Dynamic Systems with Baseline Exponential Distribution Based on Sequential Order Statistics Under a Power Trend for Hazard Rates

Majid Hashempour; Mahdi Doostparast

doi:10.2991/jsta.d.200224.005

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Volume 19, Issue 1, March 2020, Pages 1 - 9

Dynamic Systems with Baseline Exponential Distribution Based on Sequential Order Statistics Under a Power Trend for Hazard Rates

Authors

Majid Hashempour¹^{, *}, Mahdi Doostparast²

¹Department of Statistics, School of Sciences, University of Hormozgan, Bandar Abbas, Iran

²Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

^*Corresponding author. Email: ma.hashempour@hormozgan.ac.ir

Corresponding Author

Majid Hashempour

Received 6 January 2018, Accepted 19 August 2019, Available Online 2 March 2020.

DOI: 10.2991/jsta.d.200224.005 How to use a DOI?
Keywords: Proportional hazard rates; Hypotheses testing; Sequential order statistics; Estimation
Abstract: This paper deals with analyzing dynamic engineering systems consisting of independent components. The failure of a components causes more load on the surviving components. This property is modeled by a power trend conditionally proportional hazard rates. For modeling system lifetimes, the theory of sequential order statistics can be used. Sequential order statistics coming from heterogeneous exponential distributions are considered. The maximum likelihood and Bayesian estimates of the parameters are obtained in different cases. The generalized likelihood ratio and the Bayesian tests are also derived for testing homogeneity of the baseline exponential component lifetimes arising from s independent engineering systems.
Copyright: © 2020 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,⋯,Xn be independent and identically distributed (i.i.d.) random variables with a common distribution function (DF), say F, and abbreviated by X1,⋯,Xn~i.i.d.F. Denote in magnitude order of X1,⋯,Xn by X1:n≤⋯≤Xn:n, called order statistics (OSs). The theory of OSs has been widely assumption in literature. For example, in system reliability analyses, lifetimes of r-out-of- n systems coincide to Xr:n where X1,⋯,Xn stand for component lifetimes; For more information, see Barlow and Proschan [1] and David and Nagaraja [2] and references therein.

In this setting, failure of a component does not effect the surviving components. There are various practical situations in which this investigated and applied does not hold. For illustration, suppose that there exist n power generators with nominal capacity G1,⋯,Gn. The system perform satisfactory if the total generated power is at least a given threshold, say A. If a generator fails, then the remaining generators have to generate much powers to fulfill the nominal level A and hence more loads cause more pressure on the working generators. Therefore, the generator lifetimes are decreased. For statistical modeling these kinds of systems, some generalizations of OSs such as fractional OSs and generalized OSs have been introduced in literature. They are useful for providing more flexible tools and also a setting to unify similar results (David and Nagaraja [2], p. 21). In this paper, we deal with another unified concept, called the sequential order statistics (SOS), which has also a motivation in reliability analyses as discussed above. Specifically, when the component lifetimes are i.i.d., the OSs are suitable for describing the r-out-of- n system lifetime. Here failing a component does not effect the DFs of lifetimes of surviving components. Motivated by Cramer and Kamps [3], the failure of a component may result in a higher load on the surviving components and hence causes the lifetime distributions change. More precisely, suppose that Fj, for j=1,⋯n, denotes the common DF of the component lifetimes when n−j+1 components are jointly working. Then, the components begin to work independently at time t=0 with the common DF F1. If at time x1, the first component failure occurs, then the remaining n−1 components are working with the (left truncated) common DF F2 at x1. This process continues up to n−r+1(r=1,⋯,n) components with the common DF Fr work until the r-th failure occurs at time xr and hence the whole system fails. The mentioned system is called sequential r-out-of- n system or dynamic system and the system lifetime is then r-th observed component failure time, denoted by X(r)⋆. In the literature, (X(1)⋆,⋯,X(n)⋆) is called SOSs. Statistical properties of SOSs have been studied by Kamps [4,5], Cramer and Kamps [3,6], Balakrishnan et al. [7], Beutner and Kamps [8], Hashempour [9], Bedbur [10], Hashempour and Doostparast [11], and references therein. We considered the problem of estimating the parameters on the basis of s(≥2) independent SOSs samples under a new proposed power trend conditional proportional hazard rates (PTCPHR) model, defined by Fj(t)=1−(1−F0(t))aj for j=1,⋯,r and a>0, where the underlying CDF F0(t) is assumed to be the exponential distribution, i.e.,

F0(x;σ)=1−exp−xσ,x>0,σ>0.(1)

In this case, the hazard rate function of the CDF Fj, defined by hj(t)=fj(t)∕F¯j(t) for t>0 and j=1,⋯,n, is proportional to the hazard rate function of the baseline DF F0, i.e., hj(t)=ajh0(t) for t>0.

In this paper, we consider the problem of comparing non-homogeneous exponential populations on the basis of independent multiply SOS samples coming from non-homogeneous exponential populations under the abovementioned PTCPHR model. Thus, this paper is organized as follows: In Section 2, via the likelihood approach, statistical procedures including estimation, either point or interval, of the parameters as well as the problem of testing homogeneity of baseline exponential populations are considered. In Section 3, Bayesian approach is used for estimating parameters of interest. Finally, concluding remarks are given in Section 4.

2. LIKELIHOOD ANALYSIS

In the sequel, suppose that we have observed s(≥2) independent heterogeneous SOS samples. The available data may be represented as

x=[[xij]]i=1,⋯,s,j=1,⋯,r,(2)

where the i-th row of the matrix x in (2) denotes the SOS sample coming from the i-th population. The likelihood function (LF) for a single SOS sample is

L(F1,⋯,Fr;x)=Γ(n+1)Γ(n−r+1)∏j=1r−1fj(xj)F¯j(xj)F¯j+1(xj)n−jfr(xr)F¯r(xr)n−r.(3)

Therefore, the LF of the available data given by (2) is then obtained from (3) as

L(ℱ;x)=n!(n−r)!s∏i=1s∏j=1r−1fj[i](xij)F¯j[i](xij)F¯j+1[i](xij)n−jfr[i](xir)F¯r[i](xir)n−r,(4)

where F={Fj[i],i=1,⋯,s,j=1,⋯,r} and for i=1,⋯,s, j=1,⋯,r, F¯j[i](x)=1−Fj[i](x), and Fj[i] calls for the common DF of component lifetimes of the i–th dynamic system. For more details, see Cramer and Kamps [6,12] and Hashempour and Doostparast [11]. Suppose that, the baseline DF of the i-th population (i=1,⋯,s) follows the one-parameter exponential distribution with the mean σi. Substituting Equation (1) into Equation (4) and under the earlier mentioned PTCPHR model in Section 1, the LF of the available data reads

L(σ1,⋯,σs,a;x)=Bs∏j=1rajs∏i=1s1σir∏i=1s∏j=1rexp−xijmjσi =Bsasr(r+1)2∏i=1sσi−rexp−∑i=1s∑j=1rxijmjσi,(5)

where B=n!∕(n−r)! and mj=(n−j+1)aj−(n−j)aj+1 with convention (n−r)ar+1≡0. For sake of brevity, we assumed that the proportional parameter a are the same among the s sequential r-out-of- n systems. In this section, we consider the problem of homogeneity testing on the basis of independent SOS samples from different exponential populations, i.e.,

H0:σ1=⋯=σsH1:σi≠σj∃i≠j.(6)

In sequel, two cases are considered: (i) a known, and (ii) a unknown.

If a is known, the results of Cramer and Kamps ([6], Chapters 4 and 5) can be used. Also, see Schenk [13]. A summarize for this case is given in the appendix A.

Suppose that the parameter a in Equation (5) is unknown. In this case, calculations are complicated. Here, the Hessian matrix (H) is defined by

H=[[(∂2log(L)∕∂θi∂θj)1≤i,j≤s+1]],

where θi=σi(1≤i≤s) and θs+1=a. After some algebraic calculations, the Hessian matrix simplifies to

H=B11B12B21B22,(7)

where

B11=diagrσi2−2∑k=1rak(n−k+1)Dijσi3i=1,⋯,s,B22=diag−sra2jj=1,⋯,r,B12=B21T=(n−j+1)Dijσi2i=1,⋯,s,j=1,⋯,r,

where diagcii=1,⋯,m stands for a diagonal matrix with elements c1,⋯,cm on its main diagonal.

The Hessian matrix (7) is not necessary negative definite on the parameter space. Therefore, one needs to use numerically methods for maximizing the LF (5) with respect to a and σ1,⋯,σs. Note that the ML estimates of the parameters (if exist) are obtained numerically by solving the following likelihood equations:

σ̂̂i=1r∑j=1rxijm̂j=âr∑j=1r(n−j+1)jDij,i=1,⋯,s,â=rs∑i=1s∑j=1r(n−j+1)jDij∕σ̂̂i−1.

Here another approach for deriving the ML estimates of the parameters is suggested which is numerically simple. Recall that the profile LF for the parameter a is derived by substituting σ̂i given by (29), for i=1,⋯,s, into the LF (5) instead of σi, i.e., Lp(a;x)=L(σ̂1,⋯,σ̂s,a;x). The logarithm of the profile LF is

lp(a;x)=logLp(a;x) =sr(r+1)2log(a)−r∑i=1slog∑j=1r(n−j+1)ajDijr−rs.(8)

One may maximize the function (8) with respect to the parameter a numerically to derive the ML estimate of the parameter a. Then substituting this estimate into (29) (the appendix), the ML estimate of σ̂j, for i=1,⋯,s, is derived.

Consider again the hypotheses testing problem (6). It is easy to verify that the unique ML estimates of the parameters under the null hypothesis H0 are given by

σ̂̂0=1rs∑i=1s∑j=1rxijm̂0,j=â0rs∑i=1s∑j=1r(n−j+1)jDij,(9)

and

â0=rs∑i=1s∑j=1r(n−j+1)jDijσ̂̂0,(10)

where m̂0,j=(n−j+1)jâ0−(n−j)(j+1)â0, with convention â0(r+1)≡0. Therefore, the generalized likelihood ratio test (GLRT) statistic for the hypotheses testing problem (6) is

Λ2=∏j=1râ0ârs∏i=1sσ̂̂iσ̂̂0rexp∑i=1s∑j=1rm̂jσ̂̂i−m̂0,jσ̂̂0xij,(11)

where m̂j=(n−j+1)α̂j−(n−j)α̂j+1. The logarithm of the GLRT statistic Λ2 in Equation (11) reads

logΛ2=rslogâ0â+r∑i=1slogσ̂̂iσ̂̂0+∑i=1s∑j=1rm̂jσ̂̂i−m^0,jσ̂̂0xij.(12)

The null hypothesis H0 rejects if

−2logΛ2>c,(13)

where c is a positive constant and determined on the basis of the test level (Lehmann and Romano [14]).

Exact distribution of the statistic −2logΛ2 in Equation (13) under the null hypothesis H0 is complicated and we could not obtained an explicit expression. This remains as an open problem. In practice, one may use numerical methods such as Monte Carlo simulation study to derive the threshold c in the rejection region (13).

3. SOS-BASED BAYES ANALYSIS

Bayesian statistical inference on the basis of homogeneous multiply SOS samples have been considered in literature. For example, Mohie El-Din et al. [15] considered the problem of SOS-based Bayesian estimation and one-sample prediction under the CPHR model for Weibull and Pareto distributions. Schenk et al. [16] derived Bayes estimates of the parameters under the CPHR model when the baseline population follows the one- and two-parameter exponential distributions. Also, the Bayesian estimation and the two-sample Bayesian prediction based on homogeneous multiply SOS samples was studied with more details by Shafay et al. [17]. See also Hashempour and Doostparast [11]. In this section, we consider the problems of estimation, either point or interval, of parameters and hypotheses testing (6) on the basis of s independent heterogeneous SOS samples arising from the exponential populations under the PTCPHR model with known a. When the parameter a is unknown, the mentioned problems are complicated and works in this direction are under investigation and we hope to report findings in future.

3.1. Bayesian Point Estimates

We here consider the problem of estimating unknown parameters via a strict Bayesian approach. To do this, we assume that a is known and suggest the conjugate prior distributions for the scale parameters σi,i=1,⋯,s, i.e.,

σi~IG(di,ci),i=1,⋯,s,(14)

be independent random variables where IG(d,c) calls for the inverse gamma distribution with parameters d and c and density function

π(σ)=cdΓ(d)σ−(d+1)exp−cσ,σ>0,d>0,c>0.(15)

From Equation (14) and the LF (5), the joint posterior density function of σ1,…,σs is readily obtained as

π(σ1,…,σs∣x_)≡∏i=1sBsar(r+1)2cidiΓ(di)σi−(di+r)−1 ×∏i=1sexp⁡−∑j=1r(n−j+1)ajDij+ciσi,(16)

which implies

σi|x_~IGdi+r,∑j=1r(n−j+1)ajDij+ci,i=1,⋯,s.(17)

As we expected given x_, the parameter σi are independent. Under the squared error loss function, the Bayes estimate of the parameter is the mean of the associate posterior distribution. Here, the Bayes estimate of σi is

σ̂i,B=∑j=1r(n−j+1)ajDij+cidi+r−1=rσ̂i+cidi+r−1,i=1,⋯,s,(18)

where σ̂i is the ML estimate of σi given by Equation (29). Since the SOS samples are independent, the Bayes estimates in Equation (18) depend only on the respective samples.

Remark 3.1.

For i=1,⋯,s:

The mode of the posterior distribution (17) is
σ̂̂i,B=∑j=1r(n−j+1)ajDij+cidi+r+1=rσ̂i+cidi+r+1.(19)
The Bayes estimates (18) is a weighted mean of the mean of the prior (14) and the ML estimate (29); i.e., σ̂i,B=Eσiwi+1−wiσ̂i, where wi=di−1∕di+r−1;
The Bayes estimate (18) is biased, i.e., Bias(σ̂i,B)=σi(di−1)−ci∕di+r−1;
The risk function of the Bayes estimates (18) is
R(σ̂i,B,σi)=σi2r+(1−di)2+2ci(1−di)σi+ci2(di+r−1)2;(20)
The minimum of R(σ̂i,B,σi), as a function of σi, occurs at point bi(di−1)∕[(1−di)2+r];
For r=n and a=1, σ̂i,n=∑j=1nxij∕n and σ^i,B=(∑j=1nxij+ci)/(di+n−1), which are, respectively, the well-known ML and the Bayes estimates of the exponential parameters on the basis of a random sample of size n.

3.2. Bayesian Test

Under the null hypothesis H0:σ1=⋯=σs, we assume that the common value of σi(i=1,⋯,s), say σ, follows the IG(d0,c0)-distribution where d0 and c0 are known positive hyper parameters. Therefore, the Bayes factor (BF) is derived as (Berger [18])

BF=c0a0Γ(sr+d0)Γ(d0)∏i=1s(Ti+ci)di+1dicidi∑i=1sTi+c0−(sr+d0).(21)

Proof.

Under the null hypothesis H0 in (6), the LF (5) reduces to

L(σ;x)=n!(n−r)!s∏j=1rajsσ−srexp−1σ∑i=1s∑j=1rxijmj,

and

π(x|H0)=n!(n−r)!s∏j=1rajs∫0∞σ−srexp−E+c0σc0d0Γ(d0)σ−(d0+1)dσ =n!(n−r)!s∏j=1rajsc0d0Γ(d0)∫0∞exp−E+c0σσ−(sr+d0+1)dσ

where E=∑i=1s∑j=1rxijmj. If we use change of variable x=1∕σ, then

π(x|H0)=n!(n−r)!s∏j=1rajsc0d0Γ(d0)∫0∞xsr+d0−1exp−(E+c0)xdx =n!(n−r)!s∏j=1rajsc0d0Γ(d0)Γ(sr+d0)(E+c0)sr+d0.(22)

Under the alternative hypothesis H1 in (6), we have

L(σ1,…,σs;x)=n!(n−r)!s∏j=1rαjs∏i=1s1σiexp−∑j=1rxijmjσi =n!(n−r)!s∏j=1rajs∏i=1s1σiexp−Tiσi,

where Ti=∑j=1rxijmj, and

π(x∣H1)=n!(n−r)!s∏j=1rajs∏i=1s∫0∞⋯∫0∞1σiexp⁡−Tiσi.cidiΓ(di).σi−(di+1) ×exp⁡−ciσidσ1⋯dσs =n!(n−r)!s∏j=1rajs∏i=1scidiΓ(di)∏i=1s∫0∞⋯∫0∞σi−(di+2) ×exp⁡−Ti+ciσidσ1⋯dσs.

Let ui=1∕σi, then

π(x|H1)=n!(n−r)!s∏j=1rajs∏i=1scidiΓ(di)∏i=1s ×∫0∞⋯∫0∞uidi+1−1exp−(Ti+ci)du1⋯dus =n!(n−r)!s∏j=1rajs∏i=1scidiΓ(di)∏i=1sΓ(di+1)(Ti+ci)di+1 =n!(n−r)!s∏j=1rajs∏i=1sdiciai(Ti+ci)adi+1.(23)

From Equations (22) and (23), the Bayes factor is derived as

BF=c0a0Γ(sr+d0)Γ(d0)∏i=1s(Ti+ci)di+1dicidi∑i=1sTi+c0−(sr+d0).

Under the “0−Ki” loss function (Berger [18]), the Bayes test rejects the null hypothesis H0 in (6) if

BF<K0k1π1π0,

where πi and Ki, for i=1,2, are prior probability for the hypothesis Hi and the loss of the accepting Hi when Hj(j≠i) is correct, respectively.

3.3. HPD Credible Sets

In this section, the problem of finding highest posterior density (HPD) credible set is considered. To do this, from Equation (16), we see that

Ri=2∑j=1r(n−j+1)ajDij+ciσix~χ2(di+r)2.(24)

Therefore, an equi-tailed credible set at level γ for σii=1,⋯,s is obtained as

Ci(γ)=∑j=1r(n−j+1)ajDij+ciχ2(di+r),(1+γ)∕22,∑j=1r(n−j+1)ajDij+ciχ2(di+r),(1−γ)∕22.(25)

The HPD credible set for σii=1,⋯,s is

σi:σi−(di+r)−1exp−∑j=1r(n−j+1)ajDij+ciσi>k,(26)

where k is a constant determined by the condition on the level of the credible set, i.e.,

Pσi−(di+r)−1exp−∑j=1r(n−j+1)ajDij+ciσi>k|x=γ.(27)

Hence, the HPD credible set for σii=1,⋯,s in Equation (26) is an interval, say (Li,Ui), where Li and Ui are derived from the following non-linear equations:

UiLidi+r+1=expUi−LiUiLiΨi(x,a)Fχ2(di+r)22Ψi(x,a)Ui−Fχ2(di+r)22Ψi(x,a)Li=γiΓ(di)Γ(di+r)ci−diar(r+1)24Ψi(x,a)di+r

where Ψi(x,a)=∑j=1r(n−j+1)ajDij+ci, and Fχν(t) stands for the CDF of the χν-distribution. For the proof see Appendix B. In practice, one may use numerical methods for solving the above equations.

4. CONCLUSIONS

In this paper, based on independent SOSs coming from non-homogeneous exponential populations, the ML and the Bayes estimates of parameters were obtained on the basis of multiple SOS samples. The GLRT and Bayesian test were also derived for testing homogeneity of the exponential baseline populations. In Bayesian inference, when the parameter a is unknown, the mentioned problems are complicated and works in this direction are under investigation and we hope to report findings in future. The results of this paper may be extended in some directions. For example, the Bayes estimates may by obtained under some asymmetric loss functions such as linear-exponential loss function.

APPENDIX A

Suppose that the parameter a in Equation (5) is known. Under the null hypothesis H0 in (6) (Cramer and Kamps [12]), the unique ML estimate of the common mean of the s exponential populations, say σ0, is

σ̂0=1rs∑i=1s∑j=1r(n−j+1)ajDij=1rs∑i=1s∑j=1rxijmj,(28)

where Dij=xij−xi,j−1, for j=1,⋯,r. Here xi0:=0 for i=1,⋯,s. When the baseline exponential populations are heterogeneous, from Equation (28), the unique ML estimate of σi(i=1,⋯,s) is derived as

σ̂i=1r∑j=1r(n−j+1)ajDij=1r∑j=1rxijmj.(29)

Corollary 1.

Under the PTCPHR with the one-parameter exponential baseline CDF,

Ti=∑j=1r(n−j+1)ajDij~Γ(r,σi),i=1,⋯,s,(30)

where Γ(a,b) calls for the gamma distribution with shape and scale parameters a and b, respectively. Notice that ∑j=1rxijmj=∑j=1r(n−j+1)ajDij, for i=1,⋯,s.

Now consider the problem of hypotheses testing (6). The generalized likelihood ratio test (GLRT) statistic for testing the problem (6) is

Λ1=∏i=1sσ̂iσ̂0rexp∑i=1s∑j=1r1σ̂i−1σ̂0mjxij,(31)

where Ω={(σ1,⋯,σs):σi>0,i=1,⋯,s}=:ℝ+s is the whole parameter space and Ω0={(σ1,…,σs):σ1=⋯=σs} denotes the parameter space under the null hypothesis H0. After some algebraic manipulations, the logarithm of the GLRT statistic Λ1 given by Equation (31) reduces to

logΛ1=r∑i=1slogsTi∑j=1sTj,(32)

where Ti is defined by Equation (30) and “log” stands for the natural logarithm. Under the null hypothesis H0 in (6) and the usual regularity conditions (see Lehmann and Cassella [19]), −2logΛ1 has asymptotically the chi-square distribution with 1 degree of freedom. Thus, for large n, the rejection region of the GLR test of size γ is

−2logΛ1>χ1,1−γ2,(33)

where χυ,γ2 is the γ-th percentile of the chi-square distribution with υ degrees of freedom. Also, the actual level of the GLRT may be obtained by conducting a Monte Carlo simulation study for given a and σ.

The observed Fisher Information, denoted by i(σ̂1,⋯,σ̂s), on the basis of available SOSs data is equal to minus of the Hessian matrix evaluated at the MLEs of the parameters, i.e.,

i(σ^1,⋯,σ^s)=[[(−∂2log⁡(L)/∂σi∂σj)1≤i,j≤s]]|σ1=σ^1,⋯,σs=σ^s.

It is well known that the unique MLEs have asymptotically the multivariate normal distribution with mean vector (σ1,⋯,σs) and the variance–covariance matrix [i(σ̂1,⋯,σ̂s)]−1. Here, the observed Fisher Information for σi is derived as

i(σ^1,⋯,σ^s)=diagrσij^.

Notice that, by Equation (30), one can see that 2rσ̂i∕σi~χ2r2. Hence, an approximate equi-tailed confidence interval for σi is

σ^i−zγ/2σ^i2r,σ^i+zγ/2σ^i2r,(34)

where zγ stands for the γ-percentile of the standard normal distribution. Therefore, an equal-tail exact confidence interval at level 100γ% for σi(i=1,⋯,s) is

2rσ̂iχ2r,(1+γ)∕22,2rσ̂iχ2r,(1−γ)∕22,(35)

where χν,p2 calls for the p-th percentile of the χν2-distribution.

APPENDIX B

By definition, we have

∫LiUi∏j=1rajcidiΓ(di)σi−(di+r)−1exp−∑j=1rxijmj+ciσidσi=γi.

Let ti=2Ψ(x,a)∕σi, then

γi=∫2Ψ(x,a)Ui2Ψ(x,a)Li∏j=1rajcidiΓ(di)2Ψ(x,a)ti−(di+r)−1exp⁡−ti2−2Ψ(x,a)ti2dti =∏j=1rajcidiΓ(di)∫2Ψ(x,a)Ui2Ψ(x,a)Li(2Ψ(x,a))−(di+r)−1tidi+r+1exp⁡−ti2−2Ψ(x,a)ti2dti =−ar(r+1)2cidiΓ(di)(2Ψ(x,a))−(di+r)∫2Ψ(x,a)Ui2Ψ(x,a)Liti(di+r)−1exp⁡−ti2dti =−ar(r+1)2Γ(di+r)Γ(di)dici2Ψ(x,a)(di+r)∫2Ψ(x,a)Ui2Ψ(x,a)Liti(di+r)−1Γ(di+r)2di+rexp⁡−ti2dti =−Γ(di+r)Γ(di)cidi2Ψ(x,a)(di+r)ar(r+1)2Fχ2(r+di)2Ψ(x,a)Li−Fχ2(r+di)2Ψ(x,a)Ui.

After some algebraic calculations, γi is simplified to

ar(r+1)2Γ(di+r)Γ(di)cidi4Ψi(x,a)di+rFχ2(di+r)22Ψ(x,a)Ui−Fχ2(di+r)22Ψ(x,a)Li=γi,

and then

Fχ2(di+r)22Ψi(x,a)Ui−Fχ2(di+r)22Ψi(x,a)Li=γiΓ(di)Γ(di+r)ci−diar(r+1)24Ψi(x,a)di+r.

Since π(Li|x)=π(Ui|x), then

Li−(di+r)−1exp⁡−Ψi(x,a)Li=Ui−(di+r)−1exp⁡−Ψi(x,a)Uiand UiLidi+r+1=exp⁡Ψi(x,a)Li−Ψi(x,a)Ui =exp⁡Ui−LiUiLiΨi(x,a)..

Finally,

UiLidi+r+1=expUi−LiUiLiΨi(x,a)Fχ2(di+r)22Ψi(x,a)Ui−Fχ2(di+r)22Ψi(x,a)Li=γiΓ(di)Γ(di+r)ci−diar(r+1)24Ψi(x,a)di+r.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 19 - 1
Pages: 1 - 9
Publication Date: 2020/03/02
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.200224.005 How to use a DOI?
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TY  - JOUR
AU  - Majid Hashempour
AU  - Mahdi Doostparast
PY  - 2020
DA  - 2020/03/02
TI  - Dynamic Systems with Baseline Exponential Distribution Based on Sequential Order Statistics Under a Power Trend for Hazard Rates
JO  - Journal of Statistical Theory and Applications
SP  - 1
EP  - 9
VL  - 19
IS  - 1
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200224.005
DO  - 10.2991/jsta.d.200224.005
ID  - Hashempour2020
ER  -

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