Divergence-based robust inference under proportional hazards model for one-shot device life-test N. Balakrishnan∗, E. Castilla†, N. Martin‡ and L. Pardo§ Abstract In this paper, we develop robust estimators and tests for one-shot device testing under proportional hazards assumption based on divergence measures. Through a detailed Monte Carlo simulation study and a numerical example, the developed infer- ential procedures are shown to be more robust than the classical procedures, based on maximum likelihood estimators. 1 Introduction A one-shot device is a unit that performs its function only once, and after use the device either gets destroyed or must be rebuilt. For this kind of device, one can only know whether the failure time is either before or after a specific time, and consequently the lifetimes are either left- or right-censored, with the lifetime being less than the inspection time if the test outcome is a failure (resulting in left censoring) and the lifetime being more than the inspection time if the test outcome is a success (resulting in right censoring). Some examples of such one-shot devices include automobile air bags, missiles (Olwell and Sorell [2001]) and fire extinguishers (Newby [2008]). For devices with long lifetimes, accelerated life-tests (ALTs) are commonly used to induce quick failures. An ALT shortens the life span of the products by increasing the levels of stress factors, such as temperature, humidity, pressure and voltage. Then, a link function relating stress levels and lifetime is applied to extrapolate the lifetimes of units from accelerated conditions to normal operating conditions. The study of one-shot device from ALT data has been discussed considerably recently, mainly motivated by the work of Fan et al. [2009]. Under the classical parametric setup, product lifetimes are assumed to be fully de- scribed by a probability distribution involving some model parameters. This has been done with some common lifetime distributions such as exponential (Balakrishnan and Ling [2012]), gamma or Weibull (Balakrishnan and Ling [2013]). However, as data from one-shot devices do not contain actual lifetimes, parametric inferential methods can be very sensitive to violations of the model assumption. Ling et al. [2015] proposed a semi- parametric model, in which, under the proportional hazards assumption, the hazard rate is allowed to change in a non-parametric way. The simulation study carried out by Ling et al. [2015] shows that their proposed method works very well. However, this method suffer ∗McMaster University, ON, Canada. email: bala@mcmaster.ca †Complutense University of Madrid, Spain. email: elecasti@ucm.es ‡Complutense University of Madrid, Spain. email: nirian@estad.ucm.es §Complutense University of Madrid, Spain. email: lpardo@ucm.es 1 from lack of robustness, as it is based on the (non-robust) maximum likelihood estimator (MLE) of model parameters. Recently years, some work has been done for developing robust methods for one-shot device testing, most of it based on divergence measures (see, for example, Balakrishnan et al. [2019a,c,b]). In this paper, we extend the robust approach proposed in the above mentioned pa- pers and develop here robust estimators and tests for one-shot device testing based on divergence measures under proportional hazards model. Section 2 described the model and some basic concepts and results. The estimating equations and asymptotic properties of the proposed estimators are given in Section 3. Wald-type tests are then developed in Section 4 based on the proposed estimators, as a generalization of the classical Wald test. In Section 5, a simulation study is carried out to demonstrate the robustness of the proposed method. A numerical example is finally presented in Section 6, and some concluding comments are finally made in Section 7. 2 Model formulation Consider S constant-stress accelerated life-tests and I inspection times. For the i-th life-test, Ks devices are placed under stress level combinations with J stress factors, xs = (xs1, . . . , xsJ), of which Kis are tested at the i-th inspection time ITi, where Ks = ∑I i=1Kis and 0 < IT1 < · · · < ITI . Then, the numbers of devices that have failed by time ITi at stress xs are recorded as nis. One-shot device testing data obtained from such a life-test can then be represented as (nis,Kis,xs, ITi), for i = 1, 2, . . . , I and s = 1, 2, . . . , S. Instead of assuming that the true lifetimes of devices follow a specific parametric distribution such as exponential, gamma or Weibull, we assume here that the cumulative hazard function of the lifetimes of devices is of the proportional form H(t,x;η,α) = H0(t;η)λ(x;α), (1) where H0(t;η) is the baseline cumulative hazard function with η = (η1, . . . , ηI), and α = (α1 . . . , αJ) is a vector of coefficients for stress factors. The model in (1) is thus composed of two independent components, with one measuring the changes in the baseline (H0(t;η)) and the other influencing the stress factors (λ(x;α)). The corresponding reliability function is given by R(t,x;η,α) = exp (−H(t,x;η,α)) = R0(t;η)λ(x;α), (2) where R0(t;η) = exp(−H0(t;η)) is the baseline reliability function, with 0 < R0(ITI ;η) < R0(ITI−1;η) < · · · < R0(IT1;η) < 1. Therefore, we let γ(ηi) =  1−R0(ITI ;η) = 1− exp(− exp(ηI)), i = I, 1−R0(ITi;η) 1−R0(ITi+1;η) = 1− exp(− exp(ηi)), i = 1, . . . , I − 1. We then have R0(ITi;η) = 1− I∏ m=i {1− exp(− exp(ηm))} = 1−Gi, where Gi = ∏I m=i {1− exp(− exp(ηm))}. 2 We now assume a log-linear link function for relating the stress levels to the failure times of the units in the cumulative hazard function in (1), as λ(xs;α) = exp(αTxs) = exp  J∑ j=1 αjxsj  . 2.1 Maximum likelihood estimator Consider the proportional hazards model for one-shot devices in (1). The log-likelihood function based on these data is then given by `(n11, . . . , nIS ;η,α) = I∑ i=1 S∑ s=1 nis log [1−R(ITi,xs;η,α)] + (Kis − nis) log [R(ITi,xs;η,α)] + C = I∑ i=1 S∑ s=1 nis log [ 1− (1−Gi)exp( ∑J j=1 αjxsj) ] + (Kis − nis) log (1−Gi) exp  J∑ j=1 αjxsj + C, (3) where C is a constant not depending on η and α. Definition 1 Let θ = (η,α). The MLE, θ̂, of θ, is obtained by maximization of (3), i.e., θ̂ = arg min θ `(n11, . . . , nIS ;η,α). (4) In order to study the relation between the MLE, θ̂, in Definition 1, with the Kullback- Leibler divergence measure, we introduce the empirical and theoretical probability vectors, as follows: p̂is = (p̂is1, p̂is2) T = ( nis Kis , Kis − nis Kis )T , i = 1, . . . , I, s = 1, . . . , S, (5) πis(η,α) = (πis1(η,α), πis2(η,α))T , i = 1, . . . , I, s = 1, . . . , S, (6) where πis1(η,α) = 1−R(ITi,xs;η,α) and πis2(η,α) = R(ITi,xs;η,α). Definition 2 The Kullback-Leibler divergence measure between p̂is and πis(η,α) is given by dKL(p̂is,πis(η,α)) = p̂is1 log ( p̂is1 πis1(η,α) ) + p̂is2 log ( p̂is2 πis2(η,α) ) and similarly the weighted Kullback-Leibler divergence measure of all the units, where 3 K = ∑S s=1Ks is the total number of devices under the life-test, is given by I∑ i=1 S∑ s=1 Kis K dKL(p̂is,πis(η,α)) = 1 K I∑ i=1 S∑ s=1 Kis [ p̂is1 log ( p̂is1 πis1(η,α) ) + p̂is2 log ( p̂is2 πis1(η,α) )] = 1 K I∑ i=1 S∑ s=1 [ nis log ( nis Kis 1−R(ITi,xs;η,α) ) + (Kis − nis) log ( Kis−nis Kis R(ITi,xs;η,α) )] . (7) For more details, one may refer to Pardo [2005]. The relation between the MLE and the estimator obtained by minimizing the weighted Kullback-Leibler divergence measure is obtained on the basis on the following theorem. Theorem 3 The log-likelihood function `(n11, . . . , nIS ;η,α), given in (3), is related to the weighted Kullback-Leibler divergence measure through I∑ i=1 S∑ s=1 Kis K dKL(p̂is,πis(η,α)) = c− 1 K `(n11, . . . , nIS ;η,α), with c being a constant not dependent on η and α. Definition 4 The MLE, θ̂, of θ, can then be defined as θ̂ = arg min θ I∑ i=1 S∑ s=1 Kis K dKL(p̂is,πis(η,α)). (8) Remark 5 Suppose the lifetimes of one-shot devices under test follow the Weibull dis- tribution with the same shape parameter τ = exp(b) and scale parameters related to the stress levels, as = exp( ∑J j=1 cjxsj), s = 1, . . . , S. The cumulative distribution function of the Weibull distribution is then given by FT (t; as, τ) = 1− exp ( − ( t as )τ) , t > 0. If the proportional hazards assumption holds, then the baseline reliability and the coeffi- cients of stress factors are given by R0(t;β) = exp(−tτexp(−τc0)) and αs = −τcs, s = 1, . . . , S. Furthermore, we have ηi = log ( −log ( 1− 1−R0(ITi) 1−R0(ITi+1) )) , ηI = τ(log(ITI)− c0). 4 2.2 Weighted minimum DPD estimator Given the probability vectors p̂is and πis(η,α) in (5) and (6), respectively, the density power divergence (DPD) between them, as a function of a single tuning parameter β ≥ 0, is given by dβ(p̂is,πis(η,α)) = ( πβ+1 is1 (η,α) + πβ+1 is2 (η,α) ) − β + 1 β ( p̂is1π β is1(η,α) + p̂is2π β is2(η,α) ) + 1 β ( p̂β+1 is1 + p̂β+1 is2 ) , if β > 0, (9) and dβ=0(p̂is,πis(η,α)) = limβ→0+ dβ(p̂is,πis(η,α)) = dKL(p̂is,πis(η,α)), for β = 0. As the term 1 β ( p̂β+1 is1 + p̂β+1 is2 ) in (9) has no role in the minimization with respect to θ, we can consider the equivalent measure d∗β(p̂is,πis(η,α)) = ( πβ+1 is1 (η,α) + πβ+1 is2 (η,α) ) − β + 1 β ( p̂is1π β is1(η,α) + p̂is2π β is2(η,α) ) , and then can redefine the weighted minimum DPD estimator as follows. Definition 6 The weighted minimum DPD estimator for θ is given by θ̂β = arg min θ I∑ i=1 S∑ s=1 Kis K d∗β(p̂is,πis(η,α)), for β > 0, and for β = 0, we have the MLE, θ̂, as defined in (8). 3 Estimation and asymptotic distribution The estimating equations for the weighted minimum DPD estimator are as given in the following theorem. Theorem 7 For β ≥ 0, the estimating equations are given by I∑ i=1 S∑ s=1 δis(η) (Kis(1−R(ITi,xs;η,α))− nis) × [ (1−R(ITi,xs;η,α))β−1 +Rβ−1(ITi,xs;η,α) ] = 0I , I∑ i=1 S∑ s=1 δis(α) (Kis(1−R(ITi,xs;η,α))− nis) × [ (1−R(ITi,xs;η,α))β−1 +Rβ−1(ITi,xs;η,α) ] = 0J , where δis(η) = ∂R(ITi,xs;η,α) ∂η = −(1−Gi)λ(xs;α)−1λ(xs;α) ∂Gi ∂η , (10) δis(α) = ∂R(ITi,xs;η,α) ∂α = (1−Gi)λ(xs;α)log(1−Gi)λ(xs;α)xs, (11) with ∂Gi ∂ηu = { exp(ηu) exp(− exp(ηu))Gi/γ(ηu) , i ≤ u, 0 , i > u. (12) 5 Proof. The estimating equations are given by ∂ ∂η I∑ i=1 S∑ s=1 Kis K d∗β(p̂is,πis(η,α)) = I∑ i=1 S∑ s=1 Kis K ∂ ∂η d∗β(p̂is,πis(η,α)) = 0I , ∂ ∂α I∑ i=1 S∑ s=1 Kis K d∗β(p̂is,πis(η,α)) = I∑ i=1 S∑ s=1 Kis K ∂ ∂α d∗β(p̂is,πis(η,α)) = 0J , with ∂ ∂η d∗β(p̂is,πis(η,α)) = ( ∂ ∂η πβ+1 is1 (η,α) + ∂ ∂η πβ+1 is2 (η,α) ) − β + 1 β ( p̂is1 ∂ ∂η πβi1(θ) + p̂is2 ∂ ∂η πβis2(η,α) ) = (β + 1) ( πβis1(η,α)− πβis2(η,α)− p̂is1πβ−1is1 (η,α) + p̂is2π β−1 is2 (η,α) ) ∂ ∂η πis1(η,α) = (β + 1) ( (πi1(η,α)− p̂i1)πβ−1is1 (η,α)− (πis2(η,α)− p̂is2)πβ−1is2 (η,α) ) ∂ ∂η πis1(η,α) = (β + 1) ( (πis1(η,α)− p̂i1)πβ−1is1 (η,α) + (πi1(η,α)− p̂i1)πβ−1is2 (η,α) ) ∂ ∂η πis1(η,α) = (β + 1) (πis1(η,α)− p̂is1) ( πβ−1is1 (η,α) + πβ−1is2 (η,α) ) ∂ ∂η πis1(η,α) (13) and ∂ ∂α d∗β(p̂is,πis(η,α)) = (β + 1) (πis1(η,α)− p̂is1) ( πβ−1i1 (η,α) + πβ−1is2 (η,α) ) ∂ ∂α πis1(η,α). (14) But, ∂ ∂ηπis1(η,α) and ∂ ∂απis1(η,α) are as given in (10) and (11), respectively. See equa- tions (25) and (26) of Ling et al. [2015] for details. Theorem 8 Let θ∗ be the true value of the parameter θ. Then, the asymptotic distribution of the weighted minimum DPD estimator, θ̂β, is given by √ K(θ̂β − θ∗) L−→ K→∞ N ( 0I+J ,J −1 β (θ∗)Kβ(θ∗)J−1β (θ∗) ) , where Jβ(θ) and Kβ(θ) are given by Jβ(θ) = I∑ i=1 S∑ s=1 Kis K ∆is(η,α) [ (1−R(ITi,xs;η,α))β−1 +Rβ−1(ITi,xs;η,α) ] (15) Kβ(θ) = I∑ i=1 S∑ s=1 Kis K ∆is(η,α)(1−R(ITi,xs;η,α))R(ITi,xs;η,α) × [ (1−R(ITi,xs;η,α))β−1 +Rβ−1(ITi,xs;η,α) ]2 , (16) with ∆is(η,α) = ( δis(η)δTis(η) δis(η)δTis(α) δis(α)δTis(η) δis(α)δTis(α) ) , where δis(η) and δis(α) are as given in (10) and (11), respectively. 6 Proof. We denote uisj(η,α) = ( ∂ log πisj(η,α) ∂η , ∂ log πisj(η,α) ∂α )T = ( 1 πisj(η,α) ∂πisj(η,α) ∂η , 1 πisj(η,α) ∂πisj(η,α) ∂α )T = ( (−1)j+1 πisj(η,α) δis(η), (−1)j+1 πisj(η,α) δis(α) )T , with δis(η) and δis(α) as given in (10) and (11), respectively. Now, upon using Result 3.1 of Ghosh et al. [2013], we have √ K ( θ̂β − θ∗ ) L−→ K→∞ N ( 0I+J ,J −1 β (θ∗)Kβ(θ∗)J−1β (θ∗) ) , where Jβ(θ) = I∑ i=1 S∑ s=1 2∑ j=1 Kis K uisj(η,α)uTisj(η,α)πβ+1 isj (η,α), Kβ(θ) = I∑ i=1 S∑ s=1 2∑ j=1 Kis K uisj(η,α)uTisj(η,α)π2β+1 isj (η,α)− I∑ i=1 S∑ s=1 Kis K ξis,β(η,α)ξTis,β(η,α), with ξi,β(η,α) = 2∑ j=1 uisj(η,α)πβ+1 isj (η,α) = (δis(η), δis(α))T 2∑ j=1 (−1)j+1πβisj(η,α). Now, for uisj(η,α)uTisj(η,α), we have uisj(η,α)uTisj(η,α) = 1 π2isj(η,α) ( δis(η)δTis(η) δis(η)δTis(α) δis(α)δTis(η) δis(α)δTis(α) ) = 1 π2ij(θ) ∆is(η,α), with ∆is(η,α) = ( δis(η)δTis(η) δis(η)δTis(α) δis(α)δTis(η) δis(α)δTis(α) ) . It then follows that Jβ(θ) = I∑ i=1 S∑ s=1 Kis K ∆is(η,α) 2∑ j=1 πβ−1isj (η,α) = I∑ i=1 S∑ s=1 Kis K ∆is(η,α) ( πβ−1is1 (η,α) + πβ−1is2 (η,α) ) . From here on, and for simplicity, we will denoteR(ITi,x0;η,α) simply byR(ITi,x0;θ)). Based on Theorem 8, the asymptotic variance of the weighted minimum DPD estimator of the reliability at inspection time ITi under normal operating condition x0 is given by V ar(R(ITi,x0; θ̂β)) ≡ V ar(R(θ̂β)) = P TΣβ(θ̂β)P , 7 where Σβ(θ̂β) = J−1β (θ̂β)Kβ(θ̂β)J−1β (θ̂β), (17) Jβ (θ), Kβ (θ) are as given in (15) and (16), respectively, and P is a vector of the first- order derivates of R(ITi,x0;θ)) with respect to the model parameters (see (10) and (11)). Consequently, the 100(1− α)% asymptotic confidence interval for the reliability function R(θ) is given by( R(θ̂β)− z1−α/2se(R(θ̂β)), R(θ̂β) + z1−α/2se(R(θ̂β)) ) , where se(R(θ̂β)) = √ V ar(R(θ̂β)) and zγ is the uppper γ percentage point of the standard normal distribution. However, an asymptotic confidence interval may be satisfactory only for large sample sizes as it is based on the asymptotic properties of the estimators. Balakrishnan and Ling [2013] found that, in the case of small sample sizes, the distribution of the MLE of the reliability is quite skewed, and so proposed a logit-transformation for obtaining a confidence interval for the reliability function, which can be extended to the case of the weighted minimum DPD estimators of the reliabilities as well to obtain a confidence interval of the form:( R(θ̂β) R(θ̂β) + (1−R(θ̂β))T , R(θ̂β) R(θ̂β) + (1−R(θ̂β))/T ) , (18) where T = exp ( z1−α/2 se(R(θ̂β)) R(θ̂β)(1−R(θ̂β)) ) . 4 Wald-type tests Let us consider the function m : RI+J −→ Rr, where r ≤ (I + J) and m (θ) = 0r, (19) which corresponds to a composite null hypothesis. We assume that the (I +J)× r matrix M(θ) = ∂mT (θ) ∂θ exists and is continuous in θ and rank M (θ) = r. Then, for testing H0 : θ ∈ Θ0 against H1 : θ /∈ Θ0, (20) where Θ0 = { θ ∈ R(I+J) : m (θ) = 0r } , we can consider the following Wald-type test statistics: WK(θ̂β) = KmT (θ̂β) ( MT (θ̂β)Σ(θ̂β)M(θ̂β) )−1 m(θ̂β), (21) where Σβ(θ̂β) is as given in (17). Theorem 9 Under (19), we have WK(θ̂β) L−→ K→∞ χ2 r , where χ2 r denotes a central chi-square distribution with r degrees of freedom. 8 Proof. Let θ0 ∈ Θ0 be the true value of the parameter θ. It is clear that m ( θ̂β ) = m ( θ0 ) +MT ( θ̂β )( θ̂β − θ0 ) + op (∥∥∥θ̂β − θ0∥∥∥) = MT ( θ̂β )( θ̂β − θ0 ) + op ( K−1/2 ) . But, under H0, √ K ( θ̂β − θ0 ) L−→ K→∞ N ( 0(I+J),Σβ ( θ0 )) . Therefore, under H0, √ Km ( θ̂β ) L−→ K→∞ N ( 0r,M T ( θ0 ) Σβ ( θ0 ) M ( θ0 )) and taking into account that rank(M ( θ0 ) ) = r, we obtain KmT ( θ̂β ) ( MT ( θ0 ) Σβ ( θ0 ) M ( θ0 ))−1 m ( θ̂β ) L−→ K→∞ χ2 r . Because ( MT ( θ̂β ) Σβ ( θ̂β ) M ( θ̂β ))−1 is a consistent estimator of ( MT ( θ0 ) Σβ ( θ0 ) M ( θ0 ))−1 , we get WK ( θ̂β ) L−→ K→∞ χ2 r . Based on Theorem 9, we shall reject the null hypothesis in (20) if WK(θ̂β) > χ2 r,α, (22) where χ2 r,α is the upper α percentage point of χ2 r distribution. Wald-type test statistics based on weighted minimum DPD estimators have been con- sidered previously by a number of authors including Basu et al. [2016], Castilla et al. [2018] and Balakrishnan et al. [2019a,c,b]. 5 Monte Carlo Simulation Results In this section, an extensive simulation study is carried out for evaluating the proposed weighted minimum DPD estimators and Wald-type tests. The simulations results are computed based on 1, 000 simulated samples in the R statistical software. Mean square error (MSE) and bias are computed for evaluating the estimators in both balanced and unbalanced data sets, while empirical levels and powers are computed for evaluating the tests. 5.1 Weighted minimum DPD estimators Suppose the lifetimes of test units follow a Weibull distribution (see Remark 5). All the test units were divided into S = 4 groups, subject to different acceleration condi- tions with J = 2 stress factors at two elevated stress levels each, that is, (x1, x2) = {(55, 70), (55, 100), (85, 70), (85, 100)}, and were inspected at I = 3 different times, (IT1, IT2, IT3) = (2, 5, 8). 9 5.1.1 Balanced data We assume (c1, c2) = (−0.03,−0.03), c0 ∈ {6, 6.5} for different degrees of reliability and b ∈ {0, 0.5}. Note that the exponential distribution will be included as a special case when we take b = 0. In this framework, we consider outlying cells rather than outlying observations. A cell which does not follow the one-shot device model will be called an outlying cell or outlier. In this cell, the number of devices failed will be different than what is expected. This is inthe spirit of principle of inflated models in distribution theory (see Lambert (1992) and Heilbron (1994)). This outlying cell (taken to be i = 3, s = 4), is generated under the parameters (c̃1, c̃2) = (−0.027,−0.027) and b̃ ∈ {0.05, 0.45}. Bias of estimates are then computed for different (equal) samples sizesKis ∈ {50, 70, 100} and tuning parameters β ∈ {0, 0.2, 0.4, 0.6} for both pure and contaminated data. The obtained results are presented in Tables 4, 5, 6 and 7. As expected, when the sample size increases, errors tend to decrease, while in the contaminated data set, these errors are generally greater than in the case of uncontaminated data. Weighted minimum DPD estimators with β > 0 present a better behaviour than the MLE in terms of robustness. Note that reliabilities are underestimated and that the estimates are quite precise in all the cases. 5.1.2 Unbalanced data In this setting, we consider an unbalanced data set, in which at each inspection time i, (Ki1,Ki2,Ki3,Ki4) = (10r, 15r, 20r, 30r) for different values of the factor r ∈ {1, 2, . . . , 10}. We then assume (c0, c1, c2) = (6, 5,−0.03,−0.03), b = 0.5, and c̃2 = −0.027. MSEs of the parameter θ are then computed and the obtained results are presented in Figure 2. As expected, when the sample size increases, the MSE decreases, but lack of robustness of the MLE (β = 0) as compared to the weighted minimum DPD estimators with β > 0 becomes quite evident. 5.2 Wald-type tests To evaluate the performance of the proposed Wald-type tests, we consider the scenario of unbalanced data proposed discussed above. We consider the testing problem H0 : α1 = 0.04946 against H1 : α1 6= 0.04946, (23) Under the same simulation scheme as used above in Section 5.1.2, we first evaluate the empirical levels, measured as the proportion of Wald-type test statistics exceeding the corresponding chi-square critical value for a nominal size of 0.05. The empirical powers are computed in a similar manner, with α0 1 = 0.05276 (c1 = −0.032, c2 = −0.028). The obtained results are shown in Figure 3. In the case of uncontaminated data, the conventional Wald test has level to be close to nominal value and also has good power performance. The robust tests, however, has a slightly inflated level values (as compared to the nominal value), but possesses similar power as the conventional Wald test (which is evident from the Figure 3). But, when the data is contaminated, the level of the conventional Wald test turns out to be quite non-robust and takes on very high values as compared to the nominal level. This, in turn, results in higher power (see Figure 3). However, the proposed robust tests maintain levels close to the nominal value and also possesses good power values (as can be seen in the Figure 3). Thus, taking both level and power into account, the robust tests, though is slightly inferior to the conventional Wald test in the case of uncontaminated data, 10 turn out to be considerably more efficient than the conventional Walk test in the case of contaminated data 6 Application to Real Data 6.1 Testing on proportional Hazard rates Based on Balakrishnan and Ling [2012], we suggest a distance-based statistic on the form Mβ = maxi,s ∣∣∣nis −Kis(1−R(ITi,xs; θ̂β)) ∣∣∣ (24) as a discrepancy measure for evaluating the fit of the assumed model to the observed data. If the assumed model is not a good fit to the data, we will obtain a large value of Mβ. In fact, under the assumed model, we have nis ∼ Binomial(Kis, 1−R(ITi,xs;θ)), and so, by denoting Φis = dKis(1−R(ITi,xs; θ̂β))−Mβe and Ψis = bKis(1−R(ITi,xs; θ̂β))+ Mβc, the corresponding exact p-value is given by p− value = Pr ( maxi,s ∣∣∣nis −Kis(1−R(ITi,xs; θ̂β)) ∣∣∣ > Mβ ) = 1− Pr ( maxi,s ∣∣∣nis −Kis(1−R(ITi,xs; θ̂β)) ∣∣∣ ≤Mβ ) = 1− I∏ i=1 S∏ s=1 Pr (∣∣∣nis −Kis(1−R(ITi,xs; θ̂β)) ∣∣∣ ≤Mβ ) = 1− I∏ i=1 S∏ s=1 Pr (Φis ≤ nis ≤ Ψis) . (25) From (25), we can readily validate the proportional hazards assumption if the p-value is sufficiently large. 6.2 Choice of the tuning parameter In the preceding discussion, we have seen how weighted minimum DPD estimators with β > 0 tend to be more robust than the classical MLE overall whencontamination is present in the data. MLE has been shown to be more efficient when there is no contamination in the data. It is then necessary to provide a data-driven procedure for the determination of the optimal choice of the tuning parameter that would provide a trade-off between efficiency and robustness. One way to do this is as follows: In a grid of possible tuning parameters, apply a measure of discrepancy to the data. Then, the tuning parameter that leads to the minimum discrepancy-statistic can be chosen as the “optimal” one. A possible choice of the discrepancy measure could be Mβ, given in (24). Another idea may be by minimizing the estimated mean square error. This method, originally proposed by Warwick and Jones [2005], was applied in the context of one-shot devices in Balakrishnan et al. [2019a,b]. The estimation of the MSE is as follows: M̂SE(β) = (θβ − θP )T (θβ − θP ) + 1 K trace { J−1β (θβ)Kβ(θβ)J−1β (θβ) } , where θP is a pilot estimator, whose choice will affect the overall procedure. If we take θP = θ̂β, the approach coincides with that of Hong and Kim [2001], but it does not take into account the model misspecification. Note that, the need for a pilot estimator becomes a drawback of this procedure, as will be seen in the next section. 11 6.3 Electric Current data We now consider the Electric Current data (Ling et al. [2015]), in which 120 one-shot devices were divided into four accelerated conditions with higher-than-normal temperature and electric current, and inspected at three different times (see Table 1). In Table 2, estimates of the model parameters by the use of the proportional hazards model and the Weibull distribution (see Balakrishnan et al. [2019b]) are provided, for different values of the tuning parameter. Estimates of reliabilities and confidence intervals under the proportional hazards assumption are given in Table 3. Table 2 also presents the dvalues of the distance-statistic Mβ and the corresponding p- values. From these values, it seems that the proportional hazards assumption fits the data at least as well as the Weibull model. The best fit is obtained for β = 0.5. To complete the study, Warwick and Jones [2005] approach is achieved for different values of the pilot estimator in a grid of width 100. However, as pointed out before, the final choice of the optimal tuning parameter depends too much on the pilot estimator used (see Figure 1). Recently, Basak et al. [2020] proposed an “iterated Warwick and Jones algorithm” trying to solve this problem. Table 1: Electric Current data Inspection Time ITi 2 2 2 2 5 5 5 5 8 8 8 8 Temperature xs1 55 80 55 80 55 80 55 80 55 80 55 80 Electric current xs2 70 70 100 100 70 70 100 100 70 70 100 100 Number of failures nis 4 8 9 8 7 9 9 9 6 10 9 10 Number of tested items Kis 10 10 10 10 10 10 10 10 10 10 10 10 Table 2: Electric Current data: one-shot device testing data analysis by using the propor- tional hazards model and the Weibull distribution Proportional Hazards model Weibull distribution β Mβ p-value T ◦ current η1 η2 η3 Mβ p-value intercept T ◦ current shape 0 1.80 0.695 0.023 0.018 0.123 0.543 -2.182 1.80 0.695 7.022 -0.053 -0.040 -0.817 0.1 1.72 0.745 0.024 0.018 0.141 0.555 -2.283 1.72 0.745 7.398 -0.055 -0.043 -0.845 0.2 1.65 0.796 0.024 0.019 0.156 0.565 -2.399 1.65 0.796 7.803 -0.057 -0.046 -0.869 0.3 1.58 0.833 0.025 0.020 0.167 0.572 -2.534 1.57 0.833 8.254 -0.060 -0.050 -0.890 0.4 1.49 0.931 0.026 0.022 0.177 0.579 -2.695 1.49 0.931 8.747 -0.064 -0.054 -0.906 0.5 1.40 0.942 0.027 0.023 0.183 0.582 -2.887 1.40 0.942 9.324 -0.068 -0.058 -0.920 0.6 1.51 0.892 0.029 0.025 0.187 0.585 -3.130 1.51 0.892 10.026 -0.073 -0.063 -0.931 0.7 1.64 0.876 0.031 0.027 0.190 0.586 -3.438 1.64 0.876 10.868 -0.079 -0.069 -0.938 0.8 1.76 0.861 0.033 0.030 0.189 0.586 -3.798 1.76 0.861 11.827 -0.086 -0.076 -0.942 0.9 1.84 0.750 0.036 0.032 0.185 0.582 -4.106 1.84 0.750 12.575 -0.091 -0.082 -0.938 12 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0. 0 0. 2 0. 4 0. 6 0. 8 βP β o pt Figure 1: Electric Current data: estimation of the optimal tuning parameter depending on a pilot estimator by Warwick and Jones [2005] procedure Table 3: Electric Current data: estimates of reliabilities and corresponding confidence intervals β R(2, 25, 35; θ̂β) R(5, 25, 35; θ̂β) R(8, 25, 35; θ̂β) 0 0.817 (0.516, 0.949) 0.739 (0.397, 0.924) 0.689 (0.336, 0.907) 0.1 0.824 (0.526, 0.952) 0.751 (0.412, 0.928) 0.704 (0.353, 0.912) 0.2 0.833 (0.535, 0.956) 0.765 (0.427, 0.934) 0.721 (0.370, 0.919) 0.3 0.843 (0.545, 0.960) 0.780 (0.442, 0.941) 0.740 (0.387, 0.927) 0.4 0.855 (0.555, 0.965) 0.797 (0.457, 0.948) 0.760 (0.405, 0.936) 0.5 0.868 (0.566, 0.971) 0.816 (0.474, 0.956) 0.782 (0.423, 0.946) 0.6 0.884 (0.581, 0.976) 0.837 (0.493, 0.965) 0.807 (0.445, 0.956) 0.7 0.901 (0.601, 0.982) 0.861 (0.516, 0.973) 0.836 (0.471, 0.967) 0.8 0.918 (0.626, 0.987) 0.885 (0.544, 0.980) 0.863 (0.503, 0.975) 0.9 0.931 (0.649, 0.990) 0.902 (0.570, 0.985) 0.884 (0.533, 0.981) 7 Concluding Remarks In this paper, we have developed new estimators and tests for one-shot device testing under proportional hazards assumption. This semi-parametric model is presented as an alterna- tive to parametric models, by allowing the hazard rate increasing in a non-parametric way. An extensive simulation study carried out shows the robustness of the proposed methods of inference. Because model selection is an important part of reliability analysis, a test statistic for checking the proportional hazards assumption is presented as well and applied to the numerical example. As future work, it would be of interest to develop model selection criteria, and also to extend the proposed method to the case of competing risks problem, when there is more than one cause of failure of one-shot devices. 13 Table 4: Bias for the semi-parametric model with b = 0 and c0 = 6. Uncontaminated data Contaminated data Kis = 50 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -0.66688 -0.00494 -0.00276 -0.00053 -0.00372 0.09898 0.06722 0.03547 0.01708 η2 -0.01304 -0.00228 -0.00078 0.00109 -0.00131 0.06902 0.04716 0.02531 0.01286 η3 -3.92056 -0.02788 -0.01810 -0.01389 -0.01982 0.34916 0.23252 0.12087 0.05402 α1 0.03000 0.00010 0.00002 -0.00001 0.00002 -0.00281 -0.00193 -0.00107 -0.00056 α2 0.03000 0.00033 0.00027 0.00025 0.00030 -0.00259 -0.00167 -0.00081 -0.00028 R(15,x0) 0.79857 -0.00520 -0.00611 -0.00686 -0.00671 -0.03387 -0.02502 -0.01652 -0.01202 Uncontaminated data Contaminated data Kis = 70 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -0.66688 -0.00780 -0.00675 -0.00716 -0.00802 0.09876 0.06233 0.03209 0.01278 η2 -0.01304 -0.00459 -0.00386 -0.00410 -0.00465 0.06810 0.04315 0.02257 0.00948 η3 -3.92056 -0.03954 -0.03763 -0.04073 -0.04458 0.35084 0.21624 0.10254 0.03023 α1 0.03000 0.00027 0.00025 0.00026 0.00029 -0.00276 -0.00173 -0.00086 -0.00030 α2 0.03000 0.00035 0.00034 0.00038 0.00041 -0.00267 -0.00163 -0.00075 -0.00017 R(15,x0) 0.79857 -0.00197 -0.00221 -0.00223 -0.00221 -0.03131 -0.02076 -0.01246 -0.00749 Uncontaminated data Contaminated data Kis = 100 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -0.66688 -0.00778 -0.00682 -0.00711 -0.00785 0.09857 0.06207 0.03231 0.01320 η2 -0.01304 -0.00477 -0.00412 -0.00429 -0.00477 0.06776 0.04275 0.02248 0.00952 η3 -3.92056 -0.02739 -0.02332 -0.02387 -0.02586 0.36315 0.23019 0.12013 0.04993 α1 0.03000 0.00031 0.00028 0.00029 0.00031 -0.00271 -0.00169 -0.00084 -0.00029 α2 0.03000 0.00016 0.00013 0.00013 0.00015 -0.00287 -0.00185 -0.00099 -0.00045 R(15,x0) 0.79857 -0.00144 -0.00176 -0.00185 -0.00187 -0.03085 -0.02034 -0.01214 -0.00720 14 Table 5: Bias for the semi-parametric model with b = 0.5 and c0 = 6. Uncontaminated data Contaminated data Kis = 50 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -1.38827 -0.01224 -0.00988 -0.03700 -0.07648 0.03590 -0.00540 -0.03356 -0.08611 η2 -0.48138 -0.00687 -0.00537 -0.02153 -0.04394 0.02362 -0.00239 -0.01962 -0.04962 η3 -6.46391 -0.05973 -0.05148 -0.19693 -0.40632 0.14538 -0.03531 -0.17304 -0.47276 α1 0.04946 0.00032 0.00023 0.00124 0.00274 -0.00125 0.00010 0.00106 0.00325 α2 0.04946 0.00061 0.00056 0.00174 0.00361 -0.00097 0.00043 0.00155 0.00410 R(15,x0) 0.91810 -0.00348 -0.00385 -0.00348 -0.00272 -0.01136 -0.00486 -0.00361 -0.00255 Uncontaminated data Contaminated data Kis = 70 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -1.38827 -0.02188 -0.01955 -0.05972 -0.10800 0.03391 -0.01254 -0.06567 -0.12770 η2 -0.48138 -0.01318 -0.01171 -0.03481 -0.06298 0.02199 -0.00731 -0.03817 -0.07423 η3 -6.46391 -0.06923 -0.06287 -0.27869 -0.55900 0.16868 -0.03493 -0.30595 -0.67033 α1 0.04946 0.00062 0.00055 0.00202 0.00446 -0.00121 0.00033 0.00217 0.00530 α2 0.04946 0.00054 0.00051 0.00235 0.00433 -0.00128 0.00029 0.00260 0.00520 R(15,x0) 0.91810 -0.00174 -0.00199 -0.00127 -0.00004 -0.01041 -0.00289 -0.00121 0.00031 Uncontaminated data Contaminated data Kis = 100 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -1.38827 -0.01771 -0.01652 -0.06256 -0.08467 0.04334 -0.01518 -0.06228 -0.08025 η2 -0.48138 -0.01071 -0.00996 -0.03610 -0.04904 0.02774 -0.00875 -0.03612 -0.04659 η3 -6.46391 -0.05209 -0.04718 -0.27304 -0.41072 0.21133 -0.04194 -0.27462 -0.38078 α1 0.04946 0.00057 0.00053 0.00204 0.00340 -0.00145 0.00048 0.00226 0.00316 α2 0.04946 0.00034 0.00030 0.00217 0.00315 -0.00168 0.00026 0.00204 0.00295 R(15,x0) 0.91810 -0.00123 -0.00140 -0.00054 0.00003 -0.01060 -0.00223 -0.00066 -0.00012 15 Table 6: Bias for the semi-parametric model with b = 0.5 and c0 = 6.5. Uncontaminated data Contaminated data Kis = 50 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -0.66879 0.00223 0.00097 0.00020 -0.00900 0.17046 0.14959 0.12180 0.11436 η2 -0.01553 0.00320 0.00227 0.00156 -0.00465 0.11845 0.10401 0.08415 0.08029 η3 -4.42056 -0.01196 -0.00948 -0.01543 -0.03474 0.56481 0.50586 0.43705 0.36957 α1 0.03000 0.00004 0.00001 0.00004 0.00021 -0.00437 -0.00389 -0.00336 -0.00286 α2 0.03000 0.00014 0.00013 0.00018 0.00030 -0.00426 -0.00379 -0.00324 -0.00274 R(15,x0) 0.87247 -0.00529 -0.00555 -0.00534 -0.00530 -0.03609 -0.03299 -0.02952 -0.02561 Uncontaminated data Contaminated data Kis = 70 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -0.66879 0.00050 -0.00537 -0.00598 -0.00468 0.16516 0.14230 0.12082 0.10748 η2 -0.01553 0.00166 -0.00294 -0.00333 -0.00217 0.11371 0.09772 0.08279 0.07497 η3 -4.42056 -0.03260 -0.03492 -0.03697 -0.04026 0.55639 0.49446 0.42414 0.36377 α1 0.03000 0.00016 0.00017 0.00019 0.00021 -0.00434 -0.00386 -0.00329 -0.00322 α2 0.03000 0.00029 0.00032 0.00034 0.00036 -0.00418 -0.00367 -0.00314 -0.00318 R(15,x0) 0.87247 -0.00224 -0.00231 -0.00226 -0.00211 -0.03358 -0.03044 -0.02652 -0.02275 Uncontaminated data Contaminated data Kis = 100 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -0.66879 -0.00302 -0.00428 -0.00423 -0.00453 0.15788 0.14182 0.12319 0.10443 η2 -0.01553 -0.00136 -0.00242 -0.00237 -0.00256 0.10788 0.09716 0.08443 0.07154 η3 -4.42056 -0.02671 -0.02485 -0.02528 -0.02720 0.55329 0.49534 0.43019 0.36647 α1 0.03000 0.00019 0.00018 0.00019 0.00020 -0.00422 -0.00376 -0.00325 -0.00276 α2 0.03000 0.00014 0.00014 0.00014 0.00015 -0.00427 -0.00381 -0.00331 -0.00281 R(15,x0) 0.87247 -0.00126 -0.00139 -0.00140 -0.00135 -0.03191 -0.02878 -0.02528 -0.02190 16 Table 7: Bias for the semi-parametric model with b = 0.5 and c0 = 6.5. Uncontaminated data Contaminated data Kis = 50 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -1.38845 -0.00292 -0.02634 -0.06961 -0.10441 0.28565 0.19158 0.13289 0.07420 η2 -0.48171 -0.00007 -0.01454 -0.03906 -0.08819 0.18184 0.12322 0.12394 0.12467 η3 -7.28827 -0.08460 -0.15057 -0.34018 -0.97897 1.21433 0.81382 0.14850 0.33555 α1 0.04946 0.00058 0.00102 0.00237 -0.08206 -0.00889 -0.00592 -0.00116 -0.32679 α2 0.04946 0.00057 0.00108 0.00246 -0.10152 -0.00891 -0.00594 -0.00112 -0.39906 R(15,x0) 0.96322 -0.00163 -0.00176 -0.00145 0.00157 -0.02785 -0.01976 -0.01062 -0.00149 Uncontaminated data Contaminated data Kis = 70 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -1.38845 -0.01510 -0.03564 -0.05629 -0.08443 0.28682 0.18935 0.03294 0.02196 η2 -0.48171 -0.00878 -0.02069 -0.03247 -0.11914 0.18302 0.12036 0.02563 0.04542 η3 -7.28827 -0.06124 -0.15643 -0.22439 -0.99978 1.24650 0.87564 0.14490 -0.01407 α1 0.04946 0.00041 0.00110 0.00155 -0.01596 -0.00916 -0.00637 -0.00104 -0.15771 α2 0.04946 0.00047 0.00111 0.00168 -0.02075 -0.00910 -0.00631 -0.00113 -0.19334 R(15,x0) 0.96322 -0.00119 -0.00105 -0.00084 0.00146 -0.02804 -0.01977 -0.00976 -0.00119 Uncontaminated data Contaminated data Kis = 100 True value 0 0.2 0.4 0.6 0 0.2 0.4 0.6 η1 -1.38845 -0.00904 -0.01105 -0.05924 -0.23063 0.28616 0.19928 0.05644 0.03762 η2 -0.48171 -0.00531 -0.00635 -0.03368 -0.12308 0.18172 0.12619 0.03911 -0.01015 η3 -7.28827 -0.06888 -0.07584 -0.29436 -0.96654 1.22401 0.87425 0.21922 -0.38512 α1 0.04946 0.00048 0.00053 0.00202 -0.00879 -0.00897 -0.00635 -0.00168 -0.09403 α2 0.04946 0.00041 0.00047 0.00207 -0.01198 -0.00904 -0.00642 -0.00164 -0.11596 R(15,x0) 0.96322 -0.00015 -0.00021 0.00022 0.00236 -0.02602 -0.01814 -0.00878 -0.00050 17 ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 0. 1 0. 2 0. 3 0. 4 0. 5 pure data r M S E (θ ) ● β 0 0.2 0.4 0.6 ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 0. 1 0. 2 0. 3 0. 4 0. 5 contaminated data r M S E (θ ) ● β 0 0.2 0.4 0.6 Figure 2: MSEs for unbalanced data ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 0. 05 0. 06 0. 07 0. 08 pure data r em pi ric al le ve l ● β 0 0.2 0.4 0.6 ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 0. 05 0. 10 0. 15 0. 20 0. 25 0. 30 0. 35 contaminated data r em pi ric al le ve l ● β 0 0.2 0.4 0.6 ● ● ● ● ● ● ● ● ● ● 2 4 6 8 10 0. 05 0. 10 0. 15 pure data r em pi ric al p ow er ● β 0 0.2 0.4 0.6 ● ● ● ● ● ● ● ● ● 2 4 6 8 0. 05 0. 10 0. 15 0. 20 contaminated data r em pi ric al p ow er ● β 0 0.2 0.4 0.6 Figure 3: Estimated levels and powers for unbalanced data 18 A Power function of Wald-type tests In many cases, the power function of the proposed test procedure cannot be derived explic- itly. In the following theorem, we present an useful asymptotic result for approximating the power function of the Wald-type test statistics given in (22). We shall assume that θ∗ /∈ Θ0 is the true value of the parameter such that θ̂β P−→ K→∞ θ∗, and we denote `β (θ1,θ2) = mT (θ1) ( MT (θ2) Σβ (θ2)M (θ2) )−1 m (θ1) . We then have the following result. Theorem 10 We have √ K ( `β ( θ̂β,θ ∗ ) − `β (θ∗,θ∗) ) L−→ K→∞ N (0, σ2WK ,β (θ∗)) , where σ2WK ,β (θ∗) = ∂`β (θ,θ∗) ∂θT ∣∣∣∣ θ=θ∗ Σβ (θ∗) ∂`β (θ,θ∗) ∂θ ∣∣∣∣ θ=θ∗ . Proof. Under the assumption that θ̂β P−→ K→∞ θ∗, the asymptotic distribution of `β ( θ̂1, θ̂2 ) coincides with the asymptotic distribution of `β ( θ̂1,θ ∗ ) . A first-order Taylor expansion of `β ( θ̂β,θ ) at θ̂β, around θ∗, yields( `β ( θ̂β,θ ∗ ) − `β (θ∗,θ∗) ) = ∂`β (θ,θ∗) ∂θT ∣∣∣∣ θ=θ∗ ( θ̂β − θ∗ ) + op(K −1/2). Now, the result readily follows since √ K ( θ̂β − θ∗ ) L−→ K→∞ N (0J+1,Σβ (θ∗)) . Remark 11 Using Theorem 10, we can give an approximation for the power function of the Wald-type test statistic, given in (22), at θ∗, as follows: πW,K (θ∗) = Pr ( WK ( θ̂β ) > χ2 r,α ) = Pr ( K ( `β ( θ̂β,θ ∗ ) − `β (θ∗,θ∗) ) > χ2 r,α −K`β (θ∗,θ∗) ) = Pr √K ( `β ( θ̂β,θ ∗ ) − `β (θ∗,θ∗) ) σWK ,β (θ∗) > 1 σWK ,β (θ∗) ( χ2 r,α√ K − √ K`β (θ∗,θ∗) ) = 1− ΦK ( 1 σWK ,β (θ∗) ( χ2 r,α√ K − √ K`β (θ∗,θ∗) )) for a sequence of distributions functions ΦK (x) tending uniformly to the standard normal distribution Φ (x). It is clear that lim K→∞ πW,K (θ∗) = 1, i.e., the Wald-type test statistics are consistent in the sense of Fraser. 19 References N. Balakrishnan and M. H. Ling. Multiple-stress model for one-shot device testing data under exponential distribution. IEEE Transactions on Reliability, 61(3):809–821, 2012. N. Balakrishnan and M. H. Ling. Expectation maximization algorithm for one shot device accelerated life testing with weibull lifetimes, and variable parameters over stress. 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Journal of Statistical Computation and Simulation, 75(7):581–588, 2005. 21 1 Introduction 2 Model formulation 2.1 Maximum likelihood estimator 2.2 Weighted minimum DPD estimator 3 Estimation and asymptotic distribution 4 Wald-type tests 5 Monte Carlo Simulation Results 5.1 Weighted minimum DPD estimators 5.1.1 Balanced data 5.1.2 Unbalanced data 5.2 Wald-type tests 6 Application to Real Data 6.1 Testing on proportional Hazard rates 6.2 Choice of the tuning parameter 6.3 Electric Current data 7 Concluding Remarks A Power function of Wald-type tests