Characterization of Exponential Distribution Through Normalized Spacing of Generalized Order Statistics

M. J. S. Khan; S. Iqrar; M. Faizan

doi:10.2991/jsta.d.190818.004

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 18, Issue 3, September 2019, Pages 303 - 308

Characterization of Exponential Distribution Through Normalized Spacing of Generalized Order Statistics

Authors

M. J. S. Khan^*, S. Iqrar, M. Faizan

Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh-202 002, India

^*Corresponding author. Email: jahangirskhan@gmail.com

Corresponding Author

M. J. S. Khan

Received 19 October 2017, Accepted 24 December 2018, Available Online 2 September 2019.

DOI: 10.2991/jsta.d.190818.004 How to use a DOI?
Keywords: Continuous distribution; Generalized order statistics; Meijer's G-function; Characterization
Abstract: In this paper, exponential distribution is characterized by normalized spacing of generalized order statistics (gos) using Meijer's G-function. While the necessary part of the theorem was given by U. Kamps, E. Cramer, Statistics. 35 (2001), 269–280, we have given an easy proof of sufficient part in this paper. This paper contains the result of characterization of exponential distribution through normalized spacing of order statistics, sequential order statistics, progressive type II censoring and record values. Also, by simulation study, we have shown that the confidence interval based on upper records is shorter in length than asymptotic confidence interval constructed by maximum likelihood estimate.
Copyright: © 2019 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Ordered random variables like order statistics, record values, sequential order statistics and progressive type II censored order statistics frequently arise in real life situations. For application of ordered random variables, one may refer to [1], to [2] for order statistics, to [3] and [4] for record values and to [5] for progressive type II censored order statistics and reference cited therein. The concept of generalized order statistics (gos) was first introduced by [6]. Goss includes various important models related to ordered random variables like order statistics, record values, sequential order statistics, order statistics with non-integral sample size and progressive type II censored order statistics etc.

Let X1,X2,…,Xn be a sequence of independent and identically distributed (iid) random variables (rv) with the distribution function (df) F(x) and the probability density function (pdf) f(x). Let n∈N, k>0, m~=(m1,m2,…,mn−1)∈Rn−1 and Mr=∑j=rn−1mj, such that γr=k+n−r+Mr>0 for all r∈{1,2,…,n−1}, then X(1,n,m~,k),X(2,n,m~,k),⋯,X(n,n,m~,k) are said to be the gos if their joint pdf is given by

(1)k∏j=1n−1γj∏i=1n−1[1−F (xi)]mif(xi)[1−F (xn)]k−1f(xn)

on the cone F−1(0+)<x1≤x2≤⋯≤xn<F−1(1) of Rn. By adjusting the parameters of gos, one may get almost all important models related to ordered random variables. For example, at m=0, k=1 i.e. γi=n−i+1, we get ordinary order statistics. At m=−1, k∈N i.e. γi=k, we shall get kth record values. Similarly sequential order statistics (γi=(n−i+1)βi; β1,β2,⋯,βn>0), order statistics with non-integral sample size (γi=β−i+1;β>0), Pfeifer record values (γi=βi;β1,β2,⋯,βn>0) and progressive type II censored order statistics (N,n∈N,N=n+∑j=1nRj,Rj∈N0 and γj=N−∑ν=1j−1Rν−j+1,1≤j≤n) can be obtained as specific cases of gos.

Let PF stands for the probability measure on R determined by F(x). Define α=inf{x∈R: F(x)>0} and β=sup{x∈R: F(x)<1}, then the pdf of X(r,n,m~,k) with respect to a measure PF can be expressed as (see [7]):

(2)fr(x)=cr−1Gr[F¯(x)|γ1,…,γr]I(α,β)(x),

where cr−1=∏j=1rγj,F¯(x)=1−F(x) and IA denotes the indicator function of the set A. Further Gr(x)=Gr,rr,0(x|γ1,…,γr)=Gr,rr,0xγ1−1,…,γr−1γ1,…,γr is the particular Meijer's G-function defined by

(3)Gr,rr,0xγ1−1,…,γr−1γ1,…,γr=12πi∫Lx−z∏j=1r(γi−1−z)dz,| x |<1

and L is an appropriately chosen contour of integration. For the definition of G-function and its numerous properties and applications, one may refer to [8](Chapter 2). The joint PF⊗PF density of X(r,n,m~,k) and X(s,n,m~,k), 1≤r<s≤n is given by [7]

(4)fr,s(x,y)=cs−1Gs−rF¯(y)F¯(x)|γr+1,…,γsGr[F¯(x)|γ1,…,γr]F¯(x) IA1(x,y),

where the set A1 is defined as A1={(x,y)∈R2:α<x<y<β}.

Suppose X1,X2,⋯,Xn are iid random variables from exponential distribution with scale parameter θ, i.e., F(x)=1−e−xθ,x>0, θ>0 and X1:n≤X2:n≤⋯≤Xn:n are the corresponding orders statistics. The normalized spacing of order statistics (when the random variables are iid and from exponential distribution) was introduced by [9] as: D1:n=nX1:n and Dr:n=(n−r+1)(Xr:n−Xr−1:n), 2≤r≤n. [9] proved that Dr:n(1≤r≤n) are iid from exponential distribution with scale parameter θ. Utilizing the above transformation, it can be shown that Xs:n−Xr:n and Xs−r:n−r are identically distributed across all 1≤r<s≤n. Based on order statistics, the characterization of exponential distribution through normalized spacing was discussed by [10–13] and [14]. The related results of characterization of exponential distribution for record values through spacing was established by [11,13,15,16] and [14]. [17] has shown that using linear function of spacings instead of linear function of order statistics is more suitable for theoretical and applied purposes. Normalized spacing for gos was given by [6] (p. 81). Suppose X1,X2,⋯,Xn be the iid random variables from exponential distribution and X(1,n,m~,k),X(2,n,m~,k)⋯,X(n,n,m~,k) be the corresponding goss. [6] has shown that D(1,n,m~,k)=γ1X(1,n,m~,k) and D(r,n,m~,k)=γr[X(r,n,m~,k)−X(r−1,n,m~,k)], 2≤r≤n are iid from exponential distribution. Further, [18] and [7] characterized the exponential distribution using normalized spacing of goss based on increasing failure rate and decreasing failure rate distributions. [19] characterized Weibull and Uniform distributions using record values.

In this paper, we have extended the result of [16] and [14] and characterized the exponential distribution using normalized spacing of goss. While the necessary part of the main theorem was given by [20], in this paper, we have given a simple and easy proof for the sufficient part of the theorem. The paper is divided into four sections. In Section 2, based on gos, we have characterized the exponential distribution. Section 3 contains a simulation study where we have shown that confidence interval based on upper records is shorter in length than asymptotic confidence interval constructed by maximum likelihood estimate. Concluding remark focusing on the use of the result given in Section 2, is contained in Section 4.

2 CHARACTERIZATION OF EXPONENTIAL DISTRIBUTION

In this section, exponential distribution has been characterized through the normalized spacing based on goss.

The following two Lemmas are given, which have been used in the proof of the theorem. Proof of the Lemmas are straightforward and hence omitted.

Lemma 2.1.

Let X(r,n,m~,k) be the rth gos with pdf defined in (2), then

(5)1cr−1=∫αβGr[F¯(x)|γ1,…,γr]f(x)dx.

Lemma 2.2.

Let X(s−r,n−r,μ~,k) andX(s−r−1,n−r,μ~,k) be the two adjacent gos and μ~=(mr+1,…,mn−1)ϵRn−r−1, then

(6)F¯s−r,n−r,μ~,k(u)−F¯s−r−1,n−r,μ~,k(u)=cs−2cr−1Gs−r[F¯(u)|γr+1,…,γs]F¯(u).

Theorem 2.1.

Let X1,X2,…,Xn be n non-negative iid random variables with absolutely continuous df (with respect to Lebesgue measure) F(x) andpdf f(x), where F(x) is strictly increasing over the support (0,∞), then for 1≤r<s−1<n,

(7)X(l,n,m~,k)−X(r,n,m~,k)=dX(l−r,n,μ~,k), l=s−1,s

if and only if

(8)f(x)=1−e−xθ, θ>0.

Proof.

The necessary part was established by [20] as follows: X(l,n,m~,k)−X(r,n,m~,k)=∑j=r+1l(zjγj), l=s−1, s, where zj∼exp(θ).

This implies that X(l,n,m~,k)−X(r,n,m~,k)=dX(l−r,n−r,μ~,k),l=s−1,s, where the symbol =d represents equal in distribution.

To prove the sufficient part, we have for any positive and finite u

(9)P[X(s,n,m~,k)−X(r,n,m~,k)≥u]=cs−1∫0∞∫x+u∞Gs−r[F¯(y)F¯(x)|γr+1,…,γs]f(y)F¯(x)Gr[F¯(x)|γ1,…,γr]f(x)dydx.

Let

A(x,y)=Gs−rF¯(y)F¯(x)γr+1,…,γsf(y)F¯(x).

Integrating A(x,y) w.r.t, y over (x+u,∞), we have

(10)∫x+u∞A(x,y)dy=∫x+u∞Gs−rF¯(y)F¯(x)γr+1,…,γsf(y)F¯(x)dy =−Gs−rF¯(y)F¯(x)γr+1,…,γsF¯(y)F¯(x)(x+u)∞+∫x+u∞∂∂yGs−rF¯(y)F¯(x)γr+1,…,γsF¯(y)F¯(x)dy.

Now, we know from [8] (p. 130), that

(11)xaGr(x|γ1,…,γr)=Gr(x|γ1+a,…,γr+a), aϵR.

Using (11) in (10), we have

(12)A(x,y)=−Gs−rF¯(y)F¯(x)γr+1∗,…,γs∗(x+u)∞+∫x+u∞∂∂yGs−rF¯(y)F¯(x)γr+1,…,γsF¯(y)F¯(x)dy,

where γr+i∗=γr+i+1,i=1,2,…,s−r.

Let γ1:r≤...≤γr:r denote the ordered parameters of γ1,...,γr,r≥2, then

(13)limx→1−Gr(x|γ1,…,γr)=limx→0+Gr(x|γ1,…,γr)=0, if γ1:r>1.

[21] and

(14)ddxGr(x|γ1,…,γr)=1x[(γr−1)Gr(x|γ1,…,γr)−Gr−1(x|γ1,…,γr−1)].

[8] (p. 94)

Using (13) and (14) in (12), we get

A(x,y)=Gs−rF¯(x+u)F¯(x)γr+1∗,…,γs∗+∫x+u∞Gs−r−1F¯(y)F¯(x)γr+1,…,γsf(y)F¯(x)dy −(γs−1)∫x+u∞Gs−rF¯(y)F¯(x)γr+1,…,γsf(y)F¯(x)dy.

This implies that

(15)=cs−1∫0∞Gs−rF¯(x+u)F¯(x)γr+1∗,…,γs∗Gr[F¯(x)|γ1,…,γr]f(x)dx +γsP[X(s−1,n,m~,k)−X(r,n,m~,k)≥u]−(γs−1)P[X(s,n,m~,k)−X(r,n,m~,k)≥u]

that is

(16)P[X(s,n,m~,k)−X(r,n,m~,k)≥u]−P[X(s−1,n,m~,k)−X(r,n,m~,k)≥u] = cs−2∫0∞Gs−rF¯(x+u)F¯(x)|γr+1∗,…,γs∗Gr[F¯(x)|γ1,…,γr]f(x)dx.

Now in view of Lemma 2.1 together with Lemma 2.2, we have

(17)P[Xs−r,n−r,μ~,k]−P[X(s−r−1,n−r,μ~,k)≥u] = cs−2Gs−r[F¯(u)|γr+1∗,…,γs∗]∫0∞Gr[F¯(x)|γ1,…,γr]f(x)dx.

(16) and (17) will be equal if and only if

(18)cs−2∫0∞Gs−rF¯(x+u)F¯(x)|γr+1∗,…,γs∗−Gs−r[F¯(u)|γr+1∗,…,γs∗] ×Gr[F¯(x)|γ1,…,γr]f(x)dx=0.

Since both Gr[F¯(x)|γ1,…,γr] and f(x) are positive, using the generalization of Muntz-Swartz theorem (see [22]), the above integral will be zero only if

(19)Gs−rF¯(x+u)F¯(x)|γr+1∗,…,γs∗−Gs−r[F¯(u)|γr+1∗,…,γs∗]=0.

For s>r+1, the function γr+1:s∗>1, Gs−rF¯(x+u)F¯(x)|γr+1∗,…,γs∗ is unimodal, strictly increasing up to mode and then strictly decreasing. Further, the value of mode only depends upon the parameter γr+1∗,…,γs∗, hence for the functions

Gs−rF¯(x+u)F¯(x)|γr+1∗,…,γs∗

and

Gs−rF¯(u)|γr+1∗,…,γs∗

mode occurs at the same point (see [21]; Theorem 2.1). The value of Gs−rF¯(x+u)F¯(x)|γr+1∗,…,γs∗ and Gs−rF¯(u)|γr+1∗,…,γs∗ are equal only at the mode. At mode, the random variable X satisfy F¯(x+u)F¯(x)=F¯(u), which imply that X satisfies the memoryless property. Only continuous distribution over the support (0,∞), which satisfies the memoryless property is exponential distribution and hence the theorem.

3 SIMULATION STUDY

Here we have followed the procedure adopted by [19] and carried out the simulation to construct the confidence interval for scale parameter θ of exponential distribution.

Let X1,...,Xn is a random sample from pdf f(x)=1θe−xθ, x>0,θ>0. Based on maximum likelihood estimator of exponential distribution, 100(1−α)%,0<α<1, asymptotic confidence interval (CIAsymptotic) for θ is given by

(20)CIAsymptotic=X¯1+Zα∕2n,X¯1−Zα∕2n

where X¯ is sample mean of random variables and Zα∕2 is a positive constant satisfying the relation Φ(Zα∕2)−Φ(−Zα∕2)=1−α and Φ(.) is the cumulative distribution function of the standard normal distribution. Using Theorem 2.1, for record statistics, we have XU(n)−XU(m) d__ XU(n−m). As Y=XU(n−m)∼G(n−m,θ), based on Y, a 100(1−α)% confidence interval for θ is given by

(21)CI=2Yχ2n,1−α∕22,2Yχ2n,α∕22.

To construct these confidence intervals we simulate 10,000 samples of size n from (8) for n=4,5,...,100, m=1 and θ=0.2,0.5,1,2. For each sample, we calculate the limits of confidence interval given in (20) and (21). The plot of coverage lengths versus n=4,5,...,100, is shown in Figs. 1–4. From the figures, it can be seen that the confidence interval obtained from (21) is smaller than the confidence interval obtained from (20) for n≤60. Thus, we get an improved estimator based on XU(n−m).

4 CONCLUDING REMARK

In real life, a statistician is often interested in guessing the probability distribution from which the true data is obtained. Characterization problem is a theoretical approach to obtain the distribution function if certain conditions are fulfilled. A probability distribution can be characterized in many ways and the method under study here is one of them. Further this result may be used to ease statistical computation. For example, if we are computing moments of difference between two gos (order statistics, records, sequential order statistics, progressive type II censored order statistics), when the random variables X1,X2,…,Xn follow exponential distribution, then we may utilize the result of Theorem 2.1. This result may also be utilized for calculating the sample quasi range, when the random variables X1,...,Xn are iid having pdf f(x)=1θe−xθ, x>0, θ>0. Various statistical properties of quasi range can be established using Theorem 2.1. Sample quasi range is used in estimating the population standard deviation if the sample size n is small (see [23]). [24] used the sample quasi range in setting confidence intervals for the population standard deviation. Further, [25] and [26] have used the sample quasi range in estimating the scale parameter for a location scale family of distribution, if the parent distribution is symmetric about the location parameter. Sample quasi range and range are also extensively used in quality control.

REFERENCES

1.N. Balakrishnan and A.C. Cohen, Order Statsitics and Inference, Academic Press, Boston, 1991.

2.H. A. David and H. N. Nagaraja, Order Statistics, third ed., John Wiley & Sons, New York, 2003.

3.B. C. Arnold, N. Balakrishnan, and H. N. Nagaraja, Records, John Wiley & Sons, New York, 1998.

4.M. Ahsanullah, Record Values - Theory and Applications, University Press of America, Lanham, 2004.

5.N. Balakrishnan and R. Aggarwala, Progressive Censoring, Birkhäuser, Boston, 2000.

6.U. Kamps, A Concept of Generalized Order Statistics, B.G. Teubner Stuttgart, Germany, 1995.

7.E. Cramer, U. Kamps, and M.Z. Raqab, Appl. Math., Vol. 30, 2003, pp. 257-265.

8.A. M. Mathai, A Handbook of Generalized Special Functions for Statistical and Physical Sciences, Oxford Science Publications, New York, 1993.

9.P.V. Sukhatme, Ann. Eugen., Vol. 8, 1937, pp. 52-56.

10.M. Ahsanullah, Stat. Hefte., Vol. 22, 1981, pp. 316-320.

11.M. Iwińska, Fasc. Math., Vol. 16, 1986, pp. 101-107.

12.U. Gather, Probab. Lett., Vol. 7, 1988, pp. 93-96.

13.L. Gajek and U. Gather, J. Husler and R.D. Reiss (editors), Extreme Value Theory, Springer, Berlin, 1989, pp. 114-124.

14.M.J.S. Khan, M. Faizan, and S. Iqrar, Align. J. Stat., Vol. 36, 2016, pp. 57-62.

15.M. Ahsanullah, Stat. Hefte., Vol. 22, 1964, pp. 121-127.

16.M. Ahsanullah, Commun. Stat. Theory Methods, Vol. 16, 1987, pp. 375-381.

17.N. Balakrishnan and N. Papadotas, Stat. Probab. Lett., Vol. 57, 2002, pp. 193-204.

18.U. Kamps and U. Gather, Appl. Math., Vol. 24, 1997, pp. 383-391.

19.S. Nadarajah, M. Teimouri, and S.H. Shih, Braz. J. Probab. Stat., Vol. 28, 2014, pp. 209-222.

20.U. Kamps and E. Cramer, Statistics, Vol. 35, 2001, pp. 269-280.

21.E. Cramer, U. Kamps, and T. Rychlik, Ann. Inst. Stat. Math., Vol. 56, 2004, pp. 183-192.

22.J.S. Hwang and G.D. Lin, Proc. Am. Math. Soc., Vol. 91, 1984, pp. 577-580.

23.H.L. Harter, Ann. Math. Stat., Vol. 30, 1959, pp. 980-999.

24.P.C. Leone, Y.H. Rutenberg, and C.W. Topp, J. Am. Stat. Assoc., Vol. 56, 1961, pp. 260-272.

25.P.Y. Thomas, J. Indian Soc. Agric. Stat., Vol. 42, 1990, pp. 250-256.

26.N.K. Sajeevkumar, Commun. Stat. Theory Methods, Vol. 40, 2011, pp. 1780-1786.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: Journal of Statistical Theory and Applications
Volume-Issue: 18 - 3
Pages: 303 - 308
Publication Date: 2019/09/02
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.190818.004 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - M. J. S. Khan
AU  - S. Iqrar
AU  - M. Faizan
PY  - 2019
DA  - 2019/09/02
TI  - Characterization of Exponential Distribution Through Normalized Spacing of Generalized Order Statistics
JO  - Journal of Statistical Theory and Applications
SP  - 303
EP  - 308
VL  - 18
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190818.004
DO  - 10.2991/jsta.d.190818.004
ID  - Khan2019
ER  -

download .riscopy to clipboard