Inference Based on k-Record Values from Generalized Exponential Distribution

(1)

INFERENCE BASED ON

K

-RECORD VALUES FROM

GENERALIZED EXPONENTIAL DISTRIBUTION

Manoj Chacko

Department of Statistics, University of Kerala, Trivandrum 695 581, India Laji Muraleedharan1

Department of Statistics, University of Kerala, Trivandrum 695 581, India

1. INTRODUCTION

Let{X_i,i ≥1}be a sequence of independent and identically distributed (iid) random variables having an absolutely continuous cumulative distribution function (cdf)F(x) and probability density function (pdf) f(x). An observationX_jis called a lower record ifX_j<X_ifor everyi< j. An analogous definition deals with upper record values. In a number of situations, only observations that exceed or only those that fall below the current extreme value are recorded. Examples include meteorology, hydrology, athletic events and mining. Interest in records has increased steadily over the years since Chan-dler’s (1952) formulation. Useful surveys are given in Ahsanullah (1995) and Arnoldet al.(1998). Estimation of parameters using record values and prediction of future record values have been studied by several authors, for details see Balakrishnan and Chan (1998), Raqab (2002), and Sultanet al.(2002). Bayesian estimation and prediction for some life distributions based on record values have been considered by Ahamadi and Doostparast (2006).

Serious difficulties for the statistical inference based on records arise due to the fact that the occurrences of record data are very rare in practical situations and the expected waiting time is infinite for every record after the first. These problems are avoided if we consider the model ofk-record statistics introduced by Dziubdziela and Kopocinski (1976).

For a positive integerk, the lowerk-record timesT_n(k)and the lowerk-record values R_n(k)are defined as follows

T₁_(k)=k

(2)

and forn>1

T_n(k)=min¦j:j>T_n₋₁_(k),X_j<X_k_:_T

n−1(k)

©

,

whereX_i_:_mdenotes thei-th order statistic in a sample of sizem. The sequence of lower k-records are then defined by forn≥1

R_n(k)=X_k_:_T

n(k), k≥1.

In an analogous way, one can define the upperk-record values. Since the ordinary record values are contained in the k-records, by puttingk = 1, the results for usual records can be obtained as special case. Recently, a lot of works have been done on the statistical inference, based onk-records. See, for instance, Ahmadiet al.(2005), Ahmadi and Doostparast (2008), Mary and Chacko (2010),Chacko and Mary (2013a), Chacko and Mary (2013b), Paul and Thomas (2015) and Malinowska and Szynal (2004). The pdf ofnth lowerk-record valueR_n(k)forn≥1 is given by

f_n(k)(x) = k n

(n−1)![−logF(x)]

n−1_[_F₍_x_)]k−1_f₍_x₎_, _x_>_0, ₍₁₎

and the joint pdf ofmth andnth lowerk-record values,R_m(k)andR_n(k)form<nis given by

f_m_,_n(k)(x,y) = k n

(m−1)!(n−m−1)![−logF(x)]

m−1

× [−logF(y) +logF(x)]n−m−1

× [F(y)] k−1

F(x) f(x)f(y), y<x. (2)

Then the joint pdf ofR₁_(k),R₂_(k), ...,R_n(k)is given by

f(r₁,r₂, ...,r_n) =kn[F(r_n)]k n Y

i=1

f(r_i)

F(r_i),−∞<rn<rn−1<. . .<r1<∞. (3)

The generalized exponential (GE) distribution introduced by Gupta and Kundu (1999) can be used quite effectively in analyzing many lifetime data. The two param-eter generalized exponential distribution (denoted by GE(σ,β)) has the pdf given by

f(y) =β σ

1−exp

−

y σ

β−1

exp

−

y σ

, 0<y<∞,σ,β >0. (4)

The cdf corresponding to the above pdf is given by

F(y) =

1−exp

−

y σ

β

(3)

Gupta and Kundu (1999) have shown that the GE model fits better than the gamma or the Weibull model in certain cases. The GE distribution has increasing or decreasing hazard rate depending on the shape parameterβ. Raqab (2002) derived exact expres-sions for means, variances and covariances of lower record values arising from GE dis-tribution and also obtained the BLUE’s of the location and scale parameters based on lower record values. The Bayesian estimation and prediction based on lower record val-ues for the two-parameter generalized exponential distribution were discussed by Madi and Raqab (2007). The maximum likelihood and Bayesian estimation based on record values and inter-record times for the two-parameter generalized exponential distribution have been considered by Kizilaslan and Nadar (2015).

In this paper, we consider the lowerk-record values arising from generalized expo-nential distribution. The paper is organized as follows. In Section 2, we obtain the maximum likelihood estimator (MLE) for scale parameter σ and shape parameterβ

of GE distribution. In Section 3, we consider Bayes estimation of scale parameterσ

and shape parameterβof GE distribution. In Section 4, Bayesian Prediction of future record values are considered. Section 5 is devoted to some simulation studies and finally in Section 6 some concluding remarks.

2. MAXIMUM LIKELIHOOD ESTIMATION

In this section we obtain maximum likelihood estimators of the scale parameterσand shape parameterβinvolved in two parameter generalized exponential distribution GE(σ,β)using lowerk record values. LetR_i(k) i =1, 2, . . . ,n be the firstnlowerk -record values arising fromGE(σ,β)with density function given in (4).

Let D_n₍_k₎ = (R₁_(k),R₂_(k),· · ·,R_n(k))be the vector of first n lower k record values arising fromGE(σ,β). Then from (3) the likelihood function of(σ,β)is given by

L(σ,β|d_n₍_k₎) =

_kβ

σ n

1−e−rnσ kβ

n Y

i=1

e−riσ

1−e−riσ

, (6)

whered_n₍_k₎= (r₁,r₂,· · ·,r_n)is the realization ofD_n₍_k₎. The logarithm of the likelihood function is given by

logL(σ,β|d_n₍_k₎) =nlog(kβ)−nlogσ+kβlog(1−e−rnσ )₋

n X

i=1

r_i

σ −

n X

i=1

log(1−e−riσ)_.

Then

∂ logL

∂ β =

n

β+klog(1−e−

rn

(4)

and

∂ logL

∂ σ = −

n σ−

kβr_ne−rnσ

σ2₍₁₋_e−rnσ)+

n X

i=1

r_i σ2+

n X

i=1

r_ie−riσ

σ2(₁₋_e−riσ) .

Then the corresponding normal equations are

n

β+klog(1−e−

rn

σ ) = ₀ ₍₇₎

and

−n σ−

kβr_ne−rnσ

σ2(₁₋_e−rnσ)+

n X

i=1

r_i σ2+

n X

i=1

r_ie−riσ

σ2₍₁₋_e−riσ) =

0. (8)

From (7), we get

β=− n

klog(1−e−rnσ )

. (9)

Putting the value ofβin (8) we get

−n σ+

n

log(1−e−rnσ)

r_ne−rnσ

σ2(₁₋_e−rnσ) +Xn

i=1

r_i σ2+

n X

i=1

r_ie−riσ

σ2₍₁₋_e−riσ) =0.

Therefore, the MLE ofβis given by

ˆ

β_{M L}=− n

klog(1−e−σˆrnM L)

(10)

and the MLE ofσ, ˆσ_{M L}can be obtained as a solution of the non-linear equation of the form

g(σ) =σ, (11)

where

g(σ) = 1 n





n r_n

log(1−e−rnσ ) e−rnσ (1−e−rnσ )+

n X

i=1

r_i+ n X

i=1

r_ie−riσ

(1−e−riσ)



.

Since ˆσ_{M L} is a fixed point solution of non-linear equation(11),therefore ,it can be obtained by using a simple iterative scheme as follows.

(5)

were σj is the j t h iterate of ˆσM L.The iteration procedure should be stopped when |σ_j+1−σj|< εwhereεis sufficiently small positive number.Once we obtain ˆσM Lfrom

(11),and the MLE ofβbecomes

ˆ

βM L=−

n

klog(1−e−σˆrnM L)

(13)

3. BAYESIAN ESTIMATION

In this section, we consider the Bayesian estimation of parameters involved in the two parameter generalized exponential distribution with scale parameterσ and shape pa-rameterβunder symmetric as well as asymmetric loss functions.

A symmetric loss function is the squared error loss (SEL) function which is defined as

L₁d(µ), ˆd(µ)=dˆ(µ)−d(µ)2, (14)

where ˆd(µ)is an estimate ofd(µ). The Bayes estimate ofµunderL₁is the posterior mean of µ. An asymmetric loss function is the LINEX loss (LL) function which is defined as

L₂d(µ), ˆd(µ)=eh

ˆ

d(µ)−d(µ)

−hdˆ(µ)−d(µ)−1, h6=0. (15)

The Bayes estimate ofd(µ)for the loss functionL₂can be obtained as

ˆ

d_LB(µ) =−1 h log

¦

E_µe−hµ x

©

, (16)

providedE_µ(·)exists. Another asymmetric loss function is the general entropy loss (EL) function given by

L₃d(µ), ˆd(µ)= ˆ d(µ) d(µ)

!q

−qlog ˆ d(µ) d(µ) !

−1,q6=0. (17)

In this case Bayes estimate ofd(µ)is obtained as

ˆ

d_EB(µ) = E_µ µ−q x

− 1

q_. ₍₁₈₎

LetR_i(k),i =1, 2, . . . ,n be the firstnlowerk-record values arising fromGE(σ,β) with density function given in (4). Then the likelihood function is given by

L(σ,β|d_n₍_k₎) =

_k_β

σ n

1−e−rnσ kβ

n Y

i=1

e−riσ

1−e−riσ

(6)

whered_n₍_k₎ = (r₁,r₂,· · ·,r_n). Assume that the prior distributions ofσ andβfollow independent inverse gamma distribution and gamma distribution respectively with den-sity functions given by

π₁(σ|a,b) = b a

Γaσ

−(a+1)_e−_σb_,_{σ >}₀ ₍₂₀₎

and

π2(β|c,d) =

dc Γcβ

c−1_e−dβ_,_{β >}_0. ₍₂₁₎

Then the joint posterior density ofσandβgivenD_n₍_k₎=d_n₍_k₎can be written as

π∗₍_σ_,_β|_d

n(k)) =

L(σ,β|d_n₍_k₎)π₁(σ|a,b)π₂(β|c,d) R R

L(σ,β|d_n₍_k₎)π₁(σ|a,b)π₂(β|c,d)dσdβ. (22)

From (19) we have

L(σ,β|d_n₍_k₎)π1(σ|a,b)π2(β|c,d) =

kn_ba_dc

ΓaΓc β

n+c−1_σ−(a+n+1)_e−β(d−kU_σ)

exp

−_σb − n X

i=1

r_i

σ −

n X

i=1

log(1−eriσ)

,

whereU_σ=log(1−ernσ )_{. Since it is not possible to compute the posterior density (19)} explicitly, we propose MCMC method to generate samples from the posterior distribu-tions ofβandσthen find the Bayes estimates ofβandσ.

3.1. MCMC method

In this section, we consider the MCMC method to generate samples from the posterior distribution. The posterior distribution given in (22) can be written as

π∗₍_β_,_σ|_d

n(k)) ∝ β

n+c−1_e−β(d−kUσ)

σ(a+n+1) (23)

exp

−_σb − n X

i=1

r_i

σ −

n X

i=1

log(1−eriσ)

,

From (23) the conditional posterior distribution ofβgivenσanddatais given by

π∗₍_β|σ_,_d

n(k))∝βn+c−1e−β(d−kUσ). (24)

Again from (23) the conditional posterior distribution ofσgivenβanddatais given by

π∗₍_σ|β_,_d

n(k))∝

1

σ(a+n+1)e

−1_σ(b+Pn i=1ri)−

Pn i=1log(1−e

ri

σ_)+k_β_U

(7)

Thus we can see that conditional posterior distribution ofβfollows a gamma dis-tribution with parameters(n+c)and(d−kU_σ). That isβ∼gamma(n+c,d−kU_σ). Therefore one can easily generate sample from the posterior distribution ofβ. But it is not possible to generate random variables from the posterior distribution ofσgiven in (25) using standard random number generation methods. Hence we use Metropolis-Hasting (M-H) algorithm to generate sample from (25) (Chib and Greenberg , 1995). Since plot of (25) is similar to a normal plot we take normal proposal density for σ

for the M-H algorithm. For updatingσ we have used adaptive MCMC method given in Roberts and Rosenthal (2009) to get an optimum acceptance rate (44%) for the Metropo-lis sampler.

By setting initial valuesβ(0) _and_σ(0)_{, let}_β(t) _and_σ(t)_,_t₌_{1, 2, . . . ,}_N _{be the}

obser-vations generated from (24) and (25) respectively. Then the Bayes estimators under SEL function ofβandσ, by taking firstmiterations as burn-in period, are given by

ˆ

βSB=

1 N−m

N X

j=m+1

β(j) ₍₂₆₎

and

ˆ

σ_SB= 1

N−m N X

j=m+1

σ(j)_. ₍₂₇₎

The Bayes estimate ofβandσunder the LL functionL₂is obtained as

ˆ

βLB=−

1 h log

1 N−m

N X

j=m+1

e−hβ(j) !

(28)

and

ˆ

σLB=−

1 h log

1 N−m

N X

j=m+1

e−hσ(j)

!

. (29)

Finally, the Bayes estimate ofβandσunder the EL functionL₃is obtained as

ˆ

β_EB=

(

1 N−m

N X

j=m+1

β(j)−q

)−1_q

(30)

and

ˆ

αEB= (

1 N−m

N X

j=m+1

α(j)−q

)−1_q

(8)

4. BAYESIAN PREDICTION

In this section, we consider the prediction problem under Bayesian context,namely one sampling prediction. Suppose we havenlower k-recordsD_n₍_k₎= (R₁_(k),R₂_(k),· · ·,R_n(k)) from GE(σ,β)). Based on this sample we are interested to find the s-th lowerkrecord values, 1<n<s. LetR_s(k)be thes t hfuture lowerkrecord value. Then the conditional distribution ofR_(s)=r_(s)given the observed lowerkrecord valued_n₍_k₎= (r₁,r₂,· · ·,r_n) is given by

f₁(r_s|d_n₍_k₎,β,σ) = k

s−n

Γ(s−n)[logF(rn)−logF(rs)]

s−n−1 f(rs) F(r_n)

_F(_r s) F(r_n)

k−1

, (32)

0<r_s<r_n<∞,

wheref(.)andF(.)are the pdf and cdf defined in (4) and (5) respectively . Using binomial expansion, (32) can be expressed as

f₁(r_s|d_n₍_k₎,β,σ) = k

s−n

Γ(s−n) s−n−1

X

i=0

s−n−1 i

[logF(r_n)]i[−logF(r_s)]s−n−1−i f(rs) F(r_n)

_F(_r s) F(r_n)

k−1 .

(33)

Then the Bayesian predictive density function of R_s(k) given the past n lowerk record values is given by

f₁∗(rs|d_n₍_k₎) =

Z∞

0

Z∞

0

f₁(r_s|d_n₍_k₎,β,σ)π∗(σ,β|d_n₍_k₎)dσdβ, (34)

whereπ∗₍_σ_,_β_/d

n(k)) is the posterior density defined in (23). Thus the predictive

esti-mate for thes the lowerkrecord under SEL function is given by

ˆ r_s=

Zr_n

0

r_sf₁∗(r_s|d_n₍_k₎)d r_s

=

Z r_n

0

r_s

Z∞

0

Z∞

0

f₁(r_s|d_n₍_k₎,β,σ)π∗(σ,β|d_n₍_k₎)dσdβ

d r_s

=Z∞ 0

Z∞

0

Z r_n

0

r_sf₁(r_s|d_n₍_k₎,β,σ)d r_s

π∗₍_σ_,_β|_d

(9)

By puttingu=−logF(r_s)in f₁(r_s|d_n₍_k₎,β,σ)), we get

ˆ r_s=

Z∞

0

Z∞

0

I(β,σ)π∗(σ,β|d_n₍_k₎)dσdβ, (35)

where

I(β,σ) = k

s−n_σ

Γ(s−n) s−n−1

X

i=0

s−n−1 i

βi h

log(1−e−σrn)ii

h

(1−e−σrn)βik

Z∞

−βlog(1−e−rnσ)−

log(1−e−uσ)_us−n−i−1_e−uk_{d u}

(36)

One may observe that (35) can not be evaluated explicitly. So we use MCMC method to obtain ˆr_s. Letβ(t)_and_σ(t)_,_t₌_{1,2,...,N be the observation generated from conditional}

posterior densities given in (24) and (25) respectively. Then ˆr_scan be obtained by taking firstmiterations as burn period as

ˆr_s= 1 N−m

N X

i=m+1

I(β(t),σ(t)). (37)

4.1. Interval prediction

In this section we consider the equal tail prediction interval for thesth lower k record value. From (32), the one sample distribution function is obtained as

F₁(t|d_n₍_k₎,β,σ) = P(Rs(k)≤t|dn(k),β,σ)

P(R_s(k)≤r_n|d_n₍_k₎,β,σ), (38)

where

P(R_s(k)≤w|d_n₍_k₎,β,σ) =

Zw

0

f₁(r_s|d_n₍_k₎,β,σ)d r_s

= ks−n Γ(s−n)

s−n−1

X

i=0

s−n−1 i

βi h

log(1−e−σrn)ii

h

(1−e−σrn)βik

Γ(s−n−i)

ks−n−i

h

1−G(−log(1−e−σw)_,_s₋_n₋_i_,_k)i_, ₍₃₉₎

whereG(x,a,b)is the cdf of a gamma distribution given by

Zx

0

ba

Γat

(10)

Thus the posterior predictive distribution function is given by

F₁∗(r_s|d_n₍_k₎) =

Z∞

0

Z∞

0

F₁(r_s|d_n₍_k₎,β,σ)π∗(σ,β|d_n₍_k₎)dσdβ. (40)

(40) can be approximated by MCMC method. Then the prediction interval(L,U)with 1−αconfidence level is obtained by solving the non-linear equations

F₁∗(L|d_n₍_k₎) =α

2 and F

∗

1(U|dn(k)) =1−α₂. (41)

5. SIMULATION STUDY

In this section we carry out a simulation study for illustrating the estimation procedures developed in previous sections. We have obtained the bais and MSE of MLEs ofβand

σfor different values ofnusing simulated sample for different combinations ofσandβ and are given in Tables 1 and 2. For the simulation studies for Bayes estimators we take the hyper parameters for the prior distributions ofβandσasa=2,b =2,c=2 and d=2. We have obtained the Bayes estimators under squared error loss function and its MSE. For the simulation study we do the following

1. Generatenlowerk-record values from generalized exponential distribution with scale parameterβand and shape parameterσ.

2. Calculate estimators ofβandσusing the generated lowerk-record values using MCMC method as described below.

(a) start with initial valuesβ(0)_and_σ(0)

(b) sett=1

(c) generateβ(t)from gamma(n+c,d−kU_σ_(t₋₁₎)

(d) Using M-H algorithm, generateσ(t)fromπ∗

2(σ|β(t),dn(k)).

(e) sett=t+1

(f) Repeat steps (c) to (e) forN=50, 000 times

(g) Calculate the Bayes estimators ofβandσunder different loss functions by taking burn-in periodm=5000.

3. Repeat the steps 1 and 2 for 1000 times.

4. Calculate the bias and MSE of all estimators.

Repeat the simulation study forn = 5(1)8, k = 2, 3 and for different values ofσ

(11)

interval for next(n+1)t h _{future record values and average interval length (AIL) for} n=5(1)8 under different combinations ofσ andβare given in Table 7 and Table 8. From the tables one can see that the bias and MSE of Bayes estimators are smaller than bias and MSE of MLEs, therefore we conclude that Bayes estimators perform better than MLE in terms of bias and MSE. The bias and MSE of both MLE’s and Bayes estimators are decrease when n increases. Also, amoung the Bayes estimators, estimators under entropy loss function posses minimum bias and MSE.

6. CONCLUSION

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APPENDIX A. TABLES

TABLE 1

The bias and MSE of MLEs ofβandσfor k=2.

n (β,σ) βˆ_{M L} σˆ_{M L}

Bias MSE Bias MSE

(13)

TABLE 2

The bias and MSE of MLEs ofβandσfor k=3.

n (β,σ) βˆ_{M L} σˆ_{M L}

Bias MSE Bias MSE

(14)

(15)

(16)

(17)

(18)

TABLE 7

The lower and upper bound for st h record and AIL for k=2.

n s β σ Lower Upper AIL

4 5 1.5 2 0.766 0.811 0.045

4 5 1.5 2.5 0.539 0.570 0.031

4 5 2 2 1.076 1.149 0.072

4 5 2 2.5 0.908 0.965 0.057

4 5 2 3 0.585 0.620 0.035

6 7 1.5 2 0.311 0.330 0.019

6 7 1.5 2.5 0.367 0.388 0.022

6 7 2 2 0.209 0.220 0.012

6 7 2 2.5 0.345 0.366 0.020

6 7 2 3 0.323 0.341 0.018

8 9 1.5 2 0.084 0.088 0.004

8 9 1.5 2.5 0.275 0.290 0.015

8 9 2 2 0.067 0.071 0.004

8 9 2 2.5 0.171 0.180 0.009

8 9 2 3 0.664 0.703 0.039

TABLE 8

The lower and upper bound for st h record and AIL for k=3.

n s β σ Lower Upper AIL

4 5 1.5 2 0.777 0.813 0.035

4 5 1.5 2.5 0.423 0.440 0.017

4 5 2 2 0.627 0.652 0.025

4 5 2 2.5 2.599 2.723 0.125

4 5 2 3 2.313 2.434 0.121

6 7 1.5 2 0.100 0.103 0.003

6 7 1.5 2.5 0.190 0.197 0.007

6 7 2 2 1.103 1.152 0.049

6 7 2 2.5 0.585 0.608 0.023

6 7 2 3 0.819 0.853 0.035

8 9 1.5 2 0.531 0.552 0.021

8 9 1.5 2.5 0.167 0.173 0.006

8 9 2 2 0.489 0.508 0.019

8 9 2 2.5 1.100 1.149 0.048

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SUMMARY

In this paper, the lowerk-record values arising from a two parameter generalized exponential distribution is considered. The maximum likelihood estimators for the shape parameter and scale parameter are obtained. The Bayes estimates of the parameters are also developed by using Markov chain Monte Carlo method under symmetric and asymmetric loss functions. Finally, a simulation study is performed to find the performance of different estimators developed in this paper.