Parametric inference for progressive Type-I hybrid censored data on a simple step-stress accelerated life test model

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Abstract

This paper considers a simple step-stress accelerated life test model under progressive Type-I hybrid censoring scheme. The progressive Type-I hybrid censoring scheme and statistical method in synthetic accelerated stresses are provided so as to decrease the lifetime and reduce the test cost. An exponentially distributed life of test units and a cumulative exposure model are assumed. The maximum likelihood estimates of the model parameters are obtained using a pivotal quantity. Two useful lemmas and a theorem are given to construct the approximate confidence intervals for the model parameters. Finally, simulation results are provided to assess the method of inference developed in this article. The simulation results show that the method does improve for large sample size.

Introduction

Due to the rapid development of product design and manufacture in life cycle, the mean times to failure of products is sometimes too large under typical operating conditions to adopt the accelerated life test. Accelerated life test (ALT) experiments are analyzed in terms of a model to relate life length to stress for the product in reliability and survival analysis. Stress factors can include temperature, voltage, pressure, vibration, cycling rate, or any other factor that directly affects the life of the products. Further the step-stress ALT (SSALT), which is a special class way of ALT, allows the stress to be changed at pre-specified times or upon the occurrence of a fixed number of failures. In this scheme, stress levels can be choose one or more which are repeatedly changed at pre-fixed times.

The step-stress model has been studied extensively in the literature. Bagdonavičius et al. [2] discussed the applications of the additive accumulation of damages or the accelerated failure time and the proportional hazards models in accelerated life testing with step-stresses. Bai and Kim [3] studied the optimum simple time-step and failure-step stress accelerated life tests for the case where a pre-specified censoring time is involved. Khamis and Higgins [9] considered quadratic stress-life relation and derived the optimum three-step SSALT for the exponentially distributed Type-I censored data. Khamis [10] proposed an optimal m-step SSALT design with k stress variables, assuming complete knowledge of the stress-life relation with multiple stress variables. Tang [16] derived a simple SSALT for an optimum hold time under low stress and an optimum low stress level by taking the target acceleration factor into consideration.

On the other hand, progressive censoring allows to continual removal of a pre-specified number of un-failed test units at the end of testing time at each stage. Recently, progressive hybrid censoring scheme has become quite popular for analyzing highly reliable data. The apparent benefit of this scheme is that the length of the experiment can be quite large. Balakrishnan and Xie [6] considered the simple step-stress model under the exponential distribution with Type-I hybrid censored. Gouno et al. [8] proposed a k-step-stress accelerated test with equal duration steps and investigated in detail the case of progressively Type-I right-censored data with a single stress variable. Xiong [17] discussed the inferences of parameters on the simple step-stress model in accelerated life testing with Type-II censoring. For some more results on step-stress model, one may refer to Refs. [1], [4], [5], [7], [11], [12], [13], [14], [15], [18], [19], [20].

This paper deal with a simple step-stress ALT model that is subject to progressive Type-I hybrid censoring scheme (HCS). The maximum likelihood estimates (MLEs) for the model parameters are obtained in Section 3. Section 4 presents the confidence intervals estimates of the model parameters using a pivotal quantity. A simulated example to illustrate the estimation process and Monte Carlo simulation results are given in Section 5. Finally, in Section 6, some concluding remarks are discussed.

Assumptions

  • (1)

    Two stress levels s1 and s2 (s1 < s2) are used.

  • (2)

    For any level of stress, the life distribution of test unit is exponential.

  • (3)

    The mean life is assumed to be a log-linear function of the stress level s.log[θ(s)]=α+β·swhere α,β(β < 0) are unknown parameters depending on the nature of the product and the test method. Then θ1 < θ2.

  • (4)

    A cumulative exposure model holds: the remaining life of a test unit depends only on its present cumulative exposure.

Section snippets

Simple step-stress ALT

From the assumptions of cumulative exposure model and exponentially distributed life, the lifetime distributions at s1 and s2 are assumed to be exponential with failure rates θ1 and θ2. The probability density function (PDF) and cumulative distribution function (CDF) are given asfk(x;θk)=1θkexpxθk,x0,θk>0,k=1,2Fk(x;θk)=1expxθk,x0,θk>0,k=1,2.

Then, under simple step-stress test we have the cumulative exposure distribution (CED) G(x) asG(x)=G1(x)=F1(x;θ1),0<x<T1G2(x)=F2(xT1+s;θ2

Maximum likelihood estimates (MLEs) for α and β

From the CED in (3) and the corresponding PDF in (4), we obtain the likelihood function for the three cases based on the progressive Type-I hybrid censoring sample as follows:

  • Case I:

    x1,n < x2,n <  < xr,n  T1 < T, if xr,n  T1 < TL(θ1,θ2|x)k=1rg1(xk,n)[1G1(xr,n)]nr,

  • Case II:

    x1,n<<xN1,nT1<xN1+1,n<xr,nT, if T1 < xr,n  TL(θ1,θ2|x)k=1N1g1(xk,n)[1G1(T1)]ck=N1+1rg2(xk,n)[1G2(xr,n)]nrc,

  • Case III:

    x1,n<<xN1,nT1<xN1+1,n<xN1+N2,nT, if xr,n > TL(θ1,θ2|x)k=1N1g1(xk,n)[1G1(T1)]ck=N1+1N1+N2g2(xk,n)[1G2(T)]n(N1+N2)c,

Confidence interval estimations (CIE) for α and β

We construct the approximate CI for α and β in this section. Since the exact conditional PDF of α and β are quite complicated, a novelty procedure is presented to obtain the confidence intervals estimates. First we give two useful lemmas. Lemma 1 is a simpler version of [8, pp. 1271, theorem 3.5.1].

Lemma 1

Let:

  • (1)

    S1, S2, …, Sn be i.i.d. exponential r.v. with mean 1.

  • (2)

    S(1), S(2), ≤   S(r) be the r smallest ordered observations.

Then,

  • (1)

    2nS(1)  χ2(2).

  • (2)

    2i=1rS(i)+(nr)S(r)nS(1)χ2(2r2).

  • (3)

    n(r1)S(1)i=1rS(i)+(nr)S(r)

Numerical results

In order to evaluate the performance of the resulting estimators of the parameters in Section 4, a Monte Carlo simulation study is conducted and the results are presented in this section. The absolute relative bias (RABias) and mean square error (MSE) have been considered, whereRABias(θˆ)=θˆθθ,MSE(θˆ)=E(θˆθ)2

The simulation procedures are described below:

  • Step 1:

    A sample S from the exponential distribution with mean 1 is first simulated. The transformation (10) is used to obtain the sample from

Conclusions

This article proposes a simple step-stress ALT model that is subject to progressive Type-I hybrid censoring scheme, and presents the maximum likelihood estimates and the confidence intervals estimates of the model parameters. Since the exact conditional PDF of model parameters are quite complicated, two useful lemmas and a theorem are given to construct the approximate confidence intervals for the model parameters. From simulation studies, it is shown that the method does improve for large

Acknowledgements

The authors would like to thank the editor, the associate editor and the referee for their useful and valuable comments on improving the quality of the paper.

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