0.679 (0.002)0.672 (0.002)95.31(0.582, 0.769)Lemma 2.
Assume thatr≥2 and that F∈DWα has an unknown left endpoint. Then,(28)ξi=−i−1logXi:n−X1:nXi+1:n−X1:n,i=2,3,…,r−1,are asymptotically iid RVs distributed as EXP(1). Moreover,(29)Bj∗=ξj∑i=2r−1ξi,j=2,3,…,r−1,are asymptotically iid RVs having Beta 1,r−3 distribution. Furthermore, α∗→rp1, where α⋆ is defined by (27).Proof.
First, note that ifF∈DWα with a normalizing constant an>0, then the following relations are true, for any fixed i and α>0,(30)Xi:n−X1:nan=1anXi:n−x0−X1:n−x0=Xi:n−x0an1−X1:n−x0/anXi:n−x0/an=nwZ1+Z2+⋯+Zi1/α1−Z1Z1+Z2+⋯+Zi1/α.
Thus, forα=1, we get Xi:n−X1:n/an=Z2+Z3+⋯+Zi. Consequently, as in Lemma 1, we obtain(31)Xi:n−X1:nXi+1:n−X1:ni=nwZ2+Z3+⋯+ZiZ2+Z3+⋯+Zi+Zi+1i∼U0,1.
Hence, the rest of the proof can be proceeded as in Lemma1 and Theorem 1.In view of the simulation study achieved in Section4, we suggest the following slightly modified estimator for α>1:(32)α⋆⋆=r−3∑i=3r−1i−2logXi+1:n−X1:nXi:n−X1:n−1,r>3,which yields the confidence interval α⋆⋆/ν⋆⋆χδ/22ν⋆⋆,α⋆⋆/ν⋆⋆χ1−δ/22ν⋆⋆, where ν⋆⋆=2r−3. Based on the mean square error (MSE), Table 3 indicates that the performance of the estimator α⋆⋆ is better than α⋆ for α>1.Table 3
Estimates ofα attached with their MSEs when the left endpoint is known or unknown based on X1:n,…,Xr:n for different values of r and n, where Xi,i=1,2,…,r, follow the Weibull distribution Ft=1−exp−t−5/102,andt≥5witht≥5t≥10, i.e., F∈DW2.
nrThe left endpointx0 is assumed to be knownThe left endpointx0 is assumed to be unknownα˜ (MSE)α⋆ (MSE)95%CP⋆L⋆,U⋆α^ (MSE)α˜ (MSE)α⋆ (MSE)α⋆⋆ (MSE)95%CP⋆L⋆,U⋆100102.42 (1.02)2.18 (0.71)94.82(1.00, 3.82)2.16 (588.88)2.04 (0.90)1.63 (0.71)2.27 (1.34)91.71(0.91, 4.23)152.22 (0.45)2.07 (0.36)94.77(1.13, 3.29)1.55 (366.15)1.83 (0.42)1.59 (0.46)1.99 (0.49)89.33(1.03, 3.27)202.11 (0.27)2.00 (0.23)94.20(1.20, 3.00)1.44 (4.99)1.74 (0.31)1.57 (0.39)1.88 (0.31)86.95(1.09, 2.87)252.03 (0.18)1.95 (0.17)93.73(1.25, 2.80)1.37 (0.99)1.69 (0.28)1.55 (0.35)1.80 (0.25)83.47(1.13, 2.63)301.97 (0.14)1.90 (0.14)92.47(1.27, 2.65)1.33 (0.83)1.65 (0.26)1.54 (0.33)1.75 (0.22)79.56(1.15, 2.47)500102.49 (1.10)2.24 (0.76)95.17(1.02, 3.92)0.65 (9173.20)2.09 (0.92)1.67 (0.69)2.32 (1.41)91.52(0.93, 4.33)202.20 (0.32)2.09 (0.26)94.84(1.26, 3.13)1.17 (926.75)1.81 (0.32)1.63 (0.37)1.95 (0.33)88.36(1.14, 2.98)302.11 (0.17)2.04 (0.15)94.92(1.37, 2.85)1.38 (1.04)1.76 (0.23)1.64 (0.28)1.87 (0.21)84.94(1.23, 2.64)402.06 (0.11)2.01 (0.11)95.06(1.43, 2.69)1.34 (0.75)1.74 (0.19)1.65 (0.23)1.83 (0.16)81.82(1.29, 2.47)502.03 (0.08)1.99 (0.08)94.79(1.47, 2.58)1.34 (0.70)1.73 (0.17)1.66 (0.20)1.81 (0.14)79.75(1.33, 2.36)751.97 (0.05)1.95 (0.05)93.91(1.53, 2.42)1.33 (0.65)1.71 (0.14)1.67 (0.17)1.78 (0.11)73.05(1.39, 2.21)1001.93 (0.04)1.91 (0.04)91.59(1.55, 2.30)1.33 (0.62)1.70 (0.13)1.67 (0.15)1.75 (0.11)65.64(1.42, 2.12)1000102.49 (1.10)2.24 (0.75)95.39(1.02, 3.93)1.76 (409.85)2.10 (0.91)1.68 (0.68)2.34 (1.40)92.17(0.94, 4.36)202.21 (0.32)2.09 (0.26)95.40(1.26, 3.14)1.50 (7.48)1.82 (0.31)1.64 (0.35)1.97 (0.33)88.45(1.15, 3.01)302.12 (0.18)2.05 (0.15)95.39(1.37, 2.86)1.39 (0.85)1.77 (0.22)1.65 (0.26)1.89 (0.20)86.04(1.24, 2.66)402.08 (0.12)2.03 (0.11)95.18(1.44, 2.72)1.36 (0.73)1.76 (0.18)1.67 (0.22)1.86 (0.15)84.28(1.31, 2.50)502.06 (0.09)2.02 (0.08)95.20(1.49, 2.62)1.35 (0.68)1.75 (0.16)1.68 (0.19)1.84 (0.13)82.50(1.35, 2.40)602.04 (0.07)2.01 (0.07)95.07(1.53, 2.55)1.35 (0.65)1.75 (0.14)1.69 (0.17)1.83 (0.11)80.91(1.39, 2.34)702.02 (0.06)1.99 (0.06)95.11(1.55, 2.49)1.35 (0.64)1.75 (0.13)1.70 (0.15)1.82 (0.10)78.82(1.41, 2.28)802.01 (0.05)1.99 (0.05)94.94(1.57, 2.45)1.35 (0.62)1.75 (0.12)1.71 (0.14)1.82 (0.10)76.89(1.43, 2.24)902.00 (0.04)1.98 (0.04)94.98(1.59, 2.41)1.35 (0.61)1.75 (0.11)1.71 (0.13)1.81 (0.09)75.40(1.45, 2.21)1001.99 (0.04)1.97 (0.04)94.66(1.60, 2.38)1.35 (0.60)1.75 (0.11)1.72 (0.13)1.81 (0.09)73.25(1.47, 2.19)1501.95 (0.03)1.94 (0.03)92.70(1.64, 2.26)1.36 (0.56)1.75 (0.09)1.73 (0.11)1.79 (0.08)65.48(1.51, 2.09)2001.91 (0.02)1.90 (0.03)88.59(1.65, 2.18)1.37 (0.54)1.74 (0.09)1.72 (0.10)1.77 (0.08)54.79(1.53, 2.03)Smith and Weissman [13] have obtained an estimator of α based on the maximum likelihood method, which is given by(33)α˜=r∑i=1rlogXr:n−x0Xi:n−x0−1,ifx0is known,r∑i=2rlogXr:n−X1:nXi:n−X1:n−1,ifx0is unknown.We conclude this section with a theoretical comparison between the two estimatorsα⋆ and α˜. This comparison, which is based on Pitman’s measure of closeness [43], shows that the estimator α⋆ is remarkably superior whenever the assumptions of the EVT concerning r and n (n is sufficiently large and r is a lower extreme rank) are satisfied, see Remark 7.An estimatorα1 is a Pitman-closer estimate to α than α2 if Pα1−α<α2−α⩾1/2, for all values of α, with strict inequality for at least one value of α.Proposition 1.
The estimatorα⋆ is an asymptotically Pitman-closer to α than α˜.Proof.
Letx0 be known. By noting that(34)∑i=1rlogXr:n−x0Xi:n−x0=∑i=1r−1ilogXi+1:n−x0Xi:n−x0,we have Pα˜>α⋆=1, for all r⩾2. Hence,(35)P0α∗,α˜:=Pα∗−α<α˜−α=Pα˜+α∗>2α=P2r−1∑i=1r−1ilogXi+1:n−x0/Xi:n−x0>2α=Pα∑i=1r−1ilogXi+1:n−x0Xi:n−x0<2r−12⟶nwPη∗<2r−12=Γr−12r−12∈12,1−e−3/2,r≥2,where the last implication is verified by Mathematica 12.2. The proof is completed.Remark 4.
Via Mathematica 12.2, Table4 reveals that the function P0α⋆,α˜ decreases with r and it attains its maximum, 1−e−3/2≈0.777, at r=2, and its minimum 0.5. If x0 is unknown and F∈DW1, it can be shown that P1α⋆⋆,α˜:=Pα⋆⋆−α<α˜−α=Γr−2r−1, r⩾3.Table 4
Comparisons betweenα⋆ and α˜ via Pitman’s measure of closeness for different values of r.
r231025501001000P0α⋆,α˜0.7770.7130.6080.5670.5470.5330.511P1α⋆⋆,α˜—0.8650.6760.6080.5760.5530.517


## 2.1. Estimation ofα When x0 Is Known

Theorem 1.
Assume that the assumptions of Lemma1 are satisfied. Furthermore, define the statistic:(23)α⋆=r−1∑i=1r−1ilogXi+1:n−x0Xi:n−x0−1,r⩾2.
Then, a1001−δ% asymptotic confidence interval for α is(24)α⋆νχδ/22ν,α⋆νχ1−δ/22ν,where ν=2r−1 and χδ2ν, 0<δ<1, is the δ th quantile of the chi-square distribution with ν degrees of freedom. Moreover, when r is large enough (see Remark 1), statistic (23) is a consistent estimator for α.Proof.
Clearly,(25)α⋆α=r−1η1+η2+⋯+ηr−1=1η¯.
Thus, in view of Lemma1 and its Corollary 1, the DF of the pivotal quantity P=να/α⋆ weakly converges, as n⟶∞, to the chi-square distribution with ν=2r−1 degrees of freedom. On the contrary, Lemma 1 implies that, for large n, η¯ is the sample mean of a random sample of size r−1 from the EXP(1) distribution. Thus, from the law of large number, η¯→1. This completes the proof of the theorem.Remark 1.
The stipulation “r is large enough” means that r should be large enough in order that the law of large number is applicable; meanwhile, r should be chosen such that Xr:n is still a lower extreme order statistic. The simulation study was carried out to assess the performance of the estimator α⋆ (see Table 1) which reveals that, at n=1000, the best value of the estimator α⋆ is attained when r=50, while small values of rr≤5 or large values of rr≥200 give unsatisfactory estimates.Table 1
Estimates ofα attached with their MSEs when the left endpoint is known or unknown, based on X1:n,…,Xr:n for different values of r and n, where Xi,i=1,2,…,r, follow Pareto distribution Ft=1−5/t2.5,t⩾5, i.e., F∈DW1.
nrThe left endpointx0 is assumed to be knownThe left endpointx0 is assumed to be unknownα˜ (MSE)α⋆ (MSE)95%CP⋆L⋆,U⋆α^MSEα˜ (MSE)α⋆ (MSE)95%CP⋆(L⋆,U⋆)10061.461 (0.899)1.218 (0.524)95.06(0.395, 2.495)2.789 (143960.)1.960 (2.870)1.307 (0.960)94.97(0.356, 2.864)101.199 (0.236)1.079 (0.165)94.83(0.493, 1.890)1.628 (5111.2)1.375 (0.443)1.100 (0.204)94.90(0.475, 1.983)151.085 (0.099)1.013 (0.081)94.62(0.554, 1.608)1.055 (0.801)1.180 (0.153)1.022 (0.091)94.64(0.544, 1.649)201.025 (0.058)0.974 (0.052)93.90(0.586, 1.458)0.975 (0.158)1.089 (0.077)0.980 (0.056)93.98(0.581, 1.482)250.982 (0.040)0.942 (0.040)92.46(0.604, 1.355)0.938 (0.109)1.030 (0.047)0.947 (0.042)92.63(0.601, 1.372)300.946 (0.032)0.915 (0.035)90.15(0.612, 1.276)0.915 (0.089)0.984 (0.034)0.919 (0.036)90.64(0.610, 1.289)500101.243 (0.264)1.119 (0.180)95.54(0.512, 1.960)2.273 (6629.4)1.428 (0.515)1.142 (0.233)95.25(0.493, 2.060)201.098 (0.077)1.043 (0.062)95.49(0.628, 1.562)1.017 (0.187)1.166 (0.110)1.049 (0.069)94.91(0.622, 1.586)301.049 (0.041)1.014 (0.036)95.32(0.679, 1.415)0.968 (0.105)1.090 (0.053)1.017 (0.039)94.82(0.676, 1.427)401.021 (0.028)0.996 (0.026)95.05(0.708, 1.332)0.949 (0.081)1.051 (0.033)0.998 (0.027)94.54(0.706, 1.339)501.002 (0.021)0.982 (0.020)94.23(0.727, 1.276)0.939 (0.070)1.025 (0.023)0.984 (0.021)94.16(0.725, 1.281)750.971 (0.013)0.958 (0.014)92.58(0.753, 1.189)0.923 (0.058)0.986 (0.013)0.960 (0.014)92.27(0.752, 1.192)1000.944 (0.012)0.935 (0.013)88.07(0.760, 1.128)0.913 (0.051)0.955 (0.011)0.936 (0.013)88.12(0.760, 1.130)1000101.250 (0.289)1.125 (0.199)94.95(0.514, 1.970)2.534 (13135.)1.417 (0.513)1.134 (0.235)94.72(0.490, 2.044)201.103 (0.081)1.048 (0.066)95.20(0.631, 1.569)1.021 (0.205)1.163 (0.108)1.047 (0.068)95.05(0.620, 1.583)301.059 (0.045)1.024 (0.039)95.28(0.685, 1.428)0.973 (0.108)1.095 (0.054)1.022 (0.040)95.02(0.679, 1.434)401.038 (0.031)1.012 (0.028)94.70(0.720, 1.354)0.956 (0.084)1.064 (0.036)1.011 (0.029)94.73(0.716, 1.357)501.024 (0.023)1.003 (0.021)95.07(0.742, 1.303)0.947 (0.072)1.044 (0.025)1.002 (0.022)94.80(0.739, 1.305)601.012 (0.018)0.996 (0.017)94.97(0.758, 1.265)0.942 (0.065)1.029 (0.019)0.995 (0.017)94.86(0.755, 1.267)701.004 (0.015)0.989 (0.014)94.85(0.770, 1.236)0.938 (0.060)1.018 (0.016)0.989 (0.015)94.84(0.768, 1.237)800.996 (0.012)0.983 (0.012)94.71(0.779, 1.212)0.935 (0.056)1.008 (0.013)0.983 (0.012)94.62(0.777, 1.212)900.989 (0.011)0.978 (0.011)94.41(0.786, 1.192)0.932 (0.053)1.000 (0.011)0.978 (0.011)93.87(0.784, 1.193)1000.983 (0.010)0.973 (0.010)93.89(0.791, 1.174)0.930 (0.051)0.993 (0.010)0.973 (0.010)93.81(0.790, 1.175)1500.958 (0.008)0.951 (0.008)89.33(0.805, 1.110)0.923 (0.043)0.964 (0.007)0.951 (0.008)89.62(0.804, 1.111)2000.935 (0.008)0.930 (0.009)81.03(0.806, 1.064)0.917 (0.039)0.940 (0.008)0.930 (0.009)81.21(0.805, 1.064)Remark 2.
Corollary1 and Theorem 1 provide a preliminary check of the assumption F∈DWα. In other words, we can apply any goodness-of-fit test, such as the Kolmogorov–Smirnov (K-S) test, to check whether the U0,1 distribution fits the observed data u1∗,u2∗,…,ur−2∗. If the U0,1 distribution failed to fit this data, we conclude that F∈DWα. Otherwise, we can go ahead with our inference.


## 2.2. Estimation ofα when x0 is Unknown

When the left endpointx0 of F is unknown, we can invoke the strong asymptotic relation (5) combining with Theorem 1 to suggest an approximate 1001−δ% asymptotic confidence interval for α≤1 corresponding to (24) (see Remark 3) as(26)α⋆ν⋆χδ/22ν⋆,α⋆ν⋆χ1−δ/22ν⋆,where ν⋆=2r−2 and(27)α⋆=r−2∑i=2r−1i−1logXi+1:n−X1:nXi:n−X1:n−1,r>2.Remark 3.
Although there is no rigorous proof of the asymptotic properties of the confidence interval (26), its performance is discussed in Section 4 via a simulation study. This simulation study (Tables 1 and 2) demonstrates that this confidence interval performs well for α≤1. Moreover, the simulation study reveals that statistic (27) serves as a good estimator for α≤1, while this is not the case for α>1. The poor performance of estimator (27) and, consequently, the poor performance of the confidence interval (26), for α>1, may be interpreted by noting that, by increasing the value of α, we need a larger sample size to get the same accuracy from relation (5) (i.e., to get the same closeness between X1:n and x0).
It is worth mentioning that, atα=1, the asymptotic behaviour of the point estimator (27) and confidence interval (26) can be proved theoretically as in Theorem 1. This fact can be easily seen by the following lemma.Table 2
Estimates ofα attached with their MSEs when the left endpoint is known or unknown, based on X1:n,…,Xr:n for different values of r and n, where Xi,i=1,2,…,r, follow the exponentiated exponential DF Ft=1−exp−t−10/1002/3,t⩾10, i.e., F∈DW2/3.
nrThe left endpointx0 is assumed to be knownThe left endpointx0 is assumed to be unknownα˜ (MSE)α⋆ (MSE)95%CP⋆L⋆,U⋆α^ (MSE)α˜ (MSE)α⋆ (MSE)95%CP⋆L⋆,U⋆100100.830 (0.120)0.747 (0.082)95.11(0.341, 1.308)1.024 (7.711)1.067 (0.324)0.854 (0.141)95.39(0.368, 1.539)150.760 (0.055)0.709 (0.042)95.09(0.388, 1.127)0.841 (0.202)0.905 (0.123)0.784 (0.064)95.45(0.418, 1.265)200.728 (0.033)0.692 (0.027)95.03(0.416, 1.036)0.787 (0.090)0.835 (0.067)0.752 (0.039)95.44(0.446, 1.137)250.707 (0.024)0.679 (0.021)94.76(0.435, 0.976)0.761 (0.064)0.793 (0.043)0.729 (0.027)95.53(0.462, 1.056)300.692 (0.018)0.668 (0.016)94.71(0.448, 0.933)0.745 (0.050)0.763 (0.029)0.712 (0.020)95.67(0.473, 0.999)500100.838 (0.131)0.754 (0.090)95.41(0.345, 1.321)1.043 (12.095)1.072 (0.339)0.857 (0.149)95.44(0.370, 1.546)200.745 (0.038)0.707 (0.031)95.02(0.426, 1.059)0.799 (0.097)0.852 (0.075)0.767 (0.043)95.05(0.454, 1.159)300.717 (0.021)0.693 (0.018)94.99(0.464, 0.967)0.761 (0.057)0.790 (0.037)0.737 (0.024)94.73(0.490, 1.035)400.702 (0.014)0.685 (0.013)95.05(0.487, 0.916)0.744 (0.043)0.759 (0.023)0.721 (0.016)95.08(0.510, 0.968)500.694 (0.011)0.680 (0.010)94.95(0.503, 0.884)0.735 (0.036)0.741 (0.016)0.711 (0.012)94.88(0.524, 0.926)750.681 (0.007)0.672 (0.006)94.74(0.528, 0.834)0.721 (0.027)0.714 (0.009)0.695 (0.007)95.01(0.545, 0.863)1000.672 (0.005)0.665 (0.004)94.86(0.541, 0.802)0.713 (0.022)0.698 (0.006)0.684 (0.005)95.34(0.555, 0.825)1000100.840 (0.129)0.756 (0.088)95.23(0.346, 1.324)1.042 (85.092)1.077 (0.345)0.861 (0.152)95.22(0.372, 1.553)200.744 (0.038)0.707 (0.031)95.14(0.425, 1.058)0.796 (0.097)0.852 (0.077)0.767 (0.045)94.79(0.454, 1.159)300.717 (0.021)0.693 (0.018)94.92(0.464, 0.967)0.760 (0.057)0.790 (0.037)0.737 (0.025)94.87(0.490, 1.034)400.703 (0.014)0.685 (0.013)95.11(0.487, 0.917)0.743 (0.043)0.760 (0.023)0.722 (0.017)94.76(0.511, 0.969)500.696 (0.011)0.682 (0.010)95.43(0.504, 0.885)0.734 (0.036)0.743 (0.017)0.713 (0.012)95.10(0.526, 0.928)600.691 (0.008)0.679 (0.008)95.42(0.517, 0.863)0.728 (0.031)0.731 (0.013)0.707 (0.010)95.07(0.537, 0.900)700.687 (0.007)0.677 (0.007)95.41(0.527, 0.846)0.724 (0.028)0.723 (0.010)0.702 (0.008)94.80(0.545, 0.878)800.684 (0.006)0.675 (0.006)95.26(0.535, 0.832)0.720 (0.026)0.716 (0.009)0.698 (0.007)94.97(0.552, 0.861)900.681 (0.005)0.674 (0.005)95.34(0.541, 0.821)0.717 (0.024)0.710 (0.007)0.694 (0.006)94.75(0.557, 0.847)1000.679 (0.005)0.672 (0.005)95.28(0.546, 0.811)0.715 (0.023)0.705 (0.006)0.691 (0.005)95.09(0.561, 0.835)1500.671 (0.003)0.666 (0.003)94.99(0.564, 0.777)0.707 (0.018)0.689 (0.004)0.680 (0.003)95.33(0.575, 0.794)2000.664 (0.002)0.661 (0.002)94.34(0.572, 0.756)0.702 (0.016)0.679 (0.002)0.672 (0.002)95.31(0.582, 0.769)Lemma 2.
Assume thatr≥2 and that F∈DWα has an unknown left endpoint. Then,(28)ξi=−i−1logXi:n−X1:nXi+1:n−X1:n,i=2,3,…,r−1,are asymptotically iid RVs distributed as EXP(1). Moreover,(29)Bj∗=ξj∑i=2r−1ξi,j=2,3,…,r−1,are asymptotically iid RVs having Beta 1,r−3 distribution. Furthermore, α∗→rp1, where α⋆ is defined by (27).Proof.
First, note that ifF∈DWα with a normalizing constant an>0, then the following relations are true, for any fixed i and α>0,(30)Xi:n−X1:nan=1anXi:n−x0−X1:n−x0=Xi:n−x0an1−X1:n−x0/anXi:n−x0/an=nwZ1+Z2+⋯+Zi1/α1−Z1Z1+Z2+⋯+Zi1/α.
Thus, forα=1, we get Xi:n−X1:n/an=Z2+Z3+⋯+Zi. Consequently, as in Lemma 1, we obtain(31)Xi:n−X1:nXi+1:n−X1:ni=nwZ2+Z3+⋯+ZiZ2+Z3+⋯+Zi+Zi+1i∼U0,1.
Hence, the rest of the proof can be proceeded as in Lemma1 and Theorem 1.In view of the simulation study achieved in Section4, we suggest the following slightly modified estimator for α>1:(32)α⋆⋆=r−3∑i=3r−1i−2logXi+1:n−X1:nXi:n−X1:n−1,r>3,which yields the confidence interval α⋆⋆/ν⋆⋆χδ/22ν⋆⋆,α⋆⋆/ν⋆⋆χ1−δ/22ν⋆⋆, where ν⋆⋆=2r−3. Based on the mean square error (MSE), Table 3 indicates that the performance of the estimator α⋆⋆ is better than α⋆ for α>1.Table 3
Estimates ofα attached with their MSEs when the left endpoint is known or unknown based on X1:n,…,Xr:n for different values of r and n, where Xi,i=1,2,…,r, follow the Weibull distribution Ft=1−exp−t−5/102,andt≥5witht≥5t≥10, i.e., F∈DW2.
nrThe left endpointx0 is assumed to be knownThe left endpointx0 is assumed to be unknownα˜ (MSE)α⋆ (MSE)95%CP⋆L⋆,U⋆α^ (MSE)α˜ (MSE)α⋆ (MSE)α⋆⋆ (MSE)95%CP⋆L⋆,U⋆100102.42 (1.02)2.18 (0.71)94.82(1.00, 3.82)2.16 (588.88)2.04 (0.90)1.63 (0.71)2.27 (1.34)91.71(0.91, 4.23)152.22 (0.45)2.07 (0.36)94.77(1.13, 3.29)1.55 (366.15)1.83 (0.42)1.59 (0.46)1.99 (0.49)89.33(1.03, 3.27)202.11 (0.27)2.00 (0.23)94.20(1.20, 3.00)1.44 (4.99)1.74 (0.31)1.57 (0.39)1.88 (0.31)86.95(1.09, 2.87)252.03 (0.18)1.95 (0.17)93.73(1.25, 2.80)1.37 (0.99)1.69 (0.28)1.55 (0.35)1.80 (0.25)83.47(1.13, 2.63)301.97 (0.14)1.90 (0.14)92.47(1.27, 2.65)1.33 (0.83)1.65 (0.26)1.54 (0.33)1.75 (0.22)79.56(1.15, 2.47)500102.49 (1.10)2.24 (0.76)95.17(1.02, 3.92)0.65 (9173.20)2.09 (0.92)1.67 (0.69)2.32 (1.41)91.52(0.93, 4.33)202.20 (0.32)2.09 (0.26)94.84(1.26, 3.13)1.17 (926.75)1.81 (0.32)1.63 (0.37)1.95 (0.33)88.36(1.14, 2.98)302.11 (0.17)2.04 (0.15)94.92(1.37, 2.85)1.38 (1.04)1.76 (0.23)1.64 (0.28)1.87 (0.21)84.94(1.23, 2.64)402.06 (0.11)2.01 (0.11)95.06(1.43, 2.69)1.34 (0.75)1.74 (0.19)1.65 (0.23)1.83 (0.16)81.82(1.29, 2.47)502.03 (0.08)1.99 (0.08)94.79(1.47, 2.58)1.34 (0.70)1.73 (0.17)1.66 (0.20)1.81 (0.14)79.75(1.33, 2.36)751.97 (0.05)1.95 (0.05)93.91(1.53, 2.42)1.33 (0.65)1.71 (0.14)1.67 (0.17)1.78 (0.11)73.05(1.39, 2.21)1001.93 (0.04)1.91 (0.04)91.59(1.55, 2.30)1.33 (0.62)1.70 (0.13)1.67 (0.15)1.75 (0.11)65.64(1.42, 2.12)1000102.49 (1.10)2.24 (0.75)95.39(1.02, 3.93)1.76 (409.85)2.10 (0.91)1.68 (0.68)2.34 (1.40)92.17(0.94, 4.36)202.21 (0.32)2.09 (0.26)95.40(1.26, 3.14)1.50 (7.48)1.82 (0.31)1.64 (0.35)1.97 (0.33)88.45(1.15, 3.01)302.12 (0.18)2.05 (0.15)95.39(1.37, 2.86)1.39 (0.85)1.77 (0.22)1.65 (0.26)1.89 (0.20)86.04(1.24, 2.66)402.08 (0.12)2.03 (0.11)95.18(1.44, 2.72)1.36 (0.73)1.76 (0.18)1.67 (0.22)1.86 (0.15)84.28(1.31, 2.50)502.06 (0.09)2.02 (0.08)95.20(1.49, 2.62)1.35 (0.68)1.75 (0.16)1.68 (0.19)1.84 (0.13)82.50(1.35, 2.40)602.04 (0.07)2.01 (0.07)95.07(1.53, 2.55)1.35 (0.65)1.75 (0.14)1.69 (0.17)1.83 (0.11)80.91(1.39, 2.34)702.02 (0.06)1.99 (0.06)95.11(1.55, 2.49)1.35 (0.64)1.75 (0.13)1.70 (0.15)1.82 (0.10)78.82(1.41, 2.28)802.01 (0.05)1.99 (0.05)94.94(1.57, 2.45)1.35 (0.62)1.75 (0.12)1.71 (0.14)1.82 (0.10)76.89(1.43, 2.24)902.00 (0.04)1.98 (0.04)94.98(1.59, 2.41)1.35 (0.61)1.75 (0.11)1.71 (0.13)1.81 (0.09)75.40(1.45, 2.21)1001.99 (0.04)1.97 (0.04)94.66(1.60, 2.38)1.35 (0.60)1.75 (0.11)1.72 (0.13)1.81 (0.09)73.25(1.47, 2.19)1501.95 (0.03)1.94 (0.03)92.70(1.64, 2.26)1.36 (0.56)1.75 (0.09)1.73 (0.11)1.79 (0.08)65.48(1.51, 2.09)2001.91 (0.02)1.90 (0.03)88.59(1.65, 2.18)1.37 (0.54)1.74 (0.09)1.72 (0.10)1.77 (0.08)54.79(1.53, 2.03)Smith and Weissman [13] have obtained an estimator of α based on the maximum likelihood method, which is given by(33)α˜=r∑i=1rlogXr:n−x0Xi:n−x0−1,ifx0is known,r∑i=2rlogXr:n−X1:nXi:n−X1:n−1,ifx0is unknown.We conclude this section with a theoretical comparison between the two estimatorsα⋆ and α˜. This comparison, which is based on Pitman’s measure of closeness [43], shows that the estimator α⋆ is remarkably superior whenever the assumptions of the EVT concerning r and n (n is sufficiently large and r is a lower extreme rank) are satisfied, see Remark 7.An estimatorα1 is a Pitman-closer estimate to α than α2 if Pα1−α<α2−α⩾1/2, for all values of α, with strict inequality for at least one value of α.Proposition 1.
The estimatorα⋆ is an asymptotically Pitman-closer to α than α˜.Proof.
Letx0 be known. By noting that(34)∑i=1rlogXr:n−x0Xi:n−x0=∑i=1r−1ilogXi+1:n−x0Xi:n−x0,we have Pα˜>α⋆=1, for all r⩾2. Hence,(35)P0α∗,α˜:=Pα∗−α<α˜−α=Pα˜+α∗>2α=P2r−1∑i=1r−1ilogXi+1:n−x0/Xi:n−x0>2α=Pα∑i=1r−1ilogXi+1:n−x0Xi:n−x0<2r−12⟶nwPη∗<2r−12=Γr−12r−12∈12,1−e−3/2,r≥2,where the last implication is verified by Mathematica 12.2. The proof is completed.Remark 4.
Via Mathematica 12.2, Table4 reveals that the function P0α⋆,α˜ decreases with r and it attains its maximum, 1−e−3/2≈0.777, at r=2, and its minimum 0.5. If x0 is unknown and F∈DW1, it can be shown that P1α⋆⋆,α˜:=Pα⋆⋆−α<α˜−α=Γr−2r−1, r⩾3.Table 4
Comparisons betweenα⋆ and α˜ via Pitman’s measure of closeness for different values of r.
r231025501001000P0α⋆,α˜0.7770.7130.6080.5670.5470.5330.511P1α⋆⋆,α˜—0.8650.6760.6080.5760.5530.517


## 3. Asymptotic Prediction of Future Order Statistics

When the DFF is fully specified, Barakat et al. [36] used the pivotal quantity Ur,sn=Xs:n−Xr:n/Xr:n, r<s, to construct a predictive interval for the value of the future order statistic Xs:n, where the order statistic Xr:n is observed. We here slightly alter this pivotal quantity to get a more tractable asymptotic distribution of the altered pivotal quantity and less required information about the DF F. Consider the pivotal quantity:(36)Ur,s:n=Xs:n−Xr:nXs:n,r<s.The following theorem gives the exact asymptotic DF ofUr,s:n under condition (2), when α and x0 are assumed to be known.Theorem 2.
LetX1:n<X2:n<⋯<Xr:n be the first observed order statistics based on a continuous DF F with the finite known left endpoint x0. Furthermore, let condition (2) be satisfied, and α is known. Then,(37)FUr,s:nx=PUr,s:n≤x⟶nw1−I1−xαr,s−r≔Ψr,sx,0<x<1.Moreover,L,Uδ is a 100 1−δ% asymptotic one-sided predictive interval of Xs:n, where L=Xr:n, Uδ=1−uδ−1Xr:n, and uδ is the 1−δth quantile of the DF Ψr,sx.Proof.
First, without any loss of generality, we can assume thatx0=0, otherwise we treat with the new ordered sample:(38)X1:n⋆=X1:n−x0<X2:n⋆=X2:n−x0<⋯<Xr:n⋆=Xr:n−x0<⋯<Xn:n⋆=Xn:n−x0,which are order statistics based on the DF F with zero left endpoint. Now, relying on (2), we obtain(39)PXi:nanα≤x⟶nwΓix,i=r,s.
That is, the asymptotic distribution ofXi:n/anα can be expressed as the sum of iid RVs having the DF EXP(1). Thus, by using Rényi’s representation, we obtain(40)Vr,s:nα=Xr:nXs:nα=nwZ1+Z2+⋯+ZrZ1+Z2+⋯+Zs≔WrWr+Ws−r,where Z1,Z2,…,Zs are iid EXP(1) and the RVs Wr=∑i=1rZi and Ws−r=∑i=r+1sZi are independent, which have the DFs Γr. and Γs−r., respectively. Therefore, relation (40) implies(41)PVr,s:n≤x⟶nwIxαr,s−r.
In view of (36), we get Ur,s:n=d1−Vr,s:n. Therefore, the limit relation (37) follows and the proof is completed.Remark 5.
On proceeding as in Theorem2, it is easy to show that(42)FUr,snx=PUr,sn≤x⟶nw1−I1+x−αr,s−r:=Ψr,s⋆x.
Clearly, the limit DFΨr,sx is more tractable than Ψr,s⋆x.
In the next theorem, we ditch the condition thatα is known.Theorem 3.
Under the conditions of Theorem2, except that α is unknown, define the pivotal quantity(43)Wr,s:n=logXs:n−logXr:nlogX2:n−logX1:n,2≤r<s.Then, the pivotal quantityWr,s:n is free of α, and it is asymptotically distributed as(44)FWr,sw=βr+1/w,s−rβr,s−r,w>0,(note that, since βx,y∼Γyx−y, if x is large and y is fixed, we have βx,y⟶x0, which implies that the left endpoint of the limit DF defined in (44) is zero).Proof.
Again, without any loss of generality, we can assume thatx0=0, otherwise we replace the original data by Xi:n⋆=Xi:n−x0,i=1,2,…,n, which are order statistics based on F with zero left endpoint. Proceeding as we have done in the proof of Theorem 2, we obtain(45)Wr,s:n=nw−log∑i=1rZi/∑i=1sZi−logZ1/Z1+Z2:=T1T2:=−logT1∗−logT2∗.
By Lemma1, T1∗ and T2∗ are asymptotically independent beta RVs with parameter vectors r,s−r and 1,1, respectively, for any 2≤r<s. On the contrary, RV T2 follows the EXP(1) distribution, while the PDF of RV T1 is(46)fT1t=e−tfT1∗e−t=1βr,s−re−rt1−e−ts−r−1,t>0,s>r≥2.
Thus, by using the continuous total probability rule (cf. Theorem AI.3, p. 363, of [3]), the asymptotic DF of the pivotal quantity Wr,s:n is(47)FWr,sw=PT1T2⩽w=PT2T1⩾1w=PT2⩾T1w=∫0∞1−FT2twfT1tdt=1βr,s−r∫0∞e−t/we−rt1−e−ts−r−1dt=1βr,s−r∫01u1/w+r−11−us−r−1du=β1/w+r,s−rβr,s−r,w>0,(by using the transformation u=e−t). This completes the proof of the theorem.Via a simulation study (for brevity, we did not include it), it is revealed that a too large sample sizen≥10,000 is an indispensable requirement to get a good performance of the pivotal quantity Wr,s:n, which cannot be accomplished in numerous practical situations. In order to overcome this drawback, we suggest the following slightly modified pivotal quantity:(48)Wr,s:n∗=logXs:n/Xr:n∑i=1r−1ilogXi+1:n/Xi:n,which depends on all the observed lower extreme order statistics, X1:n,X2:n,…,Xr:n. The next theorem gives the exact asymptotic DF of Wr,s:n⋆.Theorem 4.
Under the same conditions of Theorem3, the asymptotic distribution of the pivotal quantity Wr,s:n⋆ is(49)Fr,s⋆w=1−1βr,s−r∑i=0s−r−1−1ir+is−r−1i1+r+iw−r−1,w>0.Proof.
First, note that(50)Wr,s:n∗=−logXr:n/Xs:n−∑i=1r−1ilogXi:n/Xi+1:n=−αlogXr:n/Xs:n−∑i=1r−1αilogXi:n/Xi+1:n=−logXr:n/Xs:nα∑i=1r−1ηi=nwT1η∗.
In view of Lemma1, RVs T1 and η⋆ are independent. Moreover, η⋆ is distributed as standard gamma distribution with shape parameter r−1, and the PDF of T1 is given by (46). Thus, we obtain(51)Fr,s⋆w=∫0∞∫0wzfT1tdtfη⋆zdz.
Therefore, the proof follows by the same argument of Theorem3, after some routine calculations.Remark 6.
Ifs=r+1, we get Fr,r+1⋆w=1−1+rw−r−1,w≥0. Again via the implemented simulation study in Section 4, it is detected that the performance of the pivotal quantity (48) and the behaviour of the DF (49) is computationally messy for large values of r or s−r except for the restrictive case s=r+1. The next theorem overcomes these issues.Theorem 5.
Letr be large enough in the sense given in Remark 1. Then, under the same conditions of Theorem 3, the asymptotic distribution of the pivotal quantity W˜r,s:n⋆=r−1Wr,s:n⋆ is(52)F˜r,sw=1−1βr,s−rIe−wr,s−r,w>0.Proof.
The pivotal quantityW˜r,s:n⋆ takes the form(53)W˜r,s:n⋆=−logXr:n/Xs:nα1/r−1∑i=1r−1ηi=−logXr:n/Xs:nαη¯.
Lemma1 implies that, for large n, η¯ is the sample mean of a random sample of size r−1 from the EXP(1) distribution. Thus, from the law of large number, η¯1. Consequently, by using Slutsky’s theorem and relation (46), relation (49) follows. This completes the proof of the theorem.


### 3.1. Asymptotic Prediction Whenα and x0 Are Unknown

We suggest the modified pivotal quantity,(54)Wr,s:n†=α⋆⋆logXs:n−X1:nXr:n−X1:n,for large enough r in the sense of Remark 1. Relying on relation (5), Lemma 2, estimator (27), and Remark 3, the asymptotic DF of Wr,s:n† may be approximated by F˜r−1,s−1w. Although there is no rigorous proof of this approximation (except for α=1), Table 5 shows that the obtained predictive interval has a good performance.Table 5
Asymptotic prediction intervals for the futureXs:n when α and the left endpoint are unknown based on the Weibull DF Ft=1−e−t−10/100α,t≥10i.e.,F∈DWα, for different values of α.
sL¯,U¯R95% CPL¯,U¯⋆95% CPL¯,U¯˜95% CPL¯,U¯T395% CPL¯,U¯T495% CPsr=12n=100α=113(22.73, 26.34)94.06(22.73, 26.50)92.10(22.73, 25.80)88.68(22.73, 27.34)94.65(22.73, 26.58)92.2714(22.73, 28.62)93.77(22.73, 28.96)91.02(22.73, 27.72)85.69(22.73, 30.86)94.68(22.73, 29.10)91.0615(22.73, 30.71)93.27(22.73, 31.25)89.48(22.73, 29.48)82.92(22.73, 34.49)94.48(22.73, 31.48)89.4516(22.73, 32.72)93.08(22.73, 33.50)88.08(22.73, 31.16)79.89(22.73, 38.34)94.49(22.73, 33.81)88.0517(22.73, 34.68)92.95(22.73, 35.73)86.87(22.73, 32.81)77.73(22.73, 42.46)94.25(22.73, 36.13)87.0818(22.73, 36.62)92.78(22.73, 37.97)85.51(22.73, 34.43)75.97(22.73, 46.89)94.29(22.73, 38.47)85.80sr=40n=500α=0.541(17.10, 18.25)94.06(17.10, 18.19)92.67(17.10, 18.14)91.69(17.10, 18.27)93.74(17.10, 18.22)93.1142(17.10, 18.98)94.52(17.10, 18.89)92.08(17.10, 18.79)90.61(17.10, 19.04)93.85(17.10, 18.93)92.6243(17.10, 19.66)94.39(17.10, 19.53)91.47(17.10, 19.39)89.73(17.10, 19.77)93.75(17.10, 19.60)92.2044(17.10, 20.33)94.19(17.10, 20.16)90.87(17.10, 19.98)88.75(17.10, 20.49)93.68(17.10, 20.25)91.7945(17.10, 20.99)94.05(17.10, 20.79)89.91(17.10, 20.57)87.57(17.10, 21.23)93.30(17.10, 20.90)90.90sr=40n=500α=1.541(200.20, 209.94)94.02(200.20, 211.26)95.24(200.20, 210.69)94.47(200.20, 210.04)93.56(200.20, 210.66)94.4642(200.20, 215.65)93.86(200.20, 217.79)95.41(200.20, 216.87)94.44(200.20, 216.00)93.19(200.20, 216.84)94.3543(200.20, 220.71)93.91(200.20, 223.60)95.22(200.20, 222.36)94.19(200.20, 221.37)93.12(200.20, 222.34)94.2344(200.20, 225.45)94.00(200.20, 229.07)94.94(200.20, 227.52)93.87200.20, 226.46)92.97(200.20, 227.52)93.7745(200.20, 229.99)93.72(200.20, 234.31)95.10(200.20, 232.47)93.55(200.20, 231.38)92.79(200.20, 232.48)93.7646(200.20, 234.37)93.49(200.20, 239.41)94.93(200.20, 237.26)93.46——(200.20, 237.30)93.4747(200.20, 238.65)93.56(200.20, 244.38)94.68(200.20, 241.94)93.01——(200.20, 242.00)93.2648(200.20, 242.82)93.21(200.20, 249.26)94.50(200.20, 246.52)92.78——(200.20, 246.62)92.9649(200.20, 246.92)93.10(200.20, 254.06)94.60(200.20, 251.03)92.68——(200.20, 251.16)92.9150(200.20, 250.94)93.08(200.20, 258.80)93.98(200.20, 255.47)92.09——(200.20, 255.64)92.55


## 3.1. Asymptotic Prediction Whenα and x0 Are Unknown

We suggest the modified pivotal quantity,(54)Wr,s:n†=α⋆⋆logXs:n−X1:nXr:n−X1:n,for large enough r in the sense of Remark 1. Relying on relation (5), Lemma 2, estimator (27), and Remark 3, the asymptotic DF of Wr,s:n† may be approximated by F˜r−1,s−1w. Although there is no rigorous proof of this approximation (except for α=1), Table 5 shows that the obtained predictive interval has a good performance.Table 5
Asymptotic prediction intervals for the futureXs:n when α and the left endpoint are unknown based on the Weibull DF Ft=1−e−t−10/100α,t≥10i.e.,F∈DWα, for different values of α.
sL¯,U¯R95% CPL¯,U¯⋆95% CPL¯,U¯˜95% CPL¯,U¯T395% CPL¯,U¯T495% CPsr=12n=100α=113(22.73, 26.34)94.06(22.73, 26.50)92.10(22.73, 25.80)88.68(22.73, 27.34)94.65(22.73, 26.58)92.2714(22.73, 28.62)93.77(22.73, 28.96)91.02(22.73, 27.72)85.69(22.73, 30.86)94.68(22.73, 29.10)91.0615(22.73, 30.71)93.27(22.73, 31.25)89.48(22.73, 29.48)82.92(22.73, 34.49)94.48(22.73, 31.48)89.4516(22.73, 32.72)93.08(22.73, 33.50)88.08(22.73, 31.16)79.89(22.73, 38.34)94.49(22.73, 33.81)88.0517(22.73, 34.68)92.95(22.73, 35.73)86.87(22.73, 32.81)77.73(22.73, 42.46)94.25(22.73, 36.13)87.0818(22.73, 36.62)92.78(22.73, 37.97)85.51(22.73, 34.43)75.97(22.73, 46.89)94.29(22.73, 38.47)85.80sr=40n=500α=0.541(17.10, 18.25)94.06(17.10, 18.19)92.67(17.10, 18.14)91.69(17.10, 18.27)93.74(17.10, 18.22)93.1142(17.10, 18.98)94.52(17.10, 18.89)92.08(17.10, 18.79)90.61(17.10, 19.04)93.85(17.10, 18.93)92.6243(17.10, 19.66)94.39(17.10, 19.53)91.47(17.10, 19.39)89.73(17.10, 19.77)93.75(17.10, 19.60)92.2044(17.10, 20.33)94.19(17.10, 20.16)90.87(17.10, 19.98)88.75(17.10, 20.49)93.68(17.10, 20.25)91.7945(17.10, 20.99)94.05(17.10, 20.79)89.91(17.10, 20.57)87.57(17.10, 21.23)93.30(17.10, 20.90)90.90sr=40n=500α=1.541(200.20, 209.94)94.02(200.20, 211.26)95.24(200.20, 210.69)94.47(200.20, 210.04)93.56(200.20, 210.66)94.4642(200.20, 215.65)93.86(200.20, 217.79)95.41(200.20, 216.87)94.44(200.20, 216.00)93.19(200.20, 216.84)94.3543(200.20, 220.71)93.91(200.20, 223.60)95.22(200.20, 222.36)94.19(200.20, 221.37)93.12(200.20, 222.34)94.2344(200.20, 225.45)94.00(200.20, 229.07)94.94(200.20, 227.52)93.87200.20, 226.46)92.97(200.20, 227.52)93.7745(200.20, 229.99)93.72(200.20, 234.31)95.10(200.20, 232.47)93.55(200.20, 231.38)92.79(200.20, 232.48)93.7646(200.20, 234.37)93.49(200.20, 239.41)94.93(200.20, 237.26)93.46——(200.20, 237.30)93.4747(200.20, 238.65)93.56(200.20, 244.38)94.68(200.20, 241.94)93.01——(200.20, 242.00)93.2648(200.20, 242.82)93.21(200.20, 249.26)94.50(200.20, 246.52)92.78——(200.20, 246.62)92.9649(200.20, 246.92)93.10(200.20, 254.06)94.60(200.20, 251.03)92.68——(200.20, 251.16)92.9150(200.20, 250.94)93.08(200.20, 258.80)93.98(200.20, 255.47)92.09——(200.20, 255.64)92.55


## 4. Simulation Study

In this section, comprehensive simulation studies are conducted to explore the efficiency of the obtained results in the preceding sections.


### 4.1. Estimation of the L-EVI

In Tables1–3, we apply Theorem 1 to obtain point estimates and 95% confidence intervals, denoted by L⋆,U⋆, with its coverage probabilities, denoted by 95%CP⋆, for α when x0 is known and relations (26) and (27) when x0 is unknown, based on 10,000 generated random samples of sizes n=100,500, and 1000 from the Pareto, exponentiated exponential, and Weibull distributions, respectively. Clearly, all the three distributions are in Weibull’s minimum domain of attraction, Wα. We compare our estimator α⋆, which is given by (23) if x0 is known and (27) if x0 is unknown, with the Smith and Weissman [13] estimator α˜, given by (33), via the MSE. When the left endpoint x0 is unknown, additional comparisons are performed with de Haan [12] estimator α^ based on the MSE, where α^−1=logr−1logXr:n−X3:n/X3:n−X2:n. All the estimates given in Tables 1–3 are actually the average values computed over these 10,000 replicates. Moreover, these replicates enable us to attach the corresponding MSE for each of those average estimates. Finally, in all tables, these average estimates will be denoted by the same symbols as the corresponding estimators.Based on Tables1–3, we note that MSEα⋆<MSEα˜<MSEα^ in almost all cases of the study. The larger MSE of the estimator α^ is expected since it depends only on three lower extreme order statistics. However, its performance is satisfactory for large values of r when α⩽1, but this is not the case if α>1. According to Pitman’s measure of closeness and MSE, α⋆ is better than α˜, whenever the assumptions of the EVT concerning r and n are satisfied, in the sense of Remark 1. When α⩽1, the performance of the estimator α⋆ is the best for small values of r (see Remark 7). On the contrary, if α>1 and the left endpoint x0 is unknown, the modified estimator α⋆⋆ defined by (32) is the best estimator followed by α˜, while the estimator α⋆ is almost the best if x0 is known.Remark 7.
The estimatorα⋆ is constructed based on the assumptions of the EVT that n is sufficiently large, while r remains constant with respect to n, i.e., Xr:n represents a lower extreme order statistic. To envision these assumptions practically, consider Table 1, as an example.(1)
Forn=100, the estimator α⋆ is better than α˜, for r≤15. The estimator α˜ is better (by a little bit value) than α⋆ if 20≤r≤25, while both the estimators become bad for r≥30.(2)
Forn=500, the estimator α⋆ is better than α˜, for r≤40; the estimator α˜ is better (by a little bit value) than α⋆ if 50≤r≤75, while both the estimators become bad for r≥100.(3)
Forn=1000, the estimator α⋆ is better than α˜, for r≤60; the estimator α˜ is better (by a little bit value) than α⋆ if 70≤r≤150, and both the estimators become bad for r≥200.


### 4.2. Asymptotic Prediction Intervals for Future Order Statistics

In this simulation study which is depicted in Table5, different asymptotic predictive intervals of Xs:n,s>r, are constructed and their corresponding converge probabilities are obtained. The simulation is implemented based on M=10,000 generated random samples of sizes n=100,500 from the three-parameter Weibull distribution with DF, Ft=1−e−t−10/100α,t≥10t≥10, for different values of α. In the first predictive interval, L¯,U¯R, which is constructed based on Theorem 2, it is assumed that the parameters α and x0 are known. The predictive interval L¯,U¯R can be considered as a reference predictive interval. For the second, L¯,U¯⋆, and the third, L¯,U¯˜, predictive intervals, we apply Theorem 2 after replacing α with α⋆ and α˜, respectively. In these cases, αi⋆ and αi˜ are estimated from the ith sample, i=1,2,…,M, their corresponding predictive intervals are constructed, and then the average limits of the predictive intervals as well as the coverage probabilities are obtained. The fourth, L¯,U¯T3, and fifth, L¯,U¯T4, predictive intervals are constructed using Theorems 4 and 5, respectively, which are dependent only on the first r lower order statistics without estimating α. Finally, the empty cells in Table 5 concerning the predictive interval L¯,U¯T3, for r=40 and s≥46 are because we are unable to obtain the quantiles corresponding to the DF (3.8) by Mathematica 12.2. (see Remark 6).Table5 reveals that all the predictive intervals are efficient and comparable, in the sense that there is no significant preference among them when x0 is known. However, when both x0 and α are unknown, the predictive pivotal quantity (54) is preferable, since it does not required the knowledge of α or x0, with no restrictions on r and s rather than they still lower extremes.Remark 8.
All the computations in this paper are implemented by Mathematica 12.2.


## 4.1. Estimation of the L-EVI

In Tables1–3, we apply Theorem 1 to obtain point estimates and 95% confidence intervals, denoted by L⋆,U⋆, with its coverage probabilities, denoted by 95%CP⋆, for α when x0 is known and relations (26) and (27) when x0 is unknown, based on 10,000 generated random samples of sizes n=100,500, and 1000 from the Pareto, exponentiated exponential, and Weibull distributions, respectively. Clearly, all the three distributions are in Weibull’s minimum domain of attraction, Wα. We compare our estimator α⋆, which is given by (23) if x0 is known and (27) if x0 is unknown, with the Smith and Weissman [13] estimator α˜, given by (33), via the MSE. When the left endpoint x0 is unknown, additional comparisons are performed with de Haan [12] estimator α^ based on the MSE, where α^−1=logr−1logXr:n−X3:n/X3:n−X2:n. All the estimates given in Tables 1–3 are actually the average values computed over these 10,000 replicates. Moreover, these replicates enable us to attach the corresponding MSE for each of those average estimates. Finally, in all tables, these average estimates will be denoted by the same symbols as the corresponding estimators.Based on Tables1–3, we note that MSEα⋆<MSEα˜<MSEα^ in almost all cases of the study. The larger MSE of the estimator α^ is expected since it depends only on three lower extreme order statistics. However, its performance is satisfactory for large values of r when α⩽1, but this is not the case if α>1. According to Pitman’s measure of closeness and MSE, α⋆ is better than α˜, whenever the assumptions of the EVT concerning r and n are satisfied, in the sense of Remark 1. When α⩽1, the performance of the estimator α⋆ is the best for small values of r (see Remark 7). On the contrary, if α>1 and the left endpoint x0 is unknown, the modified estimator α⋆⋆ defined by (32) is the best estimator followed by α˜, while the estimator α⋆ is almost the best if x0 is known.Remark 7.
The estimatorα⋆ is constructed based on the assumptions of the EVT that n is sufficiently large, while r remains constant with respect to n, i.e., Xr:n represents a lower extreme order statistic. To envision these assumptions practically, consider Table 1, as an example.(1)
Forn=100, the estimator α⋆ is better than α˜, for r≤15. The estimator α˜ is better (by a little bit value) than α⋆ if 20≤r≤25, while both the estimators become bad for r≥30.(2)
Forn=500, the estimator α⋆ is better than α˜, for r≤40; the estimator α˜ is better (by a little bit value) than α⋆ if 50≤r≤75, while both the estimators become bad for r≥100.(3)
Forn=1000, the estimator α⋆ is better than α˜, for r≤60; the estimator α˜ is better (by a little bit value) than α⋆ if 70≤r≤150, and both the estimators become bad for r≥200.


## 4.2. Asymptotic Prediction Intervals for Future Order Statistics

In this simulation study which is depicted in Table5, different asymptotic predictive intervals of Xs:n,s>r, are constructed and their corresponding converge probabilities are obtained. The simulation is implemented based on M=10,000 generated random samples of sizes n=100,500 from the three-parameter Weibull distribution with DF, Ft=1−e−t−10/100α,t≥10t≥10, for different values of α. In the first predictive interval, L¯,U¯R, which is constructed based on Theorem 2, it is assumed that the parameters α and x0 are known. The predictive interval L¯,U¯R can be considered as a reference predictive interval. For the second, L¯,U¯⋆, and the third, L¯,U¯˜, predictive intervals, we apply Theorem 2 after replacing α with α⋆ and α˜, respectively. In these cases, αi⋆ and αi˜ are estimated from the ith sample, i=1,2,…,M, their corresponding predictive intervals are constructed, and then the average limits of the predictive intervals as well as the coverage probabilities are obtained. The fourth, L¯,U¯T3, and fifth, L¯,U¯T4, predictive intervals are constructed using Theorems 4 and 5, respectively, which are dependent only on the first r lower order statistics without estimating α. Finally, the empty cells in Table 5 concerning the predictive interval L¯,U¯T3, for r=40 and s≥46 are because we are unable to obtain the quantiles corresponding to the DF (3.8) by Mathematica 12.2. (see Remark 6).Table5 reveals that all the predictive intervals are efficient and comparable, in the sense that there is no significant preference among them when x0 is known. However, when both x0 and α are unknown, the predictive pivotal quantity (54) is preferable, since it does not required the knowledge of α or x0, with no restrictions on r and s rather than they still lower extremes.Remark 8.
All the computations in this paper are implemented by Mathematica 12.2.


## 5. Applications

In this section, two real-life datasets are analyzed to explore the applicability of the obtained results.Example 1.
(failure times for a piece of electronic equipment). The following dataset represents the failure times (in hours) of 107 units of a piece of electronic equipment. The dataset was reported in Hand et al. [44] on p. 394, and it was analyzed by many authors. The ordered dataset is 1, 1.2, 1.3, 2, 2.4, 2.9, 3, 3.1, 3.3, 3.8, 4.3, 4.6, 4.7, 4.8, 5.2, 5.4, 5.9, 6.4, 6.8, 6.9, 7.2, 7.9, 8.3, 8.7, 9.2, 9.8, 10.2, 10.4, 11.9, 13.8, 14.4, 15.6, 16.2, 17, 17.5, 19.2, 28.1, 28.2, 29, 29.9, 30.6, 32.4, 33, 35, 35.3, 36.1, 40.1, 42.8, 43.7, 44.5, 50.4, 51.2, 52, 53.3, 54.2, 55.6, 56.4, 58.3, 60.2, 63.7, 64.6, 65.3, 66.2, 70.1, 71, 75.1, 75.6, 78.4, 79.2, 84.1, 86, 87.9, 88.4, 89.9, 90.8, 91.1, 91.5, 92.1, 97.9, 100.8, 102.6, 103.2, 104, 104.3, 105, 105.8, 106.5, 110.7, 112.6, 113.5, 114.8, 115.1, 117.4, 118.3, 119.7, 120.6, 121, 122.9, 123.3, 124.5, 125.8, 126.6, 127.7, 128.4, 129.2, 129.5, 129.9.The following three steps are followed in order to apply the proposed method to the above data.Step 1
: initially, letx0=0. Select r which gives the best fit of U1∗,U2∗,…,Ur−2∗ to U0,1, where U1∗,U2∗,…,Ur−2∗ are defined in Corollary 1. Based on the K-S test, it is noted that the optimal fit occurs at r=16 with a test statistic observed value of 0.1344 and a corresponding p value of 0.933. Different estimates endorse the positive value of α (α⋆=1.545 and α˜=1.648). Consequently, the assumption F∈DWα is justified.Step 2
: in order to improve our performance, we assume thatx0 is unknown. In this case, the optimal choice is r=21, after replacing ηi in Corollary 1 with ξi defined in Lemma 2. For r=21, we get α˜=1.144 and α⋆=1.035, and the observed values of the updated Ui∗,i=1,2,…,18, are 0.022, 0.153, 0.208, 0.275, 0.289, 0.305, 0.339, 0.425, 0.506, 0.553, 0.569, 0.587, 0.658, 0.693, 0.781, 0.866, 0.932, 0.949. The uniform distribution U0,1 fits the previous data very well, where the observed value of the K-S test statistic is 0.1081 and the p value is 0.969. It is worth mentioning that, as r increases, the accuracy of the fitting decreases (e.g., at r=30, the p value is 0.2108, and for r=40, the p value is 0.008).Step 3
: now, assume that the first order statistics,x1:107,…,xr:107, for r=16,21, and 26, are observed and the next xs:107 order statistics with s=r+1,…,30 ought to be predicted. Five different predictive intervals of xs:107 are obtained, and the results are presented in Table 6. The behaviour of α˜ and α⋆ for different values of r motivates us to assume that the first r order statistics follow a DF F in the Weibull domain of attraction with α=1 (i.e., F∈DW1). Since the first three predictive intervals depend on x0, one may choose x0=0.8 for comparison purposes only (the choice of x0 is not too sensitive for determining the predictive intervals).Table 6
Ninety five percentage predictive intervals of the failure timesxs:107, for s=r+1,…,30, relying on x1:107,…,xr:107 when r=16,21, and 26.
L,UsRCPL,U⋆CPL,U˜sCPL,UsT3CPL,UT4CPXs+1sr=1617(5.4, 6.35)1(5.4, 6.25)1(5.4, 6.14)1(5.4, 6.39)1(5.4, 6.28)16.418(5.4, 6.93)1(5.4, 6.78)1(5.4, 6.58)1(5.4, 7.05)1(5.4, 6.81)16.819(5.4, 7.47)1(5.4, 7.25)1(5.4, 6.98)1(5.4, 7.68)1(5.4, 7.30)16.920(5.4, 7.98)1(5.4, 7.70)1(5.4, 7.35)1(5.4, 8.31)1(5.4,7.76)17.221(5.4,8.48)1(5.4,8.13)1(5.4,7.71)1(5.4,8.94)1(5.4,8.21)17.922(5.4,8.97)1(5.4,8.55)1(5.4,8.06)1(5.4,9.58)1(5.4,8.65)18.323(5.4,9.45)1(5.4,8.97)1(5.4,8.40)1(5.4,10.22)1(5.4,9.08)18.724(5.4,9.93)1(5.4,9.38)1(5.4,8.74)1(5.4,10.88)1(5.4,9.50)19.225(5.4,10.41)1(5.4,9.79)1(5.4,9.07)0(5.4,11.55)1(5.4,9.92)19.826(5.4,10.88)1(5.4,10.19)1(5.4,9.39)0(5.4,12.23)1(5.4,10.34)110.227(5.4,11.36)1(5.4,10.59)1(5.4,9.71)0(5.4,12.91)1(5.4,10.75)110.428(5.4,11.83)1(5.4,10.98)1(5.4,10.02)0(5.4,13.61)1(5.4,11.16)111.929(5.4,12.30)1(5.4,11.38)0(5.4,10.33)0(5.4,14.32)1(5.4,11.56)013.830(5.4,12.76)0(5.4,11.77)0(5.4,10.64)0(5.4,15.05)1(5.4,11.97)014.4sr=2122(7.2,8.18)1(7.2,8.15)1(7.2,8.05)1(7.2,8.25)1(7.2,8.17)18.323(7.2,8.78)1(7.2,8.72)1(7.2,8.56)1(7.2,8.94)1(7.2,8.76)18.724(7.2,9.32)1(7.2,9.24)1(7.2,9.02)1(7.2,9.58)1(7.2,9.29)19.225(7.2,9.84)1(7.2,9.74)1(7.2,9.46)1(7.2,10.21)1(7.2,9.80)19.826(7.2,10.35)1(7.2,10.22)1(7.2,9.88)1(7.2,10.84)1(7.2,10.29)110.227(7.2,10.84)1(7.2,10.69)1(7.2,10.29)1(7.2,11.46)1(7.2,10.77)110.428(7.2,11.33)1(7.2,11.15)1(7.2,10.69)1(7.2,12.10)1(7.2,11.25)111.929(7.2,11.81)0(7.2,11.61)0(7.2,11.08)0(7.2,12.74)1(7.2,11.72)013.830(7.2,12.29)0(7.2,12.06)0(7.2,11.47)0(7.2,13.38)0(7.2,12.18)014.4sr=2627(9.8,10.90)1(9.8,11.00)1(9.8,10.90)1(9.8,11.10)1(9.8,11.02)110.428(9.8,11.56)1(9.8,11.73)1(9.8,11.56)1(9.8,11.94)1(9.8,11.77)111.929(9.8,12.17)1(9.8,12.39)1(9.8,12.17)1(9.8,12.73)1(9.8,12.44)113.830(9.8,12.74)0(9.8,13.02)0(9.8,12.74)0(9.8,13.49)0(9.8,13.09)014.4CP=1 means the predictive interval contains Xs:107.Example 2.
(breaking strengths of single carbon fibers for gauge lengths of 10 mm). The following dataset, which is due to Badar and Priest [45], represents the strength measured in GPA for single carbon fibers and impregnated 1000-carbon fiber tows. Single fibers were tested under tension at gauge lengths of 10 mm. The gauge lengths of 10 mm are 1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020.By following the same steps as in Example1, it is noted that the Weibull domain of attraction is applicable for this data. For r=12, the observed values of Ui∗,i=1,2,…,12, follow U0,1. The observed value of the K-S test statistic is 0.1401 and the corresponding p value is 0.9836. Moreover, the estimates of α when r=12 are α˜=3.501 and α⋆=2.918. Different predictive intervals for the future order statistic Xs:63 for specific values of s are given in Table 7 based on the first r order statistics, r=12,15, and 18.Table 7
Ninety five percentage predictive intervals ofXs:63, for s=r+1,…,24, based on X1:63,…,Xr:63 when r=12,15, and 18 (with the choice x0=1.67).
L,URCPL,U⋆CPL,U˜CPL,UT3CPL,UT4CPXs+1sr=1213(2.474,2.539)1(2.474,2.546)1(2.474,2.533)1(2.474,2.540)1(2.474,2.530)12.52214(2.474,2.575)1(2.474,2.586)1(2.474,2.566)1(2.474,2.581)1(2.474,2.561)12.52515(2.474,2.606)1(2.474,2.620)1(2.474,2.594)1(2.474,2.618)1(2.474,2.588)12.53216(2.474,2.633)1(2.474,2.651)1(2.474,2.619)1(2.474,2.652)1(2.474,2.611)12.57517(2.474,2.658)1(2.474,2.679)1(2.474,2.641)1(2.474,2.685)1(2.474,2.633)12.61418(2.474,2.682)1(2.474,2.705)1(2.474,2.663)1(2.474,2.717)1(2.474,2.653)12.61619(2.474,2.704)1(2.474,2.730)1(2.474,2.683)1(2.474,2.748)1(2.474,2.673)12.61820(2.474,2.725)1(2.474,2.754)1(2.474,2.701)1(2.474,2.778)1(2.474,2.691)12.62421(2.474,2.745)1(2.474,2.777)1(2.474,2.719)1(2.474,2.807)1(2.474,2.708)12.65922(2.474,2.764)1(2.474,2.798)1(2.474,2.736)1(2.474,2.835)1(2.474,2.725)12.67523(2.474,2.783)1(2.474,2.819)1(2.474,2.753)1(2.474,2.863)1(2.474,2.741)12.73824(2.474,2.800)1(2.474,2.839)1(2.474,2.769)1(2.474,2.891)1(2.474,2.756)12.74027(2.474,2.850)0(2.474,2.896)0(2.474,2.813)0(2.474,2.970)1(2.474,2.799)02.928sr=1516(2.525,2.580)1(2.525,2.585)1(2.525,2.576)1(2.525,2.578)1(2.525,2.572)12.57517(2.525,2.611)1(2.525,2.618)1(2.525,2.605)1(2.525,2.610)1(2.525,2.598)12.61418(2.525,2.637)1(2.525,2.647)1(2.525,2.630)1(2.525,2.639)1(2.525,2.620)12.61619(2.525,2.661)1(2.525,2.673)1(2.525,2.652)1(2.525,2.666)1(2.525,2.640)12.61820(2.525,2.683)1(2.525,2.697)1(2.525,2.672)1(2.525,2.691)1(2.525,2.659)12.62421(2.525,2.704)1(2.525,2.719)1(2.525,2.691)1(2.525,2.716)1(2.525,2.677)12.65922(2.525,2.724)1(2.525,2.741)1(2.525,2.709)1(2.525,2.739)1(2.525,2.693)12.67523(2.525,2.743)1(2.525,2.761)1(2.525,2.727)1(2.525,2.763)1(2.525,2.709)12.73824(2.525,2.761)1(2.525,2.781)1(2.525,2.743)1(2.525,2.785)1(2.525,2.724)02.74027(2.525,2.810)0(2.525,2.836)0(2.525,2.789)0(2.525,2.850)0(2.525,2.767)02.928sr=1819(2.614,2.675)1(2.614,2.679)1(2.614,2.672)1(2.614,2.672)1(2.614,2.666)12.61820(2.614,2.710)1(2.614,2.717)1(2.614,2.705)1(2.614,2.707)1(2.614,2.696)12.62421(2.614,2.740)1(2.614,2.749)1(2.614,2.733)1(2.614,2.739)1(2.614,2.722)12.65922(2.614,2.768)1(2.614,2.778)1(2.614,2.759)1(2.614,2.768)1(2.614,2.745)12.67523(2.614,2.794)1(2.614,2.806)1(2.614,2.783)1(2.614,2.797)1(2.614,2.767)12.73824(2.614,2.818)1(2.614,2.832)1(2.614,2.806)1(2.614,2.824)1(2.614,2.788)12.74027(2.614,2.885)0(2.614,2.904)0(2.614,2.868)0(2.614,2.902)0(2.614,2.846)02.928CP=1 means the predictive interval contains Xs:63


## 6. Conclusion

In this study, asymptotic inferential methods based on the first lower extreme order statistics, which are attracted to the Weibull distribution of known or unknown shape parameter, were proposed. Some interesting asymptotic distributional properties concerning the DFs belonging to the domain of attraction of the Weibull distribution were revealed. Several estimators of the L-EVI were suggested, as well as their corresponding asymptotic confidence intervals. Moreover, different asymptotic predictive intervals of future unobserved lower extreme order statistics were constructed. Comprehensive simulation studies were carried out for some lifetime distributions to investigate the efficiency of the suggested methods. Finally, two real-life datasets were analyzed for illustrative purposes.The real data analysis demonstrates that the prediction results are not overly sensitive to selectingr and s. That is, there is always a moderate range for which our methods can be applied, provided that Xs:n is a lower extreme order statistic. For example, in Example 1, we should have s<30, but s⩾35 is not acceptable. On the contrary, the results of this paper have a significant practical benefit because, for any fixed censoring plan, the increasing number of the items installed in any life test leads to the decreasing of the completion time of that life test. Therefore, the experimenter may put a large number of identical units into a life test, which will lead to observing a suitable number of failures in a short time. Consequently, additional failure times can now be predicted by using the proposed methods in Section 3. This enables the experimenter to terminate the experiment at a reasonable time to use the surviving units for other purposes.



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