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Volume 18, Issue 3, September 2019, Pages 236 - 243

E-Bayesian Estimation for the Exponential Model Based on Record Statistics

Authors
Hassan M. Okasha*
Department of Statistics, King Abdul Aziz University, P.O.Box: 80203, Jeddah 21589, Saudi Arabia
*Email: hokasha45@gmail.com
Home address: Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, 11884, Cairo, Egypt; E-mail: hassanokasha@yahoo.com.
Corresponding Author
Hassan M. Okasha
Received 23 March 2017, Accepted 18 March 2019, Available Online 2 September 2019.
DOI
10.2991/jsta.d.190820.001How to use a DOI?
Keywords
E-Bayes estimation; Bayes estimation; Exponential distribution; Upper record statistics values; Squared error loss function; Monte Carlo simulation
Abstract

This paper is concerned with using the E-Bayesian method for computing estimates for the parameter and reliability function of the Exponential distribution based on a set of upper record statistics values. The estimates are derived based on a conjugate prior for the scale parameter and squared error loss function. A comparisons between the new method and the corresponding Bayes technique are made using the Monte Carlo simulation.

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

Let X1,X2,⋯ be a sequence of identically independent distribution (iid) random variables with probability density function (pdf) f(x). For n≥1, define

U1=1,  Un+1=min{j:j>Un,Xj>XUn}

The sequence {XUn} ({Un}) is known as upper record statistics (record times). These statistics are of interest and important in several real-life problems involving weather, economics and sports data. The statistical study of record values started with Chandler [1] has now spread in different directions. For more details and applications in the record values, see Ahsanullah [2] and Arnold et al. [3].

Consider the one-parameter exponential (Exp (θ)) distribution with pdf

f(x)=θe−θx,  x>0, θ>0,
and the reliability function
R(t)=e−θt,  t>0.

The exponential distribution plays an important role in life testing problems. A great deal of research has been done on estimating the parameters of the exponential distribution using both classical and Bayesian techniques, and a very good summary of this work can be found in Johnson et al. [4]. There are also some papers on estimation and prediction for exponential parameters based on record and censored samples. See, for example, Balasubramanian and Balakrishnan [5], Chandrasekar et al. [6], and Ahmadi et al. [7] and references therein. Soliman [8] obtained the Bayes estimates using the symmetric (squared error) loss function.

2. BAYESIAN ESTIMATION

Suppose we observe n upper record values XU1=x1,XU2=x2,…,XUn=xn from Exp(θ) distribution with pdf given by (1). The likelihood function (LF) can be written as

L(θ∣x_)=∏i=1n−1h(xi)f(xn),
where x_=(x1,x2,…,xn) and h(.) is the hazard function corresponding to the pdf f(.). It follows, from (1), (2) and (3), that
L(θ∣x_)=θne−xnθ.

We use the following gamma conjugate prior density for the parameter θ

g(θ|α,β)=βαΓ(α)θα−1e−βθ,  θ>0,
where α>0 and β>0. This prior was first used by Papadopoulos [9]. The posterior density of θ given x_ can be obtained from (4) and (5) and written as
q(θ∣x_)=κθn+α−1e−(β+xn)θ,θ>0,
where
κ=(β+xn)n+αΓ(n+α).

Under the squared error loss function, the Bayes estimate of θ can be shown to be

θ^BS(α,β)=n+αβ+xn.

For more details about the squared error loss function, see, for example, Soliman [8].

The Bayes estimate of the reliability, RBS, based on the squared error loss function is obtained from (2) and (6) as

R^BS(t)=β+xnβ+xn+tn+α.

3. E-BAYESIAN ESTIMATION

According to Han [10], the prior parameters α and β should be selected to guarantee that g(θ|α,β) is a decreasing function of θ. The derivative of g(θ|α,β) with respect to θ is

dg(θ|α,β)dθ=βαΓ(α)θα−2e−βθ[(α−1)−βθ].

Note that α>0, β>0, and θ>0, it follows 0<α<1, β>0 due to dg(θ|α,β)θ<0, and therefore g(θ|α,β) is a decreasing function of θ.

Assuming that α and β are independent with bivariate density function

π(α,β)=π1(α)π2(β),
then, the E-Bayesian estimate of θ (expectation of the Bayesian estimate of θ) can be written as
θ^EB=E(θ|X_)=∫∫ρθ^BS(α,β)π(α,β)dαdβ,
where ρ is the domain of α and β, θ^B(α,β) is the Bayes estimate of θ given by (8). For more details, see [11–15].

3.1. E-Bayesian Estimation for θ

E-Bayesian estimation of θ is obtained based on three different distributions of the hyperparameters α and β. These distributions are used to investigate the influence of the different prior distributions on the E-Bayesian estimation of θ.

The following distributions of α and β may be used

π1(α,β)=1sB(u,v)αu−1(1−α)v−1,  0<α<1,0<β<s,π2(α,β)=2s2B(u,v)(s−β)αu−1(1−α)v−1,      0<α<1,0<β<s,π3(α,β)=2βs2B(u,v)αu−1(1−α)v−1,      0<α<1,0<β<s,.
where B(u,v) is the beta function. For π1(α,β), the E-Bayesian estimate of θ is obtained from (8) and (12) as
θ^EBS1=∫∫Dθ^BS(α,β)π1(α,β)dβdα,  =1sB(u,v)∫01∫0sn+αβ+xnαu−1(1−α)v−1dβdα.  =1sn+uu+vlns+xnxn.

Similarly, the E-Bayesian estimates of θ based on π2(α,β) and π3(α,β) are computed and given, respectively, by

θ^EBS2=2sn+uu+vs+xnslns+xnxn−1,
and
θ^EBS3=2sn+uu+v1−xnslns+xnxn.

3.2. E-Bayesian Estimation for the Reliability

Based on the squared error loss function, the E-Bayesian estimates of the reliability function is computed for the three different distributions of the hyperparameters α and β given by (12). For π1(a,b), the E-Bayesian estimate of the reliability is obtained from (9), (11) and (12) as

R^EBS1=∫∫DR^BS(t)π1(a,b)dbda  =1sB(u,v)∫01∫0sβ+xnβ+xn+tn+ααu−1(1−α)v−1dβdα.  =1sB(u,v)∫0sβ+xnβ+xn+tn∫01eαlnβ+xnβ+xn+tαu−1(1−α)v−1dαdβ.  =1s∫0sβ+xnβ+xn+tnF1:1u;u+v;lnβ+xnβ+xn+tdβ,    
where, F1:1.,.;. is the generalized hypergeometric function [see Gradshteyn and Ryzhik [16] (formula 3.383 (1))]. Similarly, the E-Bayesian estimates of the reliability based on π2(a,b) and π3(a,b) are computed and given, respectively, by
R^EBS2=2s2∫0s(s−β)β+xnβ+xn+tnF1:1u;u+v;lnβ+xnβ+xn+tdβ,    
and
R^EBS3=2s2∫0sββ+xnβ+xn+tnF1:1u;u+v;lnβ+xnβ+xn+tdβ.    

The integrals in (16), (17) and (18) can not be computed analytically in simple closed forms and numerical computations must be used for computing the E-Bayesian estimates of the reliability functions based on squared error loss function. The Maple TM12 is used for the numerical computations.

4. PROPERTIES OF E-BAYESIAN ESTIMATION

Now we discuss the relations among θ^EBSi (i=1,2,3) and the relations between R^EBSi (i=1,2,3).

1. Relations among θ^EBSi (i=1,2,3)

Lemma 4.1.

Let 0<s<xn and θ^EBSi (i=1,2,3) be given by (13), (14) and (15) then

  1. θ^EBS2<θ^EBS1<θ^EBS3.

  2. limxn→∞θ^EBS1=limxn→∞⁡θ^EBS2=limxn→∞⁡θ^EBS3.

Proof.

See Appendix A.

2. Relations among R^EBSi (i=1,2,3).

Lemma 4.2.

Let 0<s<xn and θ^EBSi (i=1,2,3) be given by (16), (17) and (18) then

limxn→∞R^EBS1=limxn→∞⁡R^EBS2=limxn→∞⁡R^EBS3.

Proof.

See Appendix A.

Remark.

From (16), (17) and (18), we have

R^EBS3−R^EBS1=R^EBS1−R^EBS2  =1s2{∫0s2β−sβ+xnβ+xn+tnF1:1u;u+v;lnβ+xnβ+xn+tdβ},    
we note that, this integral may not be computed analytically, therefore, we have solved it numerically using Maple TM12. The numerical results show that this integral usually positive. It follows that
R^EBS2<R^EBS1<R^EBS3.

5. MONTE-CARLO SIMULATION AND COMPARISONS

In this section, a Monte Carlo simulation is used for a comparison of the Bayes and E-Bayes techniques of estimation. The following steps are considered:

  • For given values of the prior parameters (u,v) and (0,s) we generate α and β from the beta and uniform priors (12), respectively.

  • For given values of (α,β) we generate θ from the gamma prior density (5).

  • For known values of θ, an upper record sample of size n is then generated from the density of the Exp(θ) distribution defined by (1) using the transformation: Xi=F−1(Ui)=−1θln(1−Ui) where Ui is the uniformly distributed random variate. The sequence of record values was generated as follows: (i) Generate the 1st value and record it as the first record value. (ii) Generate the 2nd value, if it is greater than the previous, then record it as the 2nd record value, if not generate another value and so on. The codes of Maple12 are used to generate from the gamma, beta and uniform distributions.

  • Based on the squared loss function, the estimates θ^BS, θ^EBS1, θ^EBS2 and θ^EBS3 of θ are computed from (8), (13), (14) and (15).

  • Based on the squared loss function, the estimates R^BS, R^EBS1, R^EBS2 and R^EBS3 of R are computed from (9), (16), (17) and (18).

  • The quantities (ϕ^−ϕ)2 are computed where ϕ^ stands for an estimate of ϕ.

  • The above steps are repeated 1000 times and the estimated risks (ER) of the estimates are computed by averaging the squared deviations over 1000 repetitions:

    ER(R^)=11000∑(R^−R)2.

  • The computational results are displayed in Tables 1 and 2.

n s (u,v) θ^BS θ^EBS1 θ^EBS2 θ^EBS3
5 0.1 (3, 2) 0.2017442307 0.1883297898 0.1886577740 0.1880020030
(4, 3) 0.1929137491 0.1810729415 0.1813785003 0.1807675613
0.2 (3, 2) 0.3982663752 0.3709431448 0.3734597677 0.3684320680
(4, 3) 0.3809784612 0.3568132202 0.3591615226 0.3544699350
7 0.1 (3, 2) 0.2184657662 0.2076281370 0.2078776591 0.2073786998
(4, 3) 0.2095240406 0.1999569461 0.2001895177 0.1997244518
0.2 (3, 2) 0.4329793602 0.4108522767 0.4127976468 0.4089093544
(4, 3) 0.4153715457 0.3958026592 0.3976190127 0.3939885204
10 0.1 (3, 2) 0.2444457070 0.2357117460 0.2358954264 0.2355280883
(4, 3) 0.2350112326 0.2273010337 0.2274735737 0.2271285123
0.2 (3, 2) 0.4859920654 0.4681456700 0.4695978012 0.4666940671
(4, 3) 0.4673214397 0.4515405872 0.4528968777 0.4501847724
15 0.1 (3, 2) 0.3038514648 0.2966034956 0.2967692871 0.2964377198
(4, 3) 0.2927368776 0.2927368776 0.2927368776 0.2861821773
0.2 (3, 2) 0.6051030307 0.5902370221 0.5915542680 0.5889200020
(4, 3) 0.5830482568 0.5699001678 0.5711311623 0.5686693798
20 0.1 (3, 2) 0.3721301397 0.3655446192 0.3657206211 0.3653686418
(4, 3) 0.3589087371 0.3530940504 0.3532596655 0.3529284546
0.2 (3, 2) 0.7415262048 0.7279499471 0.7293427421 0.7265573663
(4, 3) 0.7152647777 0.7032529605 0.7045548355 0.7019512817
25 0.1 (3, 2) 0.4355546074 0.4294372101 0.4296175092 0.4292569430
(4, 3) 0.4203504551 0.4149486454 0.4151222386 0.4147750800
0.2 (3, 2) 0.8682786579 0.8556138104 0.8570607075 0.8541670942
(4, 3) 0.8380568582 0.8268480177 0.8282007455 0.8254954489
30 0.1 (3, 2) 0.4860639584 0.4803990052 0.4805801273 0.4802179156
(4, 3) 0.4692934979 0.4642909207 0.4644610439 0.4641208303
0.2 (3, 2) 0.9693228105 0.9575592737 0.9589969053 0.9561217672
(4, 3) 0.9359658770 0.9255525959 0.9268965622 0.9242087416
Table 1

Estimated risks (ERs) of the estimates of θ^BS, θ^EBS1, θ^EBS2, and θ^EBS3.

n s (u,v) t R^BS R^EBS1 R^EBS2 R^EBS3
5 0.1 (3, 2) 1 0.1506261589 0.1416593141 0.1418198901 0.1414987537
(4, 3) 0.1454107429 0.1373973487 0.1375500379 0.1372446747
0.2 (3, 2) 0.2374808510 0.2240232094 0.2248199445 0.2232265374
(4, 3) 0.2307616910 0.2186234287 0.2193885079 0.2178584147
0.1 (3, 2) 2 0.2390433964 0.2258201086 0.2260214050 0.2256188162
(4, 3) 0.2322617402 0.2203489319 0.2205422289 0.2201556391
0.2 (3, 2) 0.3244785603 0.3080374905 0.3088472262 0.3072277709
(4, 3) 0.3184187738 0.3034123420 0.3042004630 0.3026242355
7 0.1 (3, 2) 1 0.1664355735 0.1591245623 0.1592575537 0.1589915789
(4, 3) 0.1610567686 0.1545229758 0.1546493432 0.1543966162
0.2 (3, 2) 0.2644564078 0.2535854752 0.2542579002 0.2529130856
(4, 3) 0.2575898017 0.2477781682 0.2484238705 0.2471325025
0.1 (3, 2) 2 0.2657684489 0.2551011427 0.2552705990 0.2549316887
(4, 3) 0.2588495258 0.2492335958 0.2493963142 0.2490708796
0.2 (3, 2) 0.3620170205 0.3491360308 0.3498112707 0.3484607975
(4, 3) 0.3561606991 0.3443832818 0.3450414239 0.3437251457
10 0.1 (3, 2) 1 0.1902154074 0.1842778547 0.1843886372 0.1841670756
(4, 3) 0.1844215968 0.1791139476 0.1792190713 0.1790088271
0.2 (3, 2) 0.3038538360 0.2951848142 0.2957580609 0.2946115861
(4, 3) 0.2965808362 0.2887456885 0.2892961584 0.2881952377
0.1 (3, 2) 2 0.3049681217 0.2964763742 0.2966205606 0.2963321890
(4, 3) 0.2976508212 0.2899859234 0.2901243769 0.1790088271
0.2 (3, 2) 0.4140555534 0.4043222408 0.4048844289 0.4037600547
(4, 3) 0.4083744874 0.3994449368 0.3999942346 0.2898474711
15 0.1 (3, 2) 1 0.2340248721 0.2292959584 0.2293955774 0.2291963409
(4, 3) 0.2274269669 0.2231936912 0.2232883215 0.2230990623
0.2 (3, 2) 0.2784629712 0.2744421332 0.2745402336 0.2743440339
(4, 3) 0.3614559775 0.3554753200 0.3559590777 0.3549915706
0.1 (3, 2) 2 0.2784629712 0.2744421332 0.2745402336 0.2743440339
(4, 3) 0.3623921863 0.3565648314 0.3566862977 0.3564433655
0.2 (3, 2) 0.4899697646 0.4832990205 0.4837533364 0.4828447059
(4, 3) 0.4848545413 0.4786939831 0.4791404803 0.4782474871
20 0.1 (3, 2) 1 0.2784629712 0.2744421332 0.2745402336 0.2743440339
(4, 3) 0.2711134801 0.2675066825 0.2676000935 0.2674132727
0.2 (3, 2) 0.4296598122 0.4243555945 0.4248220065 0.4238891869
(4, 3) 0.4214785019 0.4166494905 0.4171002457 0.4161987398
0.1 (3, 2) 2 0.4305594645 0.4254058032 0.4255227963 0.4252888103
(4, 3) 0.4223482520 0.4176644479 0.4177775148 0.4175513810
0.2 (3, 2) 0.5503792577 0.5455705727 0.5459492432 0.5451919045
(4, 3) 0.5461373796 0.5416660740 0.5420407322 0.5412914174
25 0.1 (3, 2) 1 0.3178489459 0.3143266763 0.3144231573 0.3142301961
(4, 3) 0.3098722414 0.3067067059 0.3067987487 0.3066146639
0.2 (3, 2) 0.4805653588 0.4761550800 0.4765908567 0.4757193053
(4, 3) 0.4722484669 0.4722484669 0.4722484669 0.4677955214
0.1 (3, 2) 2 0.4814044100 0.4771361598 0.4772453945 0.4770269251
(4, 3) 0.4730625065 0.4691695475 0.4692754827 0.4690636124
0.2 (3, 2) 0.5965025061 0.5929072065 0.5932271235 0.5925872917
(4, 3) 0.5931944183 0.5898286416 0.5901472156 0.5895100697
30 0.1 (3, 2) 1 0.3484771341 0.3453542827 0.3454470559 0.3452615102
(4, 3) 0.3400443226 0.3372336112 0.3373222305 0.3371449924
0.2 (3, 2) 0.5188557510 0.5151076985 0.5155110074 0.5147043908
(4, 3) 0.5105234191 0.5070875801 0.5074797408 0.5066954206
0.1 (3, 2) 2 0.5196315183 0.5160155862 0.5161166345 0.5159145379
(4, 3) 0.5112781009 0.5079703795 0.5080686382 0.5078721208
0.2 (3, 2) 0.6287009073 0.6259093266 0.6261819411 0.6256367142
(4, 3) 0.6261979056 0.6235691251 0.6238420644 0.6232961875
Table 2

Estimated risks (ERs) of the estimates of R^BS, R^EBS1, R^EBS2, and R^EBS3.

6. CONCLUDING REMARKS

In this paper, E-Bayes and Bayes methods are used for estimating the parameter and the reliability function of the exponential distribution based on record statistics. The Monte-Carlo simulation and comparisons are used for computing E-Bayes and Bayes estimates. We will present the conclusions in the following points:

  1. Generally, the ER of the E-Bayes estimate of θ and R are the smallest ERs. On the other hand, the ER of the E-Bayes estimates of θ and R based on the squared loss function are less than the ER of their corresponding Bayes estimates.

  2. It has been noticed, from Tables 1 and 2, that the E-Bayes estimates, in most cases, tend to be more efficient than the Bayes estimates in the sense of having smaller ERs of the estimates. Also, the ERs of the estimates increases as n increases and the E-Bayes estimates have the smallest ERs as compared with their corresponding Bayes estimates. By increasing n, the computations in Tables 1 and 2 show that the E-Bayes estimates (based on squared error loss) are better than the Bayes in the sense of comparing the ERs of the estimates.

  3. The author suggest take beta and uniform distribution as the priors of the hyperparameters α and β, respectively. The work in this paper showed that the E-Bayesian estimation method is both efficient and easy to perform.

ACKNOWLEDGMENTS

The authors would like to thank the editors and the referees for careful reading and for fruitful comments which greatly improved the paper.

APPENDIX A

Proof of Lemma 4.1.

  1. From (13), (14) and (15), we have

    θ^EBS2−θ^EBS1=θ^EBS1−θ^EBS3=1sn+uu+vs+2xnslnxn+sxn−2      

    For −1<x<1, we have: ln(1+x)=x−x22+x33−x44+…=∑k=1∞(−1)k−1xkk. Let x=sxn, when 0<s<xn, 0<sxn<1, we get

     s+2xnslnxn+sxn−2 =s+2xnssxn−12sxn2+13sxn3−14sxn4+15sxn5−…−2 =sxn−12sxn2+13sxn3−14sxn4+15sxn5−…−2  +2−sxn+23sxn2−24sxn3+25sxn4−… =s26xn2−s36xn3+3s46xn4−2s515xn5+… =s26xn21−sxn+s460xn49−8sxn+…>0. 

    According to (A.1) and (A.2), we have

    θ^EBS2−θ^EBS1=θ^EBS1−θ^EBS3>0,
    that is
    θ^EBS3<θ^EBS1<θ^EBS2.

  2. From (A.1) and (A.2), we get

    limxn→∞(θ^EBS2−θ^EBS1)=limxn→∞(θ^EBS1−θ^EBS3) =1s(n+uu+v)limxn→∞(s26xn2(1−sxn)+s460xn4(9−8sxn)+…) =0.

    That is, limxn→∞θ^EBS1=limxn→∞⁡θ^EBS2=limxn→∞⁡θ^EBS3.

    Thus, the proof is complete.

Proof of Lemma 4.2.

  1. From (19), we have

    limxn→∞(R^EBS3−R^EBS1)=limxn→∞(R^EBS1−R^EBS2) =limxn→∞{1s2∫0s(2β−s)β+xnβ+xn+tnF1:1(u;u+v;ln(β+xnβ+xn+t))dβ} =0.

    That is, limxn→∞R^EBS1=limxn→∞⁡R^EBS2=limxn→∞⁡R^EBS3.

    Thus, the proof is complete.

REFERENCES

2.M. Ahsanullah, Record Statistics, Nova Science, New York, 1995.
4.N.L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, second, John Wiley & Sons, New York, Vol. 1, 1994.
15.H.M. Okasha, J. Adv. Res. Sci. Comput., Vol. 44, 2012, pp. 34-45.
16.I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, Inc, London, 2007, pp. 96-99. corrected and enlarged ed., Prepared by A. Jeffrey
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Journal of Statistical Theory and Applications
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236 - 243
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2019/09/02
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risenwbib
TY  - JOUR
AU  - Hassan M. Okasha
PY  - 2019
DA  - 2019/09/02
TI  - E-Bayesian Estimation for the Exponential Model Based on Record Statistics
JO  - Journal of Statistical Theory and Applications
SP  - 236
EP  - 243
VL  - 18
IS  - 3
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.190820.001
DO  - 10.2991/jsta.d.190820.001
ID  - Okasha2019
ER  -