Statistical Inference for a Simple Step Stress Model with Competing Risks Based on Generalized Type-I Hybrid Censoring

Song Mao; Bin Liu; Yimin Shi

doi:10.21078/JSSI-2021-533-16

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Statistical Inference for a Simple Step Stress Model with Competing Risks Based on Generalized Type-I Hybrid Censoring

Song Mao , Bin Liu und Yimin Shi

Veröffentlicht/Copyright: 14. Dezember 2021

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Aus der Zeitschrift Journal of Systems Science and Information Band 9 Heft 5

Abstract

This paper investigates a simple step-stress accelerated lifetime test (SSALT) model for the inferential analysis of exponential competing risks data. A generalized type-I hybrid censoring scheme is employed to improve the efficiency and controllability of the test. Firstly, the MLEs for parameters are established based on the cumulative exposure model (CEM). Then the conditional moment generating function (MGF) for unknown parameters is set up using conditional expectation and multiple integral techniques. Thirdly, confidence intervals (CIs) are constructed by the exact MGF-based method, the approximate normality-based method, and the bias-corrected and accelerated (BCa) percentile bootstrap method. Finally, we present simulation studies and an illustrative example to compare the performances of different methods.

Keywords: step-stress accelerated lifetime testing; generalized hybrid censoring scheme; cumulative exposure model; moment generating function

1 Introduction

Accelerated lifetime test (ALT) is often implemented to collect failure data to shorten the testing time. Compared with the normal stress condition, items are exposed to more severe stress in ALT, to speed up the items’ failure. As a particular ALT case, the step stress ALT (SSALT) applies increased stress to test items in a discrete array^[1]. A simple SSALT considers only two stress levels. Generally, there is more than one factor for the failure of an item in practice. Ignoring competing risks would possibly lead to deviations or even conclusions contrary to facts. Therefore, many researchers have focused on the SSALT with competing risks data. Their papers can be divided into two categories: Statistical inference and optimal planning.

Plenty of the existing papers adopted the maximum likelihood method and the Bayesian method to discuss inferential problems under the assumption of the acceleration factor model (AF)^{[2, 4, 5, 6]}, the cumulative exposure model (CEM)^{[4, 5, 6]}, or the tampered failure rate model (TFR)^{[7, 8, 9]}. For example, Han and Balakrishnan^[4] analyzed a competing risk model in an SSALT based on the CEM model. They employed the moment generating function, the normal approximation, and the Bootstrap method to construct confidence intervals. Zhang, et al.^[7] studied inferential problems of the SSALT model with Weibull competing risks data under the TFR model. Meanwhile, optimal plans have also been a concern in the statistical field. Researches on different optimal plans can be classified according to evaluation criteria. For example, Bai and Chun^[10] presented a design scheme for an SSALT by taking the approximate variance of mean life at initial stress as a practical measurement. Based on the criteria of the minimum asymptotic variance of estimators, Pascual^{[11, 12]} discussed the design of an SSALT for Weibull and Lognormal competing risks data, respectively.

A censoring scheme is often employed to balance the tension between precision and cost in life tests. Compared with traditional censoring scheme, a generalized hybrid censoring scheme can obtain a certain amount of failure data within the scheduled time, and terminate the life test within a pre-specified time sequence^{[13, 14, 15]}. The generalized hybrid censoring scheme has attracted significant attention from statisticians due to its efficiency and controllability improvement. Balakrishnan, et al.^[16] extended the censoring scheme to a more general case and studied some inferential analyses for the exponential distribution. Park and Balakrishnan^[17] presented new results in computing Fisher information, expected censoring times, and failure number under hybrid and generalized hybrid censoring. Shafay^[18] developed an SSALT under generalized hybrid censoring from the perspective of likelihood as well as Bayes. He derived point estimators, confidence intervals, and the optimal stress-changing point. Researchers also considered the optimal plan in the context of hybrid censored life testing. Lin, et al.^[19] optimized the plan of k-level SSALT with hybrid censoring by minimizing the asymptotic variance of the pth percentile of the estimating lifetime at the normal condition. In consideration of the variance and the cost factors simultaneously, Bhattacharya, et al.^[20] proposed a multi-criteria-based optimization design for the analysis of hybrid censored data.

Extensive researches on SSALT focused on inferential problem or experimental design under traditional censored scheme, but few have considered it in the context of the generalized hybrid censoring scheme. Concerning that the generalized hybrid censoring scheme can improve the efficiency and the controllability of experiment, we have discussed inferential problems for a simple SSALT with exponential competing risks data under generalized type-I hybrid censoring scheme. The remainder of this paper is arranged as follows: Section 2 presents model assumptions, the likelihood function, and MLEs of unknown parameters. Section 3 draws an exact inference from MGF, and Section 4 derives exact CIs, asymptotic CIs, and BCa bootstrap CIs. Section 5 presents simulation studies and an illustrative instance for comparison, and Section 6 concludes.

2 Model Specification and MLEs

2.1 Description for Model and Data

Basic Assumptions

A1 There exist two independent failure factors for items at either of the two stress levels, where the two modes cannot happen simultaneously, either one can result in item failure.

A2 Lifetimes of items resulted from either failure factor follow an independent exponential distribution.

A3 SSALT in this paper is deployed under the assumption of CEM.

Let θ_jh (θ_jh>0) denote mean time to failure of a test item at stress level s_j due to risk factor h for j, h=1, 2. Then probability density function (PDF) and cumulative distribution function (CDF) of failure lifetime resulted from factor h are:

(1)ghx=1θ1hexp−xθ1h, 0<x<τ1,1θ2hexp−1θ1hτ1−1θ2hx−τ1, τ1≤x<∞,

(2)Ghx=1−exp−xθ1h, 0<x<τ1,1−exp−1θ1hτ1−1θ2hx−τ1, τ1≤x<∞.

The detailed life time procedures are:

Suppose n independent specimens are placed simultaneously on SSALT, predetermine the censoring scheme (k, r, τ₁, τ₂) , (k < r, 0 < τ₁< τ₂< ∞) . The k and r in that censoring scheme are specified minimum and maximum failure numbers. The τ₁ is the stress-change time, and the τ₂ is the pre-designated terminal time.
Let _z describe the indicator of failure factor. And then record the first failure lifetime x_1:n along with the risk factor z₁ on the initial stress level s₁.
Collect the relative information just as above until the stress changing time τ₁ is arrived. Find d₁ that satisfies xdi:n≤τ1≤xd1+1:n,then d₁ is the total failure number at stress level s₁.
Continue with the test and record failure information at the higher stress s₂ until the time τ∗=τ2∨xk:n∧xr:nterminates, where α ∨ β = max(α, β) , α ∧ β = min(α, β).

In other words, if xk:n<τ1<xr:n<τ2 or τ1<xk:n<xr:n<τ2,then terminate the test when the rth failure time is observed, and r is the total failure number; if x_k_:n< τ₁< τ₂< x_r_:n or τ₁< x_k_:n< τ₂< x_r_:n, then terminate experiment at time point τ₂, and find d₂ that satisfies xd2:n≤τ2<xd2+1:n,then d₂ is the total failure number; if τ1<τ2<xk:n<xr:n,then terminate the experiment at the failure of kth item x_k_:n, and k is the total failure number.

Noticing that if x_k_:n< x_r_:n< τ₁< τ₂, then terminate the experiment at the time point τ₁, and there exists no failure data at the higher stress s₂. Since that the object of this manuscript is competing risks items with long life-time, we need to elaborate the failure mechanism of competing risks items at the accelerated stress. That is, we have to observe at least one failure caused by each risk factor at each stress level. Obviously, this situation is not in consideration of our model.

Let v^∗ denote the total failure number, and let d_i denote the observed failure number up to τ_i (i = 1, 2). Then we can readily have the total failure number v∗=d2∨k∧ras well as eventual terminal time τ∗=τ2∨xk:n∧xr:n.So under generalized type-I hybrid censoring scheme, the observable failure lifetimes in an ascending order and corresponding failure risk factor are

x1:n,z1,x2:n,z2,⋯,xv∗:n,zv∗,

where z_i = 1 or 0 means failure x_i_:n is resulted from the risk factor 1 or 2.

Significantly, risk factor h has caused items failure, whose failure number as percentage of the total failure number at the stress level s_j is denoted by π_jh, where

πjh=1/θjh1/θj1+1/θj2,j,h=1,2.

Let N = (n₁₁, n₁₂, n₂₁, n₂₂). Here n_jh denote the failure number resulted from risk factor h at stress level s_j up to the time point τ_j.

2.2 MLEs

Since the failure is caused by only one of the factor risks, we can only observe the minimum latent lifetime under the competing risks model. Based on model assumptions and observable data above, we establish the likelihood function as

(3)Lx,z=n!n−v∗!∏i=1d1g1xi:nG¯2xi:nzig2xi:nG¯1xi:n1−zi×∏i=d1+1v∗g1xi:nG¯2xi:nzig2xi:nG¯1xi:n1−zi1−G¯1τ∗G¯2τ∗n−v∗.

Taking the logarithm of the above likelihood function, we can easily obtain the following formula after some computations:

(4)lz,x∝−∑j,h=12njhlnθjh−aθU1−bθU2,

in which a(θ)=1/θ11+1/θ12,b(θ)=1/θ21+1/θ22,U1=∑i=1d1xi:n+τ1n−d1,U2=∑i=d1+1v∗xi:n+n−v∗τ∗−τ1n−d1,v∗=d2∨k∧r,τ∗=τ2∨xk:n∧xr:n.

And we can derive MLE of θ_jh readily available from (4)

(5)θˆjh=Uj/njh,j,h=1,2.

It is obvious from (5) that there exists MLE for parameter θ_jh only when njh≠0;j,h=1,2,so we must observe no less than one failure due to any risk factor at any stress level, i.e.

ζv∗=N=n11,n12,n21,n22|∑j,h=12njh=v∗,njh≥1, for j,h=1,2.,

where ζv∗is a subset of universal set ζ.

3 Exact Statistical Inference

Lemma 1 (see [15]) Conditional on N = (n₁₁, n₁₂, n₂₁, n₂₂), the PDF ofx1:n,x2:n,...,xv∗:nis given by

fx1:n,⋯,xv∗:n|N=fx1:n,⋯,xv∗:n;D1=d1,v∗=vPN|D1=d1,v∗=v/PN=C∗∏i=1d1fxi:n∏i=d1+1vfxi:n1−Fτ∗n−v,

where C∗=Pv∗NPN×n!n−v!, and Pv∗N=d1n11π11n11π12d1−n11v∗−d1n21π21n21π22v∗−d1−n21.

Let D_i (i = 1, 2) indicate observable failure number up to τ_i. Based on models in Section 2, we can obtain the joint probability mass function (JPMF) of (D₁, D₂) as:

(6)PD1=i,D2=j=nin−ijFτ1iFτ2−Fτ1j−i1−Fτ2n−j.

In the following subsection, we derive probability mass function for the condition ζ. Since (D₁,D₂, v^∗) is discrete random variables, we can obtain

(7)PD1=i,v∗=v=∑jPD1=i,D2=j,v∗=v.

For fixed D₁ as well as the total failure number v^∗, we can readily find that

n1h|D1=d1,v∗=v∼Bind1,π1h,h=1,2.n2h|D1=d1,v∗=v,n1h∼Binv−d1,π2h,v=d2∨k∧r;h=1,2.

Conditional on D₁ = d₁, v^∗ = v, the JPMF of N = (n₁₁, n₁₂, n₂₁, n₂₂) is given by

(8)PN|D1=d1,v∗=v=d1n11π11n11π12d1−n11v−d1n21π21n21π22v−d1−n21.

Since ζr,ζd2,ζkare disjoint subsets of universal set ζ, it is clear to have

(9)Pζ=Pζr+Pζd2+Pζk.

Considering the domain of (D₁ = i,D₂ = j, v^∗ = v) in our model, we can immediately obtain P (ζ) by taking (6)∼(8) into (9).

Theorem 1Conditional on ζ, the MGF for parameter estimationθˆ1his

(10)Mθˆ1ht=1Pζ∑N∈ζr∑l=0d1Alr1−wn1haθ−d1expwn1hτd1l+∑N∈ζd2∑l=0d1Ald21−wn1haθ−d1expwn1hτd1l+∑N∈ζk∑l=0d1Alk1−wn1haθ−d1expwn1hτd1l,

in which,

Alr=−1lnd1d1l∑l1=r−d1n−d1n−d1l1p2l1q2n−d1−l1exp−aθτd1lPrN,Ald2=−1lnd1n−d1d2−d1d1lp2d2−d1q2n−d2exp−aθτd1lPd2N,Alk=−1lnd1n−d1d2−d1d1lp2d2−d1q2n−d2exp−aθτd1lPkN,τd1l=τ1n−d1+l,p2=1−exp−bθτ2−τ1,q2=1−p2.

proof Conditional on ζ, we obtain

Based on Lemma 1, conditional on N=n11,n12,n21,n22,N∈ζr,the PDF of sorted time x1:n,x2:n,⋯,xr:nis

fx1:n,⋯,xr:n|N=CrPNexp−aθ∑i=1d1xi:n−bθ∑i=d1+1r−1xi:n+n−r+1xr:n,

where Cr=n!n−r!aθd1bθr−d1exp−aθ−bθτ1n−d1PrN.

We can immediately have

(12)Eewθˆ1h|N=∫τ1τ2⋯∫τ1xd1+2:n∫0τ1⋯∫0x2:nfx1:n,⋯,xr:n|N×expwn1h∑i=1d1xi:n+τ1n−d1dx1:n⋯dxd1:ndxd1+1:n⋯dxr:n=CrPNexpwτ1n−d1n1h∫0τ1⋯∫0x2:nexp−a1hθ,w∑i=1d1xi:ndx1:n⋯dxd1:n×∫τ1τ2⋯∫τ1xd1+2:nexp−bθ∑i=d1+1r−1xi:n+n−r+1xr:ndxd1+1:n⋯dxr:n=1PN∑l=0d1Alr1−wn1haθ−d1expwn1hτd1l,

where a1hθ,w=1θ11+1θ12−wn1h;bθas before.

For N ∈ ζ^(d₂), the conditional PDF of x1:n,x2:n,⋯,xd2:nis

fx1:n,⋯,xd2:n|N=Cd2PNexp−aθ∑i=1d1xi:n−bθ∑i=d1+1d2xi:n,

where Cd2=n!n−d2!aθd1bθd2−d1exp−aθ+bθn−d1τ1−bθτ2n−d2Pd2N.

Then we can immediately arrive at

(13)Eewθˆ1h|N=∫0τ1⋯∫0x2:n∫τ1τ2⋯∫τ1xd1+2:nfx1:n,⋯,xd2:n|N×expwn1h∑i=1d1xi:n+τ1n−d1dx1:n⋯dxd1:ndxd1+1:n⋯dxd2:n=1PN∑l=0d1Ald21−wn1haθ−d1expwn1hτd1l.

Similarly, conditional on N ∈ ζ^(k), the PDF of x1:n,x2:n,⋯,xk:nis

fx1:n,⋯,xk:n|N=CkPNexp−aθ∑i=1d1xi:n−bθ∑i=d1+1k−1xi:n+n−k+1xk:n,

where Ck=n!n−k!aθd1bθk−d1exp−aθ−bθτ1n−d1PkN.

Based on the conditional PDF, it is immediate to have

(14)Eewθˆ1h|N=∫0τ1⋯∫0x2:n∫τ1τ2⋯∫τ1xd1+2:n∫τ2∞⋯∫xk−1:n∞fx1:n,⋯,xk:n|N×expwn1h∑i=1d1xi:n+τ1n−d1dxk:n⋯dxd2+1:ndxd1+1:n⋯dxd2:ndx1:n⋯dxd1:n=1PN∑l=0d1Alk1−wn1haθ−d1expwn1hτd1l.

Taking (12)∼(14) into (11), we can obtain the MGF of θˆ1has (10) shows.

Theorem 2Conditional on ζ, the MGF for parameter estimationθˆ2his

(15)Mθˆ2ht=1Pζ∑N∈ζr∑l=r−d1n−d1∑l1=0lBll1r1−wn2hbθ−r−d1×expwn2hτ2−τ1n−d1−l+l1+∑N∈ζd2∑l=0d2−d1Bld21−wn2hbθ−d2−d1expwn2hτd2l+∑N∈ζk∑l=0d2−d1Blk1−wn2hbθ−k−d1expwn2hτd2l,

where

Bll1r=−1l1nd1n−d1lll1p1d1q1n−d1q2n−d1−l+l1PrN,Bld2=−1lnd1n−d1d2−d1d2−d1lp1d1q1n−d1exp−bθτd2lPd2N,Blk=−1lnd1n−d1d2−d1d2−d1lp1d1q1n−d1exp−bθτd2lPkN,p1=1−exp−aθτ1,q1=1−p1, q2=exp−bθτ2−τ1, τd2l=τ2−τ1n−d2+l.

Proof Based on Lemma 1, Theorem 2 readily follows by simplifying the resulting expression.

4 Confidence Intervals (CIs)

In this section, CIs for vital parameters are constructed through the use of the exact method, the approximate method, and the BCa bootstrap method, as is shown in Figure 1.

Figure 1

Detailed procedures for research ideas

4.1 Exact CIs

Referring to [4], it is clear to derive the PDF of θˆjh,j,h=1,2based on the results from Theorem 1, Theorem 2.

(16)fθˆ1hx=1Pζ∑N∈ζr∑l=0d1Alrγx−τd1ln1h;d1,n1haθ+∑N∈ζd2∑l=0d1Ald2γx−τd1ln1h;d1,n1haθ+∑N∈ζk∑l=0d1Alkγx−τd1ln1h;d1,n1haθ,

(17)fθˆ2hx=1Pζ∑N∈ζr∑l=r−d1n−d1∑l1=0lBll1rγx−1n2hτ2−τ1n−d1−l+l1;r−d1,n2hbθ+∑N∈ζd2∑l=0d2−d1Bld2γx−τd2ln2h;d2−d1,n2hbθ+∑N∈ζk∑l=0d2−d1Blkγx−τd2ln2h;k−d1,n2hbθ,

wherein γx−δ;α,βindicate gamma distribution of shape parameter α, rate parameter β as well as varying shift parameter δ.

The tail probability for θˆjhcan be derived immediately from (16), (17).

(18)Pθˆ1h>b=1Pζ∑N∈ζr∑l=0d1AlrΓd1,n1haθb−τd1ln1h+∑N∈ζd2∑l=0d1Ald2Γd1,n1haθb−τd1ln1h+∑N∈ζk∑l=0d1AlkΓd1,n1haθb−τd1ln1h,

(19)Pθˆ2h>b=1Pζ∑N∈ζr∑l=r−d1n−d1∑l1=0lBll1rΓr−d1,n2hbθ×b−1n2hτ2−τ1n−d1−l+l1+∑N∈ζd2∑l=0d2−d1Bld2Γd2−d1,n2hbθb−τd2ln2h+∑N∈ζk∑l=0d2−d1BlkΓk−d1,n2hbθb−τd2ln2h,

in which b is arbitrary constant, x = max{x, 0}, and Γα,β=∫β∞1α−1!tα−1e−tdt.

By plenty of numerical simulations, it is interesting to find that when θj′h′is fixed, formula (18) is the increasing function of θ_jh, and formula (19) is also the increasing function of θ_jh. Here, j′≠j,h′≠h;j,j′,h,h′=1,2.Based on this helpful assumption, we construct the two-sided 100 (1 − α) % CI for parameter θ_jh, denoted by (θjhL,θjhU),that satisfies

PθjhLθˆjh>θˆjhobs=α/2,PθjhUθˆjh>θˆjhobs=1−α/2,j,h=1,2,

where θˆjhobsis the observed value of θˆjh.

4.2 Approximate CIs

As everyone knows, MLEs possess asymptotical optimal properties such as unbiased, efficient and asymptotic normality for enlarging sample size. Here, we construct asymptotic CIs by the above properties.

Let

Iθ=−E∂lθ∂θjh∂θj′h′=−Enjhθjh2−2Ujθjh3, when j=j′,h=h′,0,otherwise,j,h,j′,h′=1,2

denote expected Fisher information matrix. Substituting θˆjhwith θ_jh, we can immediately arrive at

Iˆjh=Iθθjh=θˆjh=njhθˆjh2,j,h=1,2.

Then we obtain the variance of θˆjhas

Vjh=Varθˆjh=θˆjh2/njh.

Based on asymptotic normality properties of MLEs, we take θˆjh−θjhVjhas a pivot for θ_jh to establish two-sided 100 (1 − α)% approximate CI for θ_jh as

θˆjh+zα/2Vjh,θˆjh+z1−α/2Vjh,j,h=1,2,

wherein z_α denotes the αth quantile of standard normal distribution.

4.3 BCa Bootstrap CIs

In this section, we establish asymptotic CIs by the BCa bootstrap method, and the detailed algorithm is as follows:

Step 1 Based on the initial sample data and pre-fixed scheme (k, r, τ₁, τ₂), calculate the MLEs θˆjhof unknown parameters θ_jh from (5).

Step 2 Generate bootstrap sample from θˆjh,then sort sample data and corresponding failure modes.

Step 3 Find the total failure number d₁ in the first stress level s₁, that satisfies xd1:n≤τ1<xd1+1:n,and record the corresponding failure modes.

Step 4 Continue with the test and record failure information at the higher stress environment until the time τ^∗ terminates. And we obtain the MLEs θˆ1=(θˆ111,θˆ121,θˆ211,θˆ221)from the above bootstrap sample.

Step 5 Repeat Steps 2∼4 B−1 times, and sort θˆiin ascending order, denoted as

θˆ∗=θˆ∗1,θˆ∗2,⋯,θˆ∗B,θˆ∗i=θˆ11∗i,θˆ12∗i,θˆ21∗i,θˆ22∗i,i=1,2,⋯,B.

Let αjh=Φ(zjh0+zjh0−zα/21−aˆjhzjh0−zα/2),βjh=Φ(zjh0+zjh0+zα/21−aˆjhzjh0+zα/2),j,h=1,2,among which zjh0=Φ−1(∑l=1BIθˆjh∗l<θˆjhB),Φ−1⋅is the converse of CDF for standard normal distribution, I (·) is the indicator function.

Here Φ⁻¹ (·) is the converse of CDF for standard normal distribution, I (·) is the indicator function. Referring to Han and Balakrishnan^[4], a good estimator for accelerated factor a_jh is

aˆjh=∑l=1njhθˆjhl−θˆjh⋅36∑l=1njhθˆjhl−θˆjh⋅23/2,j,h=1,2,

where θˆjh⋅=1njh∑l=1njhθˆjhl,θˆjhl is the MLE of θ_jh when the lth observation is deleted from the original sample for l = 1, 2, · · · , n_jh.

Noticing that the acceleration factor a_jh here is just to accelerate the bootstrap method and to reduce the bias. Finally, we can obtain 100 (1 − α)% BCa bootstrap CI for the parameter θjh as (θˆjh∗αjhB,θˆjh∗βjhB),j,h=1,2.

5 Case studies

5.1 Simulation Studies

Simulation studies using the Monte-Carlo method are conducted to assess the performances of different approaches in this section. The sample size is chosen to be n = 20, n = 40 and n = 80 to represent small, moderate and large sample cases. We also choose the true value of parameters as θ = (5, 7, 2, 3) to show the scenario where risk factor 1 is more likely to cause failure than factor 2. To show the flexibility of the hybrid censoring model, we predetermine several different choices of the minimum failure number k, maximum failure number r as well as the stress changing time point τ₁, τ₂. Adopting the coverage probability (CP) as an effective measurement to illustrate the above methods, we present numerical results based on the nominal level of 90%, 95%, 99% in Tables 1, 2. Furthermore, we extract data from Table 1 with the 95% nominal level and adopt a visualization technique to compare the performances of different methods, as shown in Figure 2, where, censoring scheme I, II, III correspond to n = 20, k = 8, r = 16, τ₁ = 0.5 and τ₂ = 1, 2, 3 respectively.

Figure 2

CP (in percentage) under different schemes based on different methods

Table 1

Estimations of CP (in percentage) for θ = (5, 7, 2, 3) with n = 20, k = 8, r = 16, τ₁ = 0.5

Nominal level		90			95			99
Values	τ₂	Exact	App	Boot	Exact	App	Boot	Exact	App	Boot
	1	89.4	90.0	76.8	94.8	89.4	84.1	97.9	92.1	94.1
θ₁₁	2	90.8	88.8	75.1	94.3	89.1	82.9	97.8	91.9	94.0
	3	89.8	88.3	78.3	94.9	89.1	85.8	98.4	91.8	94.0
	1	91.0	83.1	78.3	95.9	82.8	85.8	97.6	92.7	93.6
θ₁₂	2	88.6	82.4	76.2	93.0	82.3	84.2	96.9	91.8	92.3
	3	90.2	82.3	80.8	94.1	82.2	87.5	97.8	92.3	94.2
	1	89.4	88.2	87.9	94.9	91.3	90.8	98.6	94.9	94.6
θ₂₁	2	91.2	87.8	89.5	96.2	90.8	94.4	99.4	95.3	98.0
	3	92.7	87.6	89.5	96.6	90.7	94.4	99.9	95.0	98.5
	1	89.2	86.6	76.4	94.4	89.6	82.4	99.1	93.1	92.7
θ₂₂	2	90.5	87.2	85.4	93.5	90.2	90.2	99.1	94.1	95.2
	3	90.9	87.5	90.8	96.3	90.0	95.0	99.3	93.7	97.3

Table 2

Estimations of CP (in percentage) for θ = (5, 7, 2, 3) with medium and large samples

Values	θ₁₁		θ₁₂		θ₂₁		θ₂₂
Nominal level	App	Boot	App	Boot	App	Boot	App	Boot
	n = 40, k = 16, r = 32, τ₁ = 1, τ₂ = 4
90	89.7	87.6	89.1	88.6	89.1	89.6	88.5	89.4
95	92.9	94.4	91.8	94.2	92.1	93.4	92	94.1
99	96	97.5	95	97.4	96.2	98.2	95.9	98.6
	n = 40, k = 16, r = 32, τ₁ = 2, τ₂ = 4
90	91.1	90.4	90.2	88.8	88.2	91.8	86.8	87.6
95	94.1	94.5	93.2	93.7	90.6	95.4	89.8	93.9
99	97.4	97.9	96.9	97.6	94.6	98.8	93.8	97.1
	n = 80, k = 30, r = 60, τ₁ = 1, τ₂ = 4
90	91.1	89	90	88.4	89.6	90	90	90.7
95	93.8	94.3	93.6	93	94.2	94.8	93.4	95.4
99	97.3	98.1	96.7	98.3	97.2	99	96.8	98.8
	n = 80, k = 30, r = 60, τ₁ = 2, τ₂ = 4
90	91.1	88.8	91.1	90.4	88.7	91	88.5	91.2
95	94.5	93.8	94.1	95.9	92.4	95.4	91.8	96.1
99	97.8	98.1	97.4	98.5	96.4	99.4	95.5	98.9

Table 1 demonstrates that exact inferential methods are more stable than other methods in terms of CP in the following two aspects. Firstly, the exact CP is closer to the given confidence level under whatever censoring schemes. Moreover, when the stress changing time τ₁ increases, the results do not change much in exact CIs cases, but not so negligible in the other two approaches. The main reason is that exact method considers all the situations comprehensively which meet the needs of model, while approximate methods (App and Boot) are more random in nature especially for small sample size. We can also note that all approaches perform better at the higher stress level than at the lower stress level. It is not surprising because we expect to observe more failures occurred at the higher stress level when the stress changing time τ₁ is as small as 0.5. Furthermore, it is clear that the coverage probabilities both for θ₁₁ and θ₁₂ are closer to the nominal level with the increased stress. The primary reason for this phenomenon is that more failures are observed at lower stress levels as τ₁ increases. Last but not least, θ₁₁ performs better than θ₁₂. The possible cause may be that factor 1 observes more items than factor 2 does when θ₁₁ is smaller than θ₁₂.

When it comes to the large sample size, we compute the approximate CIs along with bootstrap CIs. The primary reason for none exact CIs is that the exact algorithm possesses complicated algorithms as well as the time-consuming codes. Table 2 indicates that both methods perform well in terms of enlarging the sample size. And the BCa percentile bootstrap method works better than the approximate method does.

Figure 2 reveals that the exact method outperforms the other two in terms of coverage probability when the sample size is small.

5.2 Illustrative Example

A set of competing risks sample is simulated to illustrate the methods above for parameters θ₁₁ = 8.96, θ₁₂ = 12.18, θ₂₁ = 4.48, θ₂₂ = 4.06 when n = 30, τ₁ = 2, τ₂ = 4. The failure lifetime and corresponding risk factor are presented in Table 3. Among which, the value j of ‘Factor’ means the failure is caused by risk factor j. Considering the flexibility of censoring scheme in this paper, we adopt k = 10, r = 20; k = 10, r = 24; k = 24, r = 28 to demonstrate various cases. And MLEs for parameters are presented in Table 4, where τ^∗, v^∗, n_jh (j, h = 1, 2) separately denotes eventual terminal time, failure numbers, and final failure number resulted from risk factor h at the stress level j.

Table 3

Data information under SSALT with two competing risks when n = 30, τ₁ = 2, τ₂ = 4 for parameters θ₁₁ = 8.96, θ₁₂ = 12.18, θ₂₁ = 4.48, θ₂₂ = 4.06

Stress level	Data
1	Lifetime	0.0879	0.1345	0.2255	0.5252	0.5321	0.7718	1.1060	1.7118	1.8415
	Factor	1	2	1	1	1	2	1	2	1

	Lifetime	2.2801	2.3770	2.4225	2.4242	2.472	2.4725	2.8488	3.1703	3.2771
	Factor	1	2	1	1	2	2	1	1	2
2	Lifetime	3.3204	3.3989	3.5175	3.7340	5.2087	5.3764	5.6431	5.8445	6.1079
	Factor	1	2	2	1	1	1	1	1	1
	Lifetime	6.3378	7.0315	7.6647
	Factor	1	2	2

Table 4

MLEs for parameters based on data in Table 3 when n = 30, τ₁ = 2, τ₂ = 4 under different censoring scheme

k	r	τ ^∗	v^∗	(n₁₁, n₁₂, n₂₁, n₂₂)	θˆ11,θˆ12,θˆ21,θˆ22
10	20	3.3989	20	(6, 3, 6, 5)	(8.1561, 16.3121, 3.7421, 4.4906)
10	24	3.7340	22	(6, 3, 7, 6)	(8.1561, 16.3121, 3.9593, 4.6192)
24	28	5.3764	24	(6, 3, 9, 6)	(8.1561, 16.3121, 4.2843, 6.4265)

Based on simulated MLEs in Table 4, we calculate CIs for parameters by the exact method, the approximate method, as well as the BCa percentile bootstrap method. And the CIs under different censoring schemes based on a nominal level of 95% are given in Table 5, where ‘Inf’ means that the 95% upper bound for θ₁₂ with k = 10, r = 24 doesnt exist or is infinite.

Table 5

Interval estimations based on data in Table 3 when n = 30, τ₁ = 2, τ₂ = 4 under different censoring scheme

Parameter	k	r	Exact CI	Approximate CI	Bootstrap CI
	10	20	(2.9536, 19.1792)	(1.3015, 19.5895)	(3.9275, 17.9864)
θ₁₁	10	24	(3.0642, 19.2020)	(1.3099, 19.2293)	(3.8261, 18.2321)
	24	28	(3.0671, 19.1913)	(1.2992, 19.3545)	(3.5082, 18.2654)
	10	20	(5.9920, 105.2128)	(0.3146, 52.1048)	(6.3969, 53.6694)
θ₁₂	10	24	(6.2255, Inf)	(0.3201, 51.5668)	(6.3793, 52.8081)
	24	28	(6.0425, 70.2365)	(0.3260, 52.2783)	(5.5852, 52.6172)
	10	20	(1.7692, 10.4090)	(0.5536, 8.3443)	(1.6146, 10.4561)
θ₂₁	10	24	(1.9950, 9.3453)	(0.3201, 8.7301)	(1.9249, 11.0384)
	24	28	(1.8187, 9.2310)	(1.3600, 7.7770)	(2.2375, 8.1590)
	10	20	(1.9858, 14.3076)	(0.4147, 11.1992)	(1.6860, 12.6403)
θ₂₂	10	24	(2.2116, 11.8925)	(0.6333, 11.1284)	(2.0715, 14.3450)
	24	28	(2.2875, 16.8124)	(0.9930, 14.6318)	(3.0491, 17.1331)

Table 5 manifests that the lengths of approximate CIs and Bootstrap CIs are sometimes longer or shorter than those of exact CIs, indicating lower or higher CP for the two methods. Furthermore, it is also obvious that the CIs for parameter θ₁₂ are considerably wider than other parameters no matter which method is used. The primary reason is that Factor 1 possesses a greater possibility to make items fail than Factor 2 at Stress level 1 does, as is evident in Table 4.

6 Conclusions

With the rapid development of the engineering and manufacturing industry, the products’ service time is getting increasingly longer. Many products do not fail or barely fail under normal stress conditions, in which case applying some stress levels that are harsher than the usage environment allows us to collect more information on failed products in a limited time. On the other hand, compared with the traditional tail truncation method, the generalized type I hybrid censoring can observe a certain amount of failure data within the scheduled time, thus ensuring that the estimation accuracy meets the test requirements. This paper incorporates this censoring scheme into simple SSALT, which significantly improves the test efficiency and has practicality and specific socio-economic effects.

This paper has discussed statistical problems for exponential competing risks items under a simple SSALT based on the generalized hybrid censoring scheme. We establish CIs with MGF-based exact approach, asymptotic normality-based approach, and BCa percentile bootstrap approach respectively. Performances of different approaches are also compared through extensive simulations, using coverage probability as a measure. The results show that the exact method outperforms the other two methods when the sample size is small. Nevertheless, when the sample size increases, BCa is the best choice due to its accuracy and simple computation.

Supported by Humanities and Social Sciences Fund in Ministry of Education in China (18YJC910009), the National Natural Science Foundation of China (12061091), Program for the Philosophy and Social Sciences Research of Higher Learning Institutions of Shanxi (201803050)

Acknowledgements

The authors gratefully acknowledge the editor and anonymous referees for their insightful comments and helpful suggestions that led to a marked improvement of the article.

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Received: 2021-01-15

Accepted: 2021-08-20

Published Online: 2021-12-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.21078/JSSI-2021-533-16

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step-stress accelerated lifetime testing; generalized hybrid censoring scheme; cumulative exposure model; moment generating function

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