
In this paper, we investigate classical and Bayesian estimation of stress-strength reliability δ=P(X>Y) under an adaptive progressive type-Ⅱ censored sample. Assume that X and Y are independent random variables that follow inverse Weibull distribution with the same shape but different scale parameters. In classical estimation, the maximum likelihood estimator and asymptotic confidence interval are deduced. An approximate maximum likelihood estimator approach is used to obtain the explicit form. In Bayesian estimation, the Bayesian estimators are derived based on symmetric entropy loss function and LINEX loss function. Due to the complexity of integrals, we proposed Lindley's approximation to get the approximate Bayesian estimates. To compare the different estimators, we performed Monte Carlo simulations. Under gamma prior, the approximate maximum likelihood estimator performs better than Bayesian estimators. Under non-informative prior, the approximate maximum likelihood estimator has the same behavior as Bayesian estimators. In the end, two data sets are used to prove the effectiveness of the proposed methods.
Citation: Xue Hu, Haiping Ren. Statistical inference of the stress-strength reliability for inverse Weibull distribution under an adaptive progressive type-Ⅱ censored sample[J]. AIMS Mathematics, 2023, 8(12): 28465-28487. doi: 10.3934/math.20231457
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[9] | Essam A. Ahmed, Laila A. Al-Essa . Inference of stress-strength reliability based on adaptive progressive type-Ⅱ censing from Chen distribution with application to carbon fiber data. AIMS Mathematics, 2024, 9(8): 20482-20515. doi: 10.3934/math.2024996 |
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In this paper, we investigate classical and Bayesian estimation of stress-strength reliability δ=P(X>Y) under an adaptive progressive type-Ⅱ censored sample. Assume that X and Y are independent random variables that follow inverse Weibull distribution with the same shape but different scale parameters. In classical estimation, the maximum likelihood estimator and asymptotic confidence interval are deduced. An approximate maximum likelihood estimator approach is used to obtain the explicit form. In Bayesian estimation, the Bayesian estimators are derived based on symmetric entropy loss function and LINEX loss function. Due to the complexity of integrals, we proposed Lindley's approximation to get the approximate Bayesian estimates. To compare the different estimators, we performed Monte Carlo simulations. Under gamma prior, the approximate maximum likelihood estimator performs better than Bayesian estimators. Under non-informative prior, the approximate maximum likelihood estimator has the same behavior as Bayesian estimators. In the end, two data sets are used to prove the effectiveness of the proposed methods.
The stress-strength model has an essential role in lifetime study and engineering application. In terms of reliability, stress-strength reliability is an interesting topic, which is defined as δ=P(X>Y), X denotes the strength of a system or unit with stress Y. The system or unit works normally when X>Y. Aziz and Chassapis [1] considered the performance δ=P(X>Y) of a gearing system, which denotes the stress on the gear tooth and X denotes the strength of the tooth root. Dong et al. [2] studied the biomechanical performance δ=P(X>Y) of the healthy and reconstructed pelvic model, which denotes the strength of the pelvic model and Y indicates daily activities such as knee bending, standing up, stair descent and stair ascent. Zhou et al. [3] studied the effect of the stress-strength ratio and fiber length on the creep properties of polypropylene fiber-reinforced alkali-activated slag concrete.
Since the application of stress-strength reliability is wide, its statistical inference has attracted the concern of many researchers. Mehdi and Mehrdad [4] assumed that strength X has the Pareto distribution within outliers but stress Y follows an unsullied Pareto distribution, and considered the stress-strength reliability estimation. They found that maximum likelihood estimation and the modified maximum likelihood estimation perform better than the method of moments and least squares. Mohamed and Reda [5] proposed a stress-strength model with a type-Ⅱ censored sample and studied in odd generalized exponential-exponential distribution. They observed that the performance of Bayesian estimation is better than maximum likelihood estimation in terms of mean square error. Based on progressive first failure censored samples, Shi and Shi [6] derived the estimators of stress-strength reliability for beta log Weibull distribution. It can be shown that the Bayesian estimation is better than the maximum likelihood estimation in terms of average absolute bias and mean squared error. For more research on stress-strength reliability, please refer to [7,8,9,10,11,12,13,14,15,16,17].
Inverse Weibull distribution (IWD) is a lifetime distribution commonly employed in reliability analysis, and its application fields include engineering, medicine and so on. Aljeddani and Mohammed [18] proposed that IWD is an effective tool for modeling wind speed characteristics, offering a deep understanding of the density function and cumulative distribution function of wind speed. IWD can also be used for statistical process control. Baklizi and Ghannam [19] proposed a control chart based on the case that the product lifetime obeys the IWD and extended the applicability of the control chart method to the case involving censored lifetime tests. The probability density function (PDF) and cumulative density function (CDF) of IWD are given by Eqs (1) and (2), respectively.
f(x;ζ,σ)=ζσx−σ−1exp(−ζx−σ) ; x>0, | (1) |
F(x;ζ,σ)=exp(−ζx−σ) ; x>0, | (2) |
where ζ>0 is the scale parameter and σ>0 is the shape parameter. For convenience, denote IWD with PDF (1) as IW(ζ,σ). In practical production, some hazard rate functions are often non-monotonic. As shown in Figure 1, the hazard rate function (hrf) of IWD exhibits an inverted bathtub shape, making it highly suitable for modeling such data. In the degradation process of diesel engine mechanical parts, Keller and Kanath [20] pointed out that IWD is more suitable for fitting failure data of pistons, crankshafts and main bearings compared to the exponential distribution and Weibull distribution.
In recent years, the statistical inference of IWD has attracted many authors. Alam and Nassar [21] considered the estimation of entropy for IWD based on improved adaptive progressive type-Ⅱ censored data. Lin et al. [22] considered the estimation of parameters and percentiles for Marshall Olkin extended IWD based on progressive type-Ⅱ censored data. They found that the least-squares estimation, maximum likelihood estimation and percentiles estimation are not stable. Therefore, Bayesian estimation is focused. Nassar and Ahmed [23] studied the constant stress partial accelerated life test using adaptive progressive type-Ⅰ censored samples. The research assumes that the life of the product under normal use conditions obeys IWD. The maximum likelihood estimation, the maximum product of the interval process and Bayesian estimation were used to estimate the point and interval estimation of model parameters and acceleration factors. Amany [24] proposed different predictive and reconstructive pivotal quantities for IWD based on dual generalized order statistics. Based on complete samples, Hassan [25] obtained a modified maximum likelihood estimator and confidence intervals of stress-strength reliabilities for IWD by ranked set sampling. Jia et al. [26] discussed the maximum likelihood and Bayesian estimation of the stress-strength model P(X>Y) under the first-failure progressive unified hybrid censored sample, which X and Y were independent random variables from IWD. Based on complete samples, Bi and Gui [27] considered the classical and Bayesian estimation of stress-strength reliability of IWD. Under the adaptive progressive type-Ⅱ (APT-Ⅱ) censored samples, Alslman and Helu [28] obtained the maximum likelihood and maximum product of spacing estimators of the stress-strength reliability for IWD. Yadav et al. [29] derived the maximum likelihood estimator and Bayesian estimator of stress-strength reliability for IWD under progressively type-Ⅱ censoring data.
In the available references, it is still not comprehensive enough in terms of the censored scheme and estimation method. Therefore, we consider the estimation of stress-strength reliability δ=P(X>Y) under APT-Ⅱ censored samples, where X and Y are two independent random variables from IWD with the same shape parameter but different scale parameters. The rest of this paper is organized as follows: Section 2 introduces the APT-Ⅱ censored scheme. Section 3 derives the maximum likelihood estimator (MLE) and asymptotic distribution of δ. Approximate maximum likelihood estimator (AMLE) and asymptotic confidence interval (ACI) are constructed. Section 4 derives the Bayesian estimators (BEs) of δ and approximates them using Lindley's approximation. Section 5 presents the Monte Carlo simulation. In Section 6, the application of the mentioned methods is illustrated by two real datasets. Section 7 contains the conclusions.
In situations where products have long life spans, obtaining failure time data can be time-consuming and costly. To address this issue, experimenters often employ censored schemes. Two commonly used censored schemes are the progressive type-Ⅰ censored scheme and progressive type-Ⅱ censored scheme. The progressive type-Ⅰ censored scheme involves ending the test at a predetermined time. However, it may result in a small number of observed failures when the product life is long. This can limit the accuracy and efficiency of statistical inference. The progressive type-Ⅱ censored scheme ends the test after a predetermined number of failures occur. While this scheme ensures a sufficient number of failures are observed, it can lead to prolonged test times, which can be costly and impractical in some cases. Ng et al. [30] developed APT-Ⅱ censored scheme to address these limitations. In this scheme, the experimenter can not only ensure to observe enough numbers of failures, but also speed up the test process, which greatly improves the efficiency of statistical inference.
Assume that n units are put into the lifetime test. Only m failure units can be observed. A censored scheme Q=(Q1,Q2,...,Qm) satisfies Q1+Q2+...+Qm+m=n. Denote the lifetime of the observed failure units by Xi (i=1,2,...,m). When the first failure X1 is observed, Q1 units are randomly removed from the residual n−1 units that have not failed. Similarly, Q2 units are randomly removed from the remaining n−Q1−2 units at the time of the second failure X2. When the m time of failure Xm is observed, all the remaining Qm units are removed. Then, (X1,X2,...,Xm) is a set of progressive type-Ⅱ censored samples.
The APT-Ⅱ censored scheme is essentially a hybrid of the type-Ⅰ censored scheme and type-Ⅱ progressive censored scheme, as detailed in Figures 2 and 3. A desired total test time T is given, but the actual test time is allowed to exceed T as well. If the number of failure units has reached m before time T, the test will be stopped before T. On the contrary, if the test time exceeds T and the failure units observed are less than m, the testers would like to terminate the test as soon as possible. To fulfill this expectation, the testers will make some changes during the test. Ensure that there is enough time to observe m failure units without the actual test time exceeding T too much. Therefore, to terminate the test as soon as possible without changing m, it is necessary to retain more surviving units in the test. The specific situations of the APT-Ⅱ censored scheme are shown below.
(1) If m failure units have been observed before T, the censored scheme is Q=(Q1,Q2,...,Qm).
(2) Suppose that J (J<m) failure units are observed before time T, that is, XJ<T<XJ+1. To retain more surviving units in the test, the testers set QJ+1=QJ+2=...=Qm−1=0 and Qm=n−m−Q1−Q2−...−QJ.
Suppose that X and Y are two independent random variables, where X∼IW(ζ1,σ) and Y∼IW(ζ2,σ). The stress-strength reliability δ=P(X>Y) is given by
δ=P(X>Y)=∫+∞0f(x;ζ1,σ)P(Y⩽x)dx=∫+∞0f(x;ζ1,σ)F(x;ζ2,σ)dx=ζ1ζ1+ζ2. | (3) |
Let X=(X1,X2,...,Xm) be an APT-Ⅱ censored sample from IW(ζ1,σ) with X1<X2<...<Xm under censored scheme Q=(Q1,...,QJ,0,...,0,Qm=n1−m−J∑i=1Qi) such that XJ<T1<XJ+1. Let Y=(Y1,Y2,...,Yt) be an APT-Ⅱ censored sample from IW(ζ2,σ) with Y1<Y2<...<Yt under censored scheme R=(R1,...,RK,0,...,0,Rt=n2−t−K∑i=1Ri) such that YK<T2<YK+1. Denote x=(x1,x2,...,xm) and y=(y1,y2,...,yt) as the observation of X and Y, respectively. The joint likelihood function can be written as
l(ζ1,ζ2,σ;x,y)=A1A2[m∏i=1f1(xi)]J∏i=1[1−F1(xi)]Qi[1−F1(xm)]Qm[t∏i=1f2(yi)]K∏i=1[1−F2(yi)]Ri[1−F2(yt)]Rt=A1A2ζm1ζt2σm+tm∏i=1x−σ−1ie−ζ1x−σi[J∏i=1(1−e−ζ1x−σi)Qi](1−e−ζ1x−σm)Qmt∏i=1y−σ−1ie−ζ2y−σi[K∏i=1(1−e−ζ2y−σi)Ri](1−e−ζ2y−σt)Rt | (4) |
where
A1=n1(n1−1−Q1)(n1−2−Q1−Q2)....(n1−m+1−m−1∑i=1Qi), |
A2=n2(n2−1−R1)(n2−2−R1−R2)....(n2−t+1−t−1∑i=1Ri), |
f1(x)=ζ1σx−σ−1e−ζ1x−σ, |
f2(y)=ζ2σy−σ−1e−ζ2y−σ, |
F1(x)=e−ζ1x−σ, |
F2(y)=e−ζ2y−σ. |
The log-likelihood function is
L(ζ1,ζ2,σ;x,y)=lnl(ζ1,ζ2,σ;x,y)=lnA1A2+mlnζ1+tlnζ2+(m+t)lnσ−(σ+1)m∑i=1lnxi−ζ1m∑i=1x−σi+J∑i=1Qiln(1−e−ζ1x−σi)+Qmln(1−e−ζ1x−σm)−(σ+1)t∑i=1lnyi−ζ2t∑i=1y−σi+K∑i=1Riln(1−e−ζ2y−σi)+Rtln(1−e−ζ2y−σt). | (5) |
The partial derivatives of the log-likelihood function L(ζ1,ζ2,σ;x,y) with respect to ζ1, ζ2 and σ are given by
∂L(ζ1,ζ2,σ;x,y)∂ζ1=mζ1−m∑i=1x−σi+J∑i=1Qix−σiexp(−ζ1x−σi)1−exp(−ζ1x−σi)+Qmx−σmexp(−ζ1x−σm)1−exp(−ζ1x−σm), | (6) |
∂L(ζ1,ζ2,σ;x,y)∂ζ2=tζ2−t∑i=1y−σi+K∑i=1Riy−σiexp(−ζ2y−σi)1−exp(−ζ2y−σi)+Rty−σtexp(−ζ2y−σt)1−exp(−ζ2y−σt), | (7) |
∂L(ζ1,ζ2,σ;x,y)∂σ=m+tσ+m∑i=1(ζ1x−σilnxi−lnxi)+t∑i=1(ζ2y−σilnyi−lnyi)−ζ1J∑i=1Qix−σiexp(−ζ1x−σi)lnxi1−exp(−ζ1x−σi)−ζ2K∑i=1Riy−σiexp(−ζ2y−σi)lnyi1−exp(−ζ2y−σi)−ζ1Qmx−σmexp(−ζ1x−σm)lnxm1−exp(−ζ1x−σm)−ζ2Rty−σtexp(−ζ2y−σt)lnyt1−exp(−ζ2y−σt), | (8) |
{∂L(ζ1,ζ2,σ;x,y)∂ζ1=0∂L(ζ1,ζ2,σ;x,y)∂ζ2=0∂L(ζ1,ζ2,σ;x,y)∂σ=0. | (9) |
The MLEs ˆζ1,ML, ˆζ2,ML and ˆσML are the solutions of likelihood Eq (9). Considering the nonlinearity, we propose an iteration method to obtain the approximate solutions. Because of the invariance of maximum likelihood estimation, the MLE ˆδML of δ can be written as
ˆδML=ˆζ1,MLˆζ1,ML+ˆζ2,ML. | (10) |
Since the explicit form of ˆδML cannot be obtained in Section 3.1, we consider the approximate maximum likelihood estimation now.
Let W=−lnX and V=−lnY. The CDFs of W and V can be obtained easily.
FW(w)=P(W⩽w)=1−P(X⩽e−w)=1−exp(−ζ1e−σw) | (11) |
FV(v)=P(V⩽v)=1−P(Y⩽e−v)=1−exp(−ζ2e−σv) | (12) |
Let σ=−β−1, ζ1=eσα1 and ζ2=eσα2. The CDFs (11) and (12) can be rewritten as
FW(w)=1−exp[−exp(w−α1β)] , w>0, | (13) |
FV(v)=1−exp[−exp(v−α2β)] , v>0. | (14) |
It's obvious that W and V follow the extreme value distribution. Denote that W∼EV(α1,β) and V∼EV(α2,β). We assume that wj=−lnxj (j=1,2,...,m) and vk=−lnyk (k=1,2,...,t).
Given the observations w1,w2,...,wm and v1,v2,...,vt, the log-likelihood function of α1, α2 and β is
LWV=A3−(m+t)lnβ+m∑j=1ωj−m∑j=1eωj−J∑j=1Qjeωj−Qmeωm+t∑k=1υk−t∑k=1eυk−K∑k=1Rkeυk−Rteυt, | (15) |
where ωj=β−1(wj−α1), υk=β−1(vk−α2) and A3 is a constant.
Next, expending the function eωj and eυk at ω0j=ln[−ln(1−pj)] and υ0k=ln[−ln(1−qk)], respectively, and retaining the first derivative.
eωj≃ax,j+bx,jωj, | (16) |
eυk≃ay,k+by,kυk, | (17) |
where
pj=1−m∏i=m−j+1i+Qm−i+1+...+Qmi+1+Qm−i+1+...+Qm , ax,j=eω0j(1−ω0j) , bx,j=eω0j |
qk=1−t∏i=t−k+1i+Rt−i+1+...+Rti+1+Rt−i+1+...+Rt , ay,k=eυ0k(1−υ0k) , by,k=eυ0k. |
Thus,
∂LWV∂α1≃−1β[m−∑mj=1(ax,j+bx,jωj)−∑Jj=1Qj(ax,j+bx,jωj)−Qm(ax,m+bx,mωm)], | (18) |
∂LWV∂α2≃−1β[t−∑tk=1(ay,k+by,kυk)−∑Kk=1Rk(ay,k+by,kυk)−Rt(ay,t+by,tυt)], | (19) |
∂LWV∂β≃−1β[m+t+m∑j=1ωj+t∑k=1υk−m∑j=1ωj(ax,j+bx,jωj)−t∑k=1υk(ay,k+by,kυk)−J∑j=1Qjωj(ax,j+bx,jωj)−K∑k=1Rkυk(ay,k+by,kυk)−Qmωm(ax,m+bx,mωm)−Rtυt(ay,t+by,tυt)], | (20) |
{∂LWV∂α1=0∂LWV∂α2=0∂LWV∂β=0. | (21) |
The solutions of likelihood Eq (21) are
{ˆα1=(Bx−Axˆβ)C−1xˆα2=(By−Ayˆβ)C−1yˆβ=[√(DC2x−AxBxCx)2−4mC2x(B2xCx−EC2x)+AxBxCx−DC2x](2mC2x)−1, | (22) |
where
Ax=m−m∑j=1ax,j−J∑j=1Qjax,j−Qmax,m, |
Ay=t−t∑k=1ay,k−K∑k=1Rkay,k−Rtay,t, |
Bx=m∑j=1bx,jwj+J∑j=1Qjbx,jwj+Qmbx,mwm, |
By=t∑k=1by,kvk+K∑k=1Rkby,kvk+Rtby,tvt, |
Cx=m∑j=1bx,j+J∑j=1Qjbx,j+Qmbx,m, |
Cy=t∑k=1by,k+K∑k=1Rkby,k+Rtby,t, |
D=m∑j=1wj−m∑j=1ax,jwj−J∑j=1Qjax,jwj−Qmax,mwm, |
E=m∑j=1bx,jw2j+J∑j=1Qjbx,jw2j+Qmbx,mw2m. |
Hence, the AMLE of δ is given by
ˆδAML=ˆζ1,AMLˆζ1,AML+ˆζ2,AML, | (23) |
where
ˆσAML=−ˆβ , ˆζ1,AML=exp(ˆσAMLˆα1) , ˆζ2,AML=exp(ˆσAMLˆα2). |
It can be seen from Section 3.1 that the MLE of δ cannot be given in an explicit form. Therefore, we cannot construct the exact confidence interval. Based on the asymptotically normal property of maximum likelihood estimation, we construct the ACI of δ in this subsection.
Denote θ=(ζ1,ζ2,σ) and ˆθML=(ˆζ1,ML,ˆζ2,ML,ˆσML). The observed Fisher information matrix can be expressed as
H=[−H11(ˆθML)−H12(ˆθML)−H13(ˆθML)−H21(ˆθML)−H22(ˆθML)−H23(ˆθML)−H31(ˆθML)−H32(ˆθML)−H33(ˆθML)]. | (24) |
Here,
H11(θ)=∂2L(ζ1,ζ2,σ;x,y)∂ζ21=−mζ1−J∑i=1Qix−2σie−ζ1x−σi(1−e−ζ1x−σi)2−Qmx−2σme−ζ1x−σm(1−e−ζ1x−σm)2, |
H22(θ)=∂2L(ζ1,ζ2,σ;x,y)∂ζ22=−tζ2−K∑i=1Riy−2σie−ζ2y−σi(1−e−ζ2y−σi)2−Rty−2σte−ζ2y−σt(1−e−ζ2y−σt)2, |
H13(θ)=∂2L(ζ1,ζ2,σ;x,y)∂ζ1∂σ=m∑i=1x−σilnxi+ζ1J∑i=1Qix−σie−ζ1x−σilnxi(1−e−ζ1x−σi)2+ζ1Qmx−σme−ζ1x−σmlnxm(1−e−ζ1x−σm)2, |
H23(θ)=∂2L(ζ1,ζ2,σ;x,y)∂ζ2∂σ=t∑i=1y−σilnyi+ζ2K∑i=1Riy−σie−ζ2y−σilnyi(1−e−ζ2y−σi)2+ζ2Rty−σte−ζ2y−σtlnyt(1−e−ζ2y−σt)2, |
H33(θ)=∂2L(ζ1,ζ2,σ;x,y)∂σ2=−m+tσ2−ζ1m∑i=1x−σi(lnxi)2−ζ2t∑i=1y−σi(lnyi)2+ζ1J∑i=1Qix−σie−ζ1x−σi(lnxi)2+ζ2K∑i=1Riy−σie−ζ2y−σi(lnyi)2+ζ1Qmx−σme−ζ1x−σm(lnxm)2+ζ2Rty−σte−ζ2y−σt(lnyt)2−ζ21J∑i=1Qix−2σie−ζ1x−σi(lnxi)2(1+e−ζ1x−σi)1−e−ζ1x−σi−ζ21Qmx−2σme−ζ1x−σm(lnxm)2(1+e−ζ1x−σm)1−e−ζ1x−σm−ζ22K∑i=1Riy−2σie−ζ2y−σi(lnyi)2(1+e−ζ2y−σi)1−e−ζ2y−σi−ζ22Rty−2σte−ζ2y−σt(lnyt)2(1+e−ζ2y−σt)1−e−ζ2y−σt, |
H12(θ)=H21(θ)=0 , H31(θ)=H13(θ) , H32(θ)=H23(θ). |
Next, the Delta method is used to derive the ACI of δ. Let ϕ=(ϕ1(ˆθML),ϕ2(ˆθML),ϕ3(ˆθML))T, and
ϕ1(θ)=∂δ∂ζ1=ζ2(ζ1+ζ2)2 , ϕ2(θ)=∂δ∂ζ2=−ζ1(ζ1+ζ2)2 , ϕ3(θ)=∂δ∂σ=0. |
According to the Delta method, the estimate of variance Var(ˆδML) is approximated by Eq (25), where H−1 is the inverse matrix of Fisher information matrix H.
Var(ˆδML)=ϕTH−1ϕ. | (25) |
Then, the 100(1−λ)% ACI of δ is present by Eq (26), where zλ2 is the upper λ2th quantile of the standardized normal distribution.
(ˆδML−zλ2√Var(ˆδML) , ˆδML+zλ2√Var(ˆδML)). | (26) |
In this section, we assume that ζ1 and ζ2 are independent random variables and follow gamma priors. The BEs of ζ1 and ζ2 are derived under symmetric entropy loss function and LINEX loss function.
In Bayesian estimation, selecting prior distribution for unknown parameter is a significant matter. First, the gamma prior is versatile for adjusting different shapes of the distribution density function. Second, the gamma prior is relatively simple, and there will not be too complicated computational issues. Its advantage is to provide conjugacy and mathematical ease. As a result, we investigate the gamma prior. Then, the prior distributions of ζ1 and ζ2 are given as
π(ζ1)∝ζa1−11exp(−b1ζ1), a1,b1>0, | (27) |
π(ζ2)∝ζa2−12exp(−b2ζ2), a2,b2>0. | (28) |
Denote that ζ1∼G(a1,b1) and ζ2∼G(a2,b2). The joint prior is
π(ζ1,ζ2)∝ζa1−11ζa2−12exp(−b1ζ1−b2ζ2). | (29) |
Therefore, the joint posterior distribution given observation data is
π(ζ1,ζ2,σ|x,y)=A4ζm+a1−11ζt+a2−12σm+te−b1ζ1−b2ζ2m∏i=1x−σ−1ie−ζ1x−σit∏i=1y−σ−1ie−ζ2y−σiJ∏i=1(1−e−ζ1x−σi)Qi[K∏i=1(1−e−ζ2y−σi)Ri](1−e−ζ1x−σm)Qm(1−e−ζ2y−σt)Rt, | (30) |
and A−14=∭π(ζ1,ζ2,σ|x,y)l(ζ1,ζ2,σ;x,y)dζ1dζ2dσ.
Let ˆρ be the estimator of ρ. The symmetric entropy loss function (Xu et al. [31]) and LINEX loss function (Varian [32]) are defined as
LS(ρ,ˆρ)=ˆρρ+ρˆρ−2, | (31) |
LE(ρ,ˆρ)=exp[d(ˆρ−ρ)]−d(ˆρ−ρ)−1, | (32) |
where d is the hype-parameter of LINEX loss function. Given observations x and y, the BEs of ρ under symmetric entropy loss function and LINEX loss function are presented by Eqs (33) and (34), where E(⋅|x,y) denotes the posterior expectation.
ˆρS=[E(ρ|x,y)E(ρ−1|x,y)]12 | (33) |
ˆρE=−1dln[E(e−dρ|x,y)] | (34) |
Thus, based on APT-Ⅱ censored samples, the BE ˆδS of δ under symmetric entropy loss function is given by
ˆδS=[E(δ|x,y)E(δ−1|x,y)]12=[∫+∞0∫+∞0∫+∞0δπ(ζ1,ζ2,σ|x,y)dζ1dζ2dσ∫+∞0∫+∞0∫+∞0δ−1π(ζ1,ζ2,σ|x,y)dζ1dζ2dσ]12={(∫+∞0∫+∞0∫+∞0ζm+a11ζt+a2−12(ζ1+ζ2)−1σm+te−b1ζ1−b2ζ2m∏i=1x−σ−1ie−ζ1x−σit∏i=1y−σ−1ie−ζ2y−σiJ∏i=1(1−e−ζ1x−σi)Qi[K∏i=1(1−e−ζ2y−σi)Ri](1−e−ζ1x−σm)Qm(1−e−ζ2y−σt)Rtdζ1dζ2dσ)[∫+∞0∫+∞0∫+∞0ζm+a1−21ζt+a2−12(ζ1+ζ2)σm+te−b1ζ1−b2ζ2m∏i=1x−σ−1ie−ζ1x−σit∏i=1y−σ−1ie−ζ2y−σiJ∏i=1(1−e−ζ1x−σi)Qi[K∏i=1(1−e−ζ2y−σi)Ri](1−e−ζ1x−σm)Qm(1−e−ζ2y−σt)Rtdζ1dζ2dσ]−1}12. | (35) |
The BE ˆδE of δ under LINEX loss function is given by
ˆδE=−1dln[E(e−dδ|x,y)]=−1dln[A4∫+∞0∫+∞0∫+∞0ζm+a1−11ζt+a2−12σm+te−(b1+d)ζ1−b2ζ2−d(ζ1+ζ2)−1m∏i=1x−σ−1ie−ζ1x−σit∏i=1y−σ−1ie−ζ2y−σiJ∏i=1(1−e−ζ1x−σi)Qi[K∏i=1(1−e−ζ2y−σi)Ri](1−e−ζ1x−σm)Qm(1−e−ζ2y−σt)Rtdζ1dζ2dσ]. | (36) |
It can be seen that both Eqs (35) and (36) involve the ratio of two integrals, and the form of integral is complex. Hence, we use Lindley's approximation (Lindley [33]) to compute the approximate Bayesian estimates. Lindley's approximation provides a method to obtain an approximation of the posterior expectation like the following form.
E[η(ζ1,ζ2,σ)|x,y]=∫η(ζ1,ζ2,σ)eL(ζ1,ζ2,σ;x,y)+π∗(ζ1,ζ2,σ)d(ζ1,ζ2,σ)∫eL(ζ1,ζ2,σ;x,y)+π∗(ζ1,ζ2,σ)d(ζ1,ζ2,σ). | (37) |
In Eq (37), η(ζ1,ζ2,σ) is a function of ζ1, ζ2 and σ, and π∗(ζ1,ζ2,σ)=lnπ(ζ1,ζ2,σ). According to Lindley's approximation, the form of posterior expectation (37) can be rewritten as
E[η(ζ1,ζ2,σ)|x,y]=η+12[(η11+2η1π∗1)φ11+(η21+2η2π∗1)ˆφ21+(η12+2η1π∗2)φ12+(η22+2η2π∗2)φ22+(η1φ11+η2φ12)(L111φ11+L121φ12+L211φ21+L221φ22)+(η1φ21+η2φ22)(L112φ11+L122φ12+L212φ21+L222φ22)], | (38) |
where
L111=∂3L(ζ1,ζ2,σ;x,y)∂ζ31=2mζ31+J∑i=1Qix−3σie−ζ1x−σi(1−e−ζ1x−σi)3+Qmx−3σme−ζ1x−σm(1−e−ζ1x−σm)3, |
L222=∂3L(ζ1,ζ2,σ;x,y)∂ζ32=2mtζ32+K∑i=1Riy−3σie−ζ2y−σi(1−e−ζ2y−σi)3+Rty−3σte−ζ2y−σt(1−e−ζ2y−σt)3, |
π∗1=a1−1ζ1−b1 , π∗2=a2−1ζ2−b2, |
L121=L211=L221=L112=L122=L212=0, |
φ=[−∂2L(ζ1,ζ2,σ;x,y)∂ζ2100−∂2L(ζ1,ζ2,σ;x,y)∂ζ22]−1, |
and φij (i,j=1,2) is the element of φ.
Under symmetric entropy loss function, we need to approximate E(δ|x,y) and E(δ−1|x,y) referring Eq (38). Let η=η(ζ1,ζ2) be a function of ζ1 and ζ2, and we denote
η1=∂η∂ζ1,η2=∂η∂ζ2,η11=∂2η∂ζ21,η22=∂2η∂ζ22,η12=∂2η∂ζ1∂ζ2 ,η21=∂2η∂ζ2∂ζ1. |
When the function mentioned in Eq (37) is η=ζ1(ζ1+ζ2)−1, the partial derivatives are
η1=ζ2(ζ1+ζ2)2 , η2=−ζ1(ζ1+ζ2)2,η11=−2ζ2(ζ1+ζ2)3 , η12=ζ1−ζ2(ζ1+ζ2)3 , η22=2ζ1(ζ1+ζ2)3 , η21=η12. | (39) |
Therefore,
E(δ|x,y)=ζ1ζ1+ζ2+[−ζ2(ζ1+ζ2)3+ζ2π∗1(ζ1+ζ2)2]φ11+[ζ1(ζ1+ζ2)3−ζ1π∗2(ζ1+ζ2)2]φ22+12[(ζ2(ζ1+ζ2)2φ211L111−ζ1(ζ1+ζ2)2φ222L222]. | (40) |
When the function mentioned in Eq (37) is η=ζ−11(ζ1+ζ2), the partial derivatives are
η1=−ζ2ζ21 , η2=1ζ1 , η11=2ζ2ζ31 , η12=−1ζ21 , η22=0 , η21=η12. | (41) |
Therefore,
E(δ−1|x,y)=1+ζ2ζ1+(ζ2ζ31−ζ2ζ21π∗1)φ11+1ζ1π∗2φ22+12(1ζ1L222φ222−ζ2ζ21φ211L111). | (42) |
The BE ˆδS is given by Eq (43).
ˆδS=[E(δ|x,y)E(δ−1|x,y)]12|(ζ1,ζ2,σ)=(ˆζ1,ML,ˆζ2,ML,ˆσML). | (43) |
Under the LINEX loss function, we only need to approximate E(e−dδ|x,y). When η=exp(−dζ1ζ1+ζ2), the partial derivatives are
η1=−dζ2(ζ1+ζ2)2exp(−dζ1ζ1+ζ2) , η2=dζ1(ζ1+ζ2)2exp(−dζ1ζ1+ζ2) ,η11=[2dζ2(ζ1+ζ2)3+d2ζ22(ζ1+ζ2)4]exp(−dζ1ζ1+ζ2) ,η22=[−2dζ1(ζ1+ζ2)3+d2ζ21(ζ1+ζ2)4]exp(−dζ1ζ1+ζ2) ,η12=[dζ2−dζ1(ζ1+ζ2)3−d2ζ1ζ2(ζ1+ζ2)4]exp(−dζ1ζ1+ζ2) , η21=η12 . | (44) |
Thus,
E(e−dδ|x,y)=exp(−dζ1ζ1+ζ2)+12exp(−dζ1ζ1+ζ2){[2dζ2(ζ1+ζ2)3+d2ζ22(ζ1+ζ2)4−2dζ2(ζ1+ζ2)2π∗1]φ11+[−2dζ1(ζ1+ζ2)3+d2ζ21(ζ1+ζ2)4+2dζ1(ζ1+ζ2)2π∗2]φ22+−dζ2(ζ1+ζ2)2φ211L111+dζ1(ζ1+ζ2)2L222φ222}. | (45) |
The BE ˆδE is given by Eq (46)
ˆδE=−1dln[E(e−dδ|x,y)]|(ζ1,ζ2,σ)=(ˆζ1,ML,ˆζ2,ML,ˆσML). | (46) |
In this section, Monte Carlo simulation is used to evaluate the behavior of different estimators under different APT-Ⅱ censored schemes. We take the true values are (ζ1,real,ζ2,real,σreal)=(2,3,5). Hence, the true value δreal is 0.4000. Consider two priors, namely, Priors 1 and 2. The hyper-parameters of Prior 1 are (a1,b1)=(5,2) and (a2,b2)=(3,6). Prior 2 is non-informative prior, that is, a1=a2=b1=b2=0. Without loss of generality, let T1=xm/m22 and T2=yt/5. On this basis, the trails are N at 10,000 times. We consider two cases with different censored schemes, which are detailed in Table 1. The point estimates are compared by average bias (AB) and mean squared error (MSE). The performance of confidence interval is represented by the average width (AW) and coverage probability (CP). All the results are displayed in Tables 2–8. It is necessary to select initial values using iteration method, so we take AMLE ˆδAML to substitute for MLE ˆδML. The algorithm of generating APT-Ⅱ censored data is shown in Algorithm 1. Finally, the AB, MSE and AW are calculated by the following formulas:
AB=1NN∑i=1(ˆδi−δreal),MSE=1NN∑i=1(ˆδi−δreal)2andAW=1NN∑i=1(ˆδi,up−ˆδi,low). |
(n1,m) | Q | (n2,t) | R | |
Case 1 | (30,10) | Q1=(0∗8,10∗2) | (40,20) | R1=(2∗10,0∗10) |
Q2=(20∗1,0∗9) | R2=(0∗19,20∗1) | |||
Q3=((0,5)∗5) | R3=((0,0,0,0,5)∗4) | |||
Case 2 | (50,20) | Q1=(10∗1,0∗18,20∗1) | (50,30) | R1=(5∗2,0∗13,5∗2,0∗13) |
Q2=(0∗19,30∗1) | R2=(0∗20,2∗10) | |||
Q3=(0∗10,2∗15) | R3=(10∗1,0∗28,10∗1) | |||
Case 3 | (100,50) | Q1=(10∗5,0∗45) | (150,70) | R1=(2∗40,0∗30) |
Q2=(20∗1,0∗24,30∗1,0∗24) | R2=(30∗1,0∗30,50∗1,0∗38) | |||
Q3=(0∗20,50∗1,0∗29) | R3=(0∗45,80∗1,0∗24) |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0165 | 0.0495 | 0.0406 | 0.0397 | 0.1129 | 0.2181 | 0.2003 | 0.1957 |
Q1, R2 | 0.0064 | 0.0044 | 0.0868 | 0.0291 | -0.0152 | 0.3182 | 0.3529 | 0.1627 |
Q1, R3 | 0.0060 | 0.0031 | 0.0663 | 0.0331 | 0.0400 | 0.2613 | 0.2569 | 0.1765 |
Q2, R1 | 0.0008 | 0.0034 | 0.0031 | 0.0030 | -0.0140 | 0.0543 | 0.0514 | 0.0492 |
Q2, R2 | 0.0032 | 0.0025 | 0.0034 | 0.0019 | -0.0481 | 0.0571 | 0.0532 | 0.0373 |
Q2, R3 | 0.0021 | 0.0029 | 0.0025 | 0.0018 | -0.0374 | 0.0473 | 0.0447 | 0.0368 |
Q3, R1 | 0.0032 | 0.0151 | 0.0137 | 0.0135 | 0.0422 | 0.1189 | 0.1139 | 0.1114 |
Q3, R2 | 0.0027 | 0.0078 | 0.0167 | 0.0107 | -0.0011 | 0.1285 | 0.1247 | 0.0968 |
Q3, R3 | 0.0021 | 0.0185 | 0.0130 | 0.0106 | 0.0130 | 0.1155 | 0.1099 | 0.0973 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0021 | 0.0096 | 0.0090 | 0.0090 | 0.0385 | 0.0955 | 0.0923 | 0.0922 |
Q1, R2 | 0.0008 | 0.0076 | 0.0072 | 0.0061 | 0.0047 | 0.0850 | 0.0824 | 0.0751 |
Q1, R3 | 0.0017 | 0.0094 | 0.0088 | 0.0086 | 0.0331 | 0.0944 | 0.0913 | 0.0899 |
Q2, R1 | 0.0136 | 0.0320 | 0.0288 | 0.0298 | 0.1113 | 0.1775 | 0.1683 | 0.1709 |
Q2, R2 | 0.0029 | 0.0349 | 0.0273 | 0.0222 | 0.0530 | 0.1707 | 0.1640 | 0.1464 |
Q2, R3 | 0.0113 | 0.0311 | 0.0281 | 0.0284 | 0.1008 | 0.1749 | 0.1662 | 0.1665 |
Q3, R1 | 0.0129 | 0.0318 | 0.0286 | 0.0292 | 0.1068 | 0.1767 | 0.1676 | 0.1685 |
Q3, R2 | 0.0035 | 0.0297 | 0.0281 | 0.0203 | 0.0377 | 0.1712 | 0.1661 | 0.1390 |
Q3, R3 | 0.0102 | 0.0306 | 0.0276 | 0.0272 | 0.0941 | 0.1731 | 0.1646 | 0.1625 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 3.61E-4 | 0.0012 | 0.0011 | 0.0012 | 0.0137 | 0.0318 | 0.0304 | 0.0326 |
Q1, R2 | 7.16E-4 | 0.0018 | 0.0017 | 0.0019 | 0.0229 | 0.0403 | 0.0389 | 0.0412 |
Q1, R3 | 6.24E-4 | 1.60E-4 | 1.56E-4 | 1.65E-4 | -0.0208 | 0.0013 | 2.01E-4 | 0.0019 |
Q2, R1 | 0.0052 | 0.0082 | 0.0078 | 0.0083 | 0.0708 | 0.0893 | 0.0872 | 0.0899 |
Q2, R2 | 0.0065 | 0.0096 | 0.0092 | 0.0097 | 0.0793 | 0.0972 | 0.0950 | 0.0977 |
Q2, R3 | 0.0016 | 0.0038 | 0.0035 | 0.0038 | 0.0373 | 0.0597 | 0.0579 | 0.0601 |
Q3, R1 | 0.0024 | 0.0046 | 0.0040 | 0.0046 | 0.0421 | 0.0638 | 0.0620 | 0.0641 |
Q3, R2 | 0.0039 | 0.0066 | 0.0063 | 0.0067 | 0.0574 | 0.0775 | 0.0757 | 0.0780 |
Q3, R3 | 9.39E-4 | 9.27E-4 | 8.63 E-4 | 9.09E-4 | -0.0144 | 0.0205 | 0.0192 | 0.0193 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0165 | 0.0179 | 0.0129 | 0.0175 | 0.0385 | 0.1173 | 0.0967 | 0.1177 |
Q1, R2 | 0.0064 | 0.0061 | 0.0059 | 0.0059 | 0.0047 | -0.0063 | -0.0235 | -0.0015 |
Q1, R3 | 0.0060 | 0.0066 | 0.0044 | 0.0066 | 0.0331 | 0.0447 | 0.0240 | 0.0483 |
Q2, R1 | 0.0008 | 0.0009 | 0.0010 | 0.0007 | 0.1113 | -0.0162 | -0.0204 | -0.0100 |
Q2, R2 | 0.0032 | 0.0035 | 0.0038 | 0.0027 | 0.0530 | -0.0504 | -0.0538 | -0.0427 |
Q2, R3 | 0.0021 | 0.0023 | 0.0025 | 0.0017 | 0.1008 | -0.0397 | -0.0434 | -0.0327 |
Q3, R1 | 0.0032 | 0.0032 | 0.0026 | 0.0037 | 0.1068 | 0.0424 | 0.0353 | 0.0480 |
Q3, R2 | 0.0027 | 0.0028 | 0.0027 | 0.0027 | 0.0377 | -0.0019 | -0.0082 | 0.0059 |
Q3, R3 | 0.0021 | 0.0022 | 0.0019 | 0.0023 | 0.0941 | 0.0134 | 0.0063 | 0.0201 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0021 | 0.0021 | 0.0018 | 0.0025 | 0.0385 | 0.0386 | 0.0342 | 0.0433 |
Q1, R2 | 0.0008 | 0.0008 | 0.0007 | 0.0009 | 0.0047 | 0.0048 | 0.0009 | 0.0101 |
Q1, R3 | 0.0017 | 0.0018 | 0.0015 | 0.0021 | 0.0331 | 0.0340 | 0.0295 | 0.0380 |
Q2, R1 | 0.0136 | 0.0141 | 0.0118 | 0.0145 | 0.1113 | 0.1139 | 0.1036 | 0.1161 |
Q2, R2 | 0.0029 | 0.0049 | 0.0037 | 0.0053 | 0.0530 | 0.0565 | 0.0465 | 0.0605 |
Q2, R3 | 0.0113 | 0.0118 | 0.0097 | 0.0123 | 0.1008 | 0.1026 | 0.0923 | 0.1052 |
Q3, R1 | 0.0129 | 0.0134 | 0.0110 | 0.0136 | 0.1068 | 0.1097 | 0.0987 | 0.1111 |
Q3, R2 | 0.0035 | 0.0037 | 0.0029 | 0.0040 | 0.0377 | 0.0408 | 0.0313 | 0.0451 |
Q3, R3 | 0.0102 | 0.0108 | 0.0087 | 0.0111 | 0.0941 | 0.0964 | 0.0855 | 0.0988 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 3.61E-4 | 3.50E-4 | 3.11E-4 | 3.95E-4 | 0.0137 | 0.0132 | 0.0117 | 0.0149 |
Q1, R2 | 7.16E-4 | 6.92E-4 | 6.24E-4 | 7.66E-4 | 0.0229 | 0.0223 | 0.0208 | 0.0239 |
Q1, R3 | 6.24E-4 | 6.54E-4 | 7.08E-4 | 5.76E-4 | -0.0208 | -0.0215 | -0.0228 | -0.0196 |
Q2, R1 | 0.0052 | 0.0053 | 0.0049 | 0.0055 | 0.0708 | 0.0710 | 0.0688 | 0.0724 |
Q2, R2 | 0.0065 | 0.0066 | 0.0062 | 0.0068 | 0.0793 | 0.0798 | 0.0775 | 0.0811 |
Q2, R3 | 0.0016 | 0.0016 | 0.0015 | 0.0017 | 0.0373 | 0.0371 | 0.0350 | 0.0387 |
Q3, R1 | 0.0024 | 0.0024 | 0.0022 | 0.0025 | 0.0421 | 0.0426 | 0.0407 | 0.0442 |
Q3, R2 | 0.0039 | 0.0039 | 0.0037 | 0.0041 | 0.0574 | 0.0574 | 0.0554 | 0.0589 |
Q3, R3 | 9.39E-4 | 9.52E-4 | 9.92E-4 | 8.85E-4 | -0.0144 | -0.0151 | -0.0168 | -0.0131 |
Censored scheme | CP | AW | ||||
Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | |
Q1, R1 | 0.8838 | 0.9564 | 0.9993 | 0.3178 | 0.2518 | 0.1287 |
Q1, R2 | 0.8384 | 0.9075 | 0.9951 | 0.3723 | 0.2999 | 0.1288 |
Q1, R3 | 0.8855 | 0.9289 | 0.9976 | 0.3001 | 0.2714 | 0.1315 |
Q2, R1 | 0.9681 | 0.9456 | 0.9869 | 0.2493 | 0.2657 | 0.1341 |
Q2, R2 | 0.8468 | 0.9702 | 0.9840 | 0.3669 | 0.2500 | 0.1339 |
Q2, R3 | 0.9238 | 0.9396 | 0.9707 | 0.2783 | 0.2672 | 0.1374 |
Q3, R1 | 0.9059 | 0.9422 | 0.9766 | 0.2456 | 0.2599 | 0.1238 |
Q3, R2 | 0.8831 | 0.9747 | 0.9873 | 0.2831 | 0.2588 | 0.1287 |
Q3, R3 | 0.8298 | 0.9657 | 0.9634 | 0.2730 | 0.2588 | 0.1283 |
Algorithm 1.
(1) Generate two sets of random numbers (wx,1,wx,2,...,wx,m) and (wy,1,wy,2,...,wy,t) from U(0,1).
(2) Let vx,i=wx,i(i+Qm+Qm−1+...+Qm−i+1)−1 (i=1,2,...,m) and vy,j=wy,j(j+Rt+Rt−1+...+Rt−j+1)−1 (j=1,2,...,t). Set ux,i=1−vx,mvx,m−1...vx,m−i+1 and uy,j=1−vy,tvy,t−1...vy,t−j+1.
(3) Let xi=F−1(ux,i;ζ1,real,σreal) and yj=F−1(uy,j;ζ2,real,σreal), where F is the CDF of IWD. Then, (x1,x2,...,xm) is the progressive type-Ⅱ censored data from IW(ζ1,real,σreal) with censored scheme (Q1,Q2,...,Qm) and (y1,y2,...,yt) is the progressive type-Ⅱ censored data from IW(ζ2,real,σreal) with censored scheme (R1,R2,...,Rt).
(4) Determine J and K such that xJ<T1<xJ+1 and yK<T2<yK+1. Remove xJ+2,xJ+3,...,xm and yK+2,yK+3,...,yt.
(5) Generate the first m−J−1 order statistics from the truncated distribution f(x;ζ1,real,σreal)1−F(xJ+1;ζ1,real,σreal) and denote them as xJ+2,xJ+3,...,xm. Then, the censored scheme changes to (Q1,...,QJ,0,...,0,Qm=n1−m−J∑i=1Qi). Similarly, generate the first t−K+1 order statistics from f(y;ζ2,real,σreal)1−F(yK+1;ζ2,real,σreal) as yK+2,yK+3,...,yt. Then, the censored scheme changes to (R1,...,RK,0,...,0,Rt=n2−t−K∑i=1Ri).
From Tables 2–8, the following conclusions may be drawn:
(1) When the effective sample sizes (m and t) increase, the MSEs of AMLE and BE decrease. Therefore, enlarging the effective sample size can appropriately enhance the accuracy of the estimation.
(2) The BEs under Prior 2 perform similarly to the AMLE in terms of MSEs. However, the BEs under Prior 1 perform worse than AMLE.
(3) Under the same prior, as the sample size increases, the available information increases. Therefore, the MSEs show a decreasing trend.
(4) Under Prior 1, the BE based on LINEX loss function with d=−3 has better behavior than the other BEs. Under Prior 2, the performance of all the BEs is comparable.
(5) With the increase of the effective sample sizes, the CPs gradually reach the confidence level of 95%.
In this section, two real data sets are used to validate the feasibility of the proposed method. These data sets are reported by Nelson [34], indicating the time when the electrodes are broken down by the insulating fluids at different voltages. X represents the insulating fluid at a voltage of 34kV, and Y represents the insulating fluid at a voltage of 36kV. The data sets are:
X: 0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91, 36.71, 72.89;
Y: 0.35, 0.59, 0.96, 0.99, 1.69, 1.97, 2.07, 2.58, 2.71, 2.90, 3.67, 3.99, 5.35, 13.77, 25.50.
First, we need to check whether the IWD can fit these data sets. We know that if a random variable T follows Weibull distribution, X=T−1 follows IWD. Set X=T−1 and Y=Z−1. The transformed data sets, Anderson-Darling (A-D) statistics and p-values are presented in Table 9.
Data sets | A-D | p-values | ||||||||
T | 0.0137 | 0.0272 | 0.0295 | 0.0308 | 0.0315 | 0.0829 | 0.1209 | 0.1248 | 0.6006 | 0.1132 |
0.1361 | 0.1538 | 0.2062 | 0.2141 | 0.2410 | 0.3165 | 0.3597 | 0.7634 | |||
1.0417 | 1.2821 | 5.2632 | ||||||||
Z | 0.0392 | 0.0726 | 0.1869 | 0.2506 | 0.2725 | 0.3448 | 0.3690 | 0.3876 | 0.3121 | 0.5858 |
0.4831 | 0.5076 | 0.5917 | 1.0101 | 1.0417 | 1.6949 | 2.8571 |
As we can see, the p-values are all greater than a 5% significance level, which means that the Weibull distribution can fit these data sets T and Z effectively. In other words, the IWD is suitable for fitting data sets X and Y. Figures 4 and 5 give the probability-probability (P-P) plot and quantile-quantile (Q-Q) plot to visually show the fitting.
Next, Table 10 presents the different APT-Ⅱ censored schemes. Since we cannot obtain any prior information, we take the hyperparameters of the prior distribution as a1=b1=a2=b2=0. The approximate maximum likelihood estimates, the Bayesian estimates under symmetric entropy loss function and LINEX loss function with d=3 and d=−3, and 95% ACIs are given in Table 11. We illustrate the existence and uniqueness of MLE through visual representations. Without the loss of generality, we choose censored scheme 3 in Table 10 to plot, as shown in Figure 6.
Censored scheme | Q | R |
1 | (2*5, 0*2, 1*1) | (1*5, 0*5) |
2 | (0*7, 11*1) | (0*9, 5*1) |
3 | (0*3, 5*1, 0*3, 6*1) | (0*4, 5*1, 0*3) |
Censored scheme | ˆδAML | ˆδS | ˆδE | ACI | |
d=3 | d=−3 | ||||
1 | 0.5179 | 0.5228 | 0.5080 | 0.5335 | (0.3815, 0.6579) |
2 | 0.5055 | 0.5131 | 0.4912 | 0.5244 | (0.2808, 0.7303) |
3 | 0.4425 | 0.4507 | 0.4315 | 0.4634 | (0.2852, 0.5998) |
The APT-Ⅱ censored scheme allows more flexibility during the lifetime test, thereby providing more control on the test, leading to shorter test time and more failed observations. In this paper, we investigate the classical and Bayesian estimation of stress-strength reliability based on APT-Ⅱ censored sample for IWD with the same shape but different scale parameters. The MLE can be obtained by the iteration algorithm. Note that the form of MLE is not explicit, and we propose AMLE and construct ACI. The BEs are also derived based on gamma prior under symmetric entropy loss function and LINEX loss function. Lindley's approximation is used to obtain the approximate Bayesian estimates. The simulation results show that MLE has the smaller MSE than BE under gamma prior. In addition, the censored scheme has a significant impact on the estimates. Yan et al. [35] proposed an improved adaptive progressive type-Ⅱ censored scheme. Based on this censored scheme, we will consider the statistical inference of multi-component stress-strength reliability for other distributions, such as Weighted Exponential distribution and improved Lomax distribution.
This research was funded by the National Natural Science Foundation of China, grant number 71661012.
The authors declare no conflict of interest.
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2. | Mustafa M. Hasaballah, Yusra A. Tashkandy, Oluwafemi Samson Balogun, M. E. Bakr, Reliability analysis for two populations Nadarajah-Haghighi distribution under Joint progressive type-II censoring, 2024, 9, 2473-6988, 10333, 10.3934/math.2024505 | |
3. | Sunita Sharma, Vinod Kumar, Multicomponent Stress-Strength Reliability Estimation of Weighted Exponential-Lindley Lifetime Model Under Progressive Censoring, 2025, 2364-9569, 10.1007/s41096-025-00238-8 |
(n1,m) | Q | (n2,t) | R | |
Case 1 | (30,10) | Q1=(0∗8,10∗2) | (40,20) | R1=(2∗10,0∗10) |
Q2=(20∗1,0∗9) | R2=(0∗19,20∗1) | |||
Q3=((0,5)∗5) | R3=((0,0,0,0,5)∗4) | |||
Case 2 | (50,20) | Q1=(10∗1,0∗18,20∗1) | (50,30) | R1=(5∗2,0∗13,5∗2,0∗13) |
Q2=(0∗19,30∗1) | R2=(0∗20,2∗10) | |||
Q3=(0∗10,2∗15) | R3=(10∗1,0∗28,10∗1) | |||
Case 3 | (100,50) | Q1=(10∗5,0∗45) | (150,70) | R1=(2∗40,0∗30) |
Q2=(20∗1,0∗24,30∗1,0∗24) | R2=(30∗1,0∗30,50∗1,0∗38) | |||
Q3=(0∗20,50∗1,0∗29) | R3=(0∗45,80∗1,0∗24) |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0165 | 0.0495 | 0.0406 | 0.0397 | 0.1129 | 0.2181 | 0.2003 | 0.1957 |
Q1, R2 | 0.0064 | 0.0044 | 0.0868 | 0.0291 | -0.0152 | 0.3182 | 0.3529 | 0.1627 |
Q1, R3 | 0.0060 | 0.0031 | 0.0663 | 0.0331 | 0.0400 | 0.2613 | 0.2569 | 0.1765 |
Q2, R1 | 0.0008 | 0.0034 | 0.0031 | 0.0030 | -0.0140 | 0.0543 | 0.0514 | 0.0492 |
Q2, R2 | 0.0032 | 0.0025 | 0.0034 | 0.0019 | -0.0481 | 0.0571 | 0.0532 | 0.0373 |
Q2, R3 | 0.0021 | 0.0029 | 0.0025 | 0.0018 | -0.0374 | 0.0473 | 0.0447 | 0.0368 |
Q3, R1 | 0.0032 | 0.0151 | 0.0137 | 0.0135 | 0.0422 | 0.1189 | 0.1139 | 0.1114 |
Q3, R2 | 0.0027 | 0.0078 | 0.0167 | 0.0107 | -0.0011 | 0.1285 | 0.1247 | 0.0968 |
Q3, R3 | 0.0021 | 0.0185 | 0.0130 | 0.0106 | 0.0130 | 0.1155 | 0.1099 | 0.0973 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0021 | 0.0096 | 0.0090 | 0.0090 | 0.0385 | 0.0955 | 0.0923 | 0.0922 |
Q1, R2 | 0.0008 | 0.0076 | 0.0072 | 0.0061 | 0.0047 | 0.0850 | 0.0824 | 0.0751 |
Q1, R3 | 0.0017 | 0.0094 | 0.0088 | 0.0086 | 0.0331 | 0.0944 | 0.0913 | 0.0899 |
Q2, R1 | 0.0136 | 0.0320 | 0.0288 | 0.0298 | 0.1113 | 0.1775 | 0.1683 | 0.1709 |
Q2, R2 | 0.0029 | 0.0349 | 0.0273 | 0.0222 | 0.0530 | 0.1707 | 0.1640 | 0.1464 |
Q2, R3 | 0.0113 | 0.0311 | 0.0281 | 0.0284 | 0.1008 | 0.1749 | 0.1662 | 0.1665 |
Q3, R1 | 0.0129 | 0.0318 | 0.0286 | 0.0292 | 0.1068 | 0.1767 | 0.1676 | 0.1685 |
Q3, R2 | 0.0035 | 0.0297 | 0.0281 | 0.0203 | 0.0377 | 0.1712 | 0.1661 | 0.1390 |
Q3, R3 | 0.0102 | 0.0306 | 0.0276 | 0.0272 | 0.0941 | 0.1731 | 0.1646 | 0.1625 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 3.61E-4 | 0.0012 | 0.0011 | 0.0012 | 0.0137 | 0.0318 | 0.0304 | 0.0326 |
Q1, R2 | 7.16E-4 | 0.0018 | 0.0017 | 0.0019 | 0.0229 | 0.0403 | 0.0389 | 0.0412 |
Q1, R3 | 6.24E-4 | 1.60E-4 | 1.56E-4 | 1.65E-4 | -0.0208 | 0.0013 | 2.01E-4 | 0.0019 |
Q2, R1 | 0.0052 | 0.0082 | 0.0078 | 0.0083 | 0.0708 | 0.0893 | 0.0872 | 0.0899 |
Q2, R2 | 0.0065 | 0.0096 | 0.0092 | 0.0097 | 0.0793 | 0.0972 | 0.0950 | 0.0977 |
Q2, R3 | 0.0016 | 0.0038 | 0.0035 | 0.0038 | 0.0373 | 0.0597 | 0.0579 | 0.0601 |
Q3, R1 | 0.0024 | 0.0046 | 0.0040 | 0.0046 | 0.0421 | 0.0638 | 0.0620 | 0.0641 |
Q3, R2 | 0.0039 | 0.0066 | 0.0063 | 0.0067 | 0.0574 | 0.0775 | 0.0757 | 0.0780 |
Q3, R3 | 9.39E-4 | 9.27E-4 | 8.63 E-4 | 9.09E-4 | -0.0144 | 0.0205 | 0.0192 | 0.0193 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0165 | 0.0179 | 0.0129 | 0.0175 | 0.0385 | 0.1173 | 0.0967 | 0.1177 |
Q1, R2 | 0.0064 | 0.0061 | 0.0059 | 0.0059 | 0.0047 | -0.0063 | -0.0235 | -0.0015 |
Q1, R3 | 0.0060 | 0.0066 | 0.0044 | 0.0066 | 0.0331 | 0.0447 | 0.0240 | 0.0483 |
Q2, R1 | 0.0008 | 0.0009 | 0.0010 | 0.0007 | 0.1113 | -0.0162 | -0.0204 | -0.0100 |
Q2, R2 | 0.0032 | 0.0035 | 0.0038 | 0.0027 | 0.0530 | -0.0504 | -0.0538 | -0.0427 |
Q2, R3 | 0.0021 | 0.0023 | 0.0025 | 0.0017 | 0.1008 | -0.0397 | -0.0434 | -0.0327 |
Q3, R1 | 0.0032 | 0.0032 | 0.0026 | 0.0037 | 0.1068 | 0.0424 | 0.0353 | 0.0480 |
Q3, R2 | 0.0027 | 0.0028 | 0.0027 | 0.0027 | 0.0377 | -0.0019 | -0.0082 | 0.0059 |
Q3, R3 | 0.0021 | 0.0022 | 0.0019 | 0.0023 | 0.0941 | 0.0134 | 0.0063 | 0.0201 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0021 | 0.0021 | 0.0018 | 0.0025 | 0.0385 | 0.0386 | 0.0342 | 0.0433 |
Q1, R2 | 0.0008 | 0.0008 | 0.0007 | 0.0009 | 0.0047 | 0.0048 | 0.0009 | 0.0101 |
Q1, R3 | 0.0017 | 0.0018 | 0.0015 | 0.0021 | 0.0331 | 0.0340 | 0.0295 | 0.0380 |
Q2, R1 | 0.0136 | 0.0141 | 0.0118 | 0.0145 | 0.1113 | 0.1139 | 0.1036 | 0.1161 |
Q2, R2 | 0.0029 | 0.0049 | 0.0037 | 0.0053 | 0.0530 | 0.0565 | 0.0465 | 0.0605 |
Q2, R3 | 0.0113 | 0.0118 | 0.0097 | 0.0123 | 0.1008 | 0.1026 | 0.0923 | 0.1052 |
Q3, R1 | 0.0129 | 0.0134 | 0.0110 | 0.0136 | 0.1068 | 0.1097 | 0.0987 | 0.1111 |
Q3, R2 | 0.0035 | 0.0037 | 0.0029 | 0.0040 | 0.0377 | 0.0408 | 0.0313 | 0.0451 |
Q3, R3 | 0.0102 | 0.0108 | 0.0087 | 0.0111 | 0.0941 | 0.0964 | 0.0855 | 0.0988 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 3.61E-4 | 3.50E-4 | 3.11E-4 | 3.95E-4 | 0.0137 | 0.0132 | 0.0117 | 0.0149 |
Q1, R2 | 7.16E-4 | 6.92E-4 | 6.24E-4 | 7.66E-4 | 0.0229 | 0.0223 | 0.0208 | 0.0239 |
Q1, R3 | 6.24E-4 | 6.54E-4 | 7.08E-4 | 5.76E-4 | -0.0208 | -0.0215 | -0.0228 | -0.0196 |
Q2, R1 | 0.0052 | 0.0053 | 0.0049 | 0.0055 | 0.0708 | 0.0710 | 0.0688 | 0.0724 |
Q2, R2 | 0.0065 | 0.0066 | 0.0062 | 0.0068 | 0.0793 | 0.0798 | 0.0775 | 0.0811 |
Q2, R3 | 0.0016 | 0.0016 | 0.0015 | 0.0017 | 0.0373 | 0.0371 | 0.0350 | 0.0387 |
Q3, R1 | 0.0024 | 0.0024 | 0.0022 | 0.0025 | 0.0421 | 0.0426 | 0.0407 | 0.0442 |
Q3, R2 | 0.0039 | 0.0039 | 0.0037 | 0.0041 | 0.0574 | 0.0574 | 0.0554 | 0.0589 |
Q3, R3 | 9.39E-4 | 9.52E-4 | 9.92E-4 | 8.85E-4 | -0.0144 | -0.0151 | -0.0168 | -0.0131 |
Censored scheme | CP | AW | ||||
Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | |
Q1, R1 | 0.8838 | 0.9564 | 0.9993 | 0.3178 | 0.2518 | 0.1287 |
Q1, R2 | 0.8384 | 0.9075 | 0.9951 | 0.3723 | 0.2999 | 0.1288 |
Q1, R3 | 0.8855 | 0.9289 | 0.9976 | 0.3001 | 0.2714 | 0.1315 |
Q2, R1 | 0.9681 | 0.9456 | 0.9869 | 0.2493 | 0.2657 | 0.1341 |
Q2, R2 | 0.8468 | 0.9702 | 0.9840 | 0.3669 | 0.2500 | 0.1339 |
Q2, R3 | 0.9238 | 0.9396 | 0.9707 | 0.2783 | 0.2672 | 0.1374 |
Q3, R1 | 0.9059 | 0.9422 | 0.9766 | 0.2456 | 0.2599 | 0.1238 |
Q3, R2 | 0.8831 | 0.9747 | 0.9873 | 0.2831 | 0.2588 | 0.1287 |
Q3, R3 | 0.8298 | 0.9657 | 0.9634 | 0.2730 | 0.2588 | 0.1283 |
Data sets | A-D | p-values | ||||||||
T | 0.0137 | 0.0272 | 0.0295 | 0.0308 | 0.0315 | 0.0829 | 0.1209 | 0.1248 | 0.6006 | 0.1132 |
0.1361 | 0.1538 | 0.2062 | 0.2141 | 0.2410 | 0.3165 | 0.3597 | 0.7634 | |||
1.0417 | 1.2821 | 5.2632 | ||||||||
Z | 0.0392 | 0.0726 | 0.1869 | 0.2506 | 0.2725 | 0.3448 | 0.3690 | 0.3876 | 0.3121 | 0.5858 |
0.4831 | 0.5076 | 0.5917 | 1.0101 | 1.0417 | 1.6949 | 2.8571 |
Censored scheme | Q | R |
1 | (2*5, 0*2, 1*1) | (1*5, 0*5) |
2 | (0*7, 11*1) | (0*9, 5*1) |
3 | (0*3, 5*1, 0*3, 6*1) | (0*4, 5*1, 0*3) |
Censored scheme | ˆδAML | ˆδS | ˆδE | ACI | |
d=3 | d=−3 | ||||
1 | 0.5179 | 0.5228 | 0.5080 | 0.5335 | (0.3815, 0.6579) |
2 | 0.5055 | 0.5131 | 0.4912 | 0.5244 | (0.2808, 0.7303) |
3 | 0.4425 | 0.4507 | 0.4315 | 0.4634 | (0.2852, 0.5998) |
(n1,m) | Q | (n2,t) | R | |
Case 1 | (30,10) | Q1=(0∗8,10∗2) | (40,20) | R1=(2∗10,0∗10) |
Q2=(20∗1,0∗9) | R2=(0∗19,20∗1) | |||
Q3=((0,5)∗5) | R3=((0,0,0,0,5)∗4) | |||
Case 2 | (50,20) | Q1=(10∗1,0∗18,20∗1) | (50,30) | R1=(5∗2,0∗13,5∗2,0∗13) |
Q2=(0∗19,30∗1) | R2=(0∗20,2∗10) | |||
Q3=(0∗10,2∗15) | R3=(10∗1,0∗28,10∗1) | |||
Case 3 | (100,50) | Q1=(10∗5,0∗45) | (150,70) | R1=(2∗40,0∗30) |
Q2=(20∗1,0∗24,30∗1,0∗24) | R2=(30∗1,0∗30,50∗1,0∗38) | |||
Q3=(0∗20,50∗1,0∗29) | R3=(0∗45,80∗1,0∗24) |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0165 | 0.0495 | 0.0406 | 0.0397 | 0.1129 | 0.2181 | 0.2003 | 0.1957 |
Q1, R2 | 0.0064 | 0.0044 | 0.0868 | 0.0291 | -0.0152 | 0.3182 | 0.3529 | 0.1627 |
Q1, R3 | 0.0060 | 0.0031 | 0.0663 | 0.0331 | 0.0400 | 0.2613 | 0.2569 | 0.1765 |
Q2, R1 | 0.0008 | 0.0034 | 0.0031 | 0.0030 | -0.0140 | 0.0543 | 0.0514 | 0.0492 |
Q2, R2 | 0.0032 | 0.0025 | 0.0034 | 0.0019 | -0.0481 | 0.0571 | 0.0532 | 0.0373 |
Q2, R3 | 0.0021 | 0.0029 | 0.0025 | 0.0018 | -0.0374 | 0.0473 | 0.0447 | 0.0368 |
Q3, R1 | 0.0032 | 0.0151 | 0.0137 | 0.0135 | 0.0422 | 0.1189 | 0.1139 | 0.1114 |
Q3, R2 | 0.0027 | 0.0078 | 0.0167 | 0.0107 | -0.0011 | 0.1285 | 0.1247 | 0.0968 |
Q3, R3 | 0.0021 | 0.0185 | 0.0130 | 0.0106 | 0.0130 | 0.1155 | 0.1099 | 0.0973 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0021 | 0.0096 | 0.0090 | 0.0090 | 0.0385 | 0.0955 | 0.0923 | 0.0922 |
Q1, R2 | 0.0008 | 0.0076 | 0.0072 | 0.0061 | 0.0047 | 0.0850 | 0.0824 | 0.0751 |
Q1, R3 | 0.0017 | 0.0094 | 0.0088 | 0.0086 | 0.0331 | 0.0944 | 0.0913 | 0.0899 |
Q2, R1 | 0.0136 | 0.0320 | 0.0288 | 0.0298 | 0.1113 | 0.1775 | 0.1683 | 0.1709 |
Q2, R2 | 0.0029 | 0.0349 | 0.0273 | 0.0222 | 0.0530 | 0.1707 | 0.1640 | 0.1464 |
Q2, R3 | 0.0113 | 0.0311 | 0.0281 | 0.0284 | 0.1008 | 0.1749 | 0.1662 | 0.1665 |
Q3, R1 | 0.0129 | 0.0318 | 0.0286 | 0.0292 | 0.1068 | 0.1767 | 0.1676 | 0.1685 |
Q3, R2 | 0.0035 | 0.0297 | 0.0281 | 0.0203 | 0.0377 | 0.1712 | 0.1661 | 0.1390 |
Q3, R3 | 0.0102 | 0.0306 | 0.0276 | 0.0272 | 0.0941 | 0.1731 | 0.1646 | 0.1625 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 3.61E-4 | 0.0012 | 0.0011 | 0.0012 | 0.0137 | 0.0318 | 0.0304 | 0.0326 |
Q1, R2 | 7.16E-4 | 0.0018 | 0.0017 | 0.0019 | 0.0229 | 0.0403 | 0.0389 | 0.0412 |
Q1, R3 | 6.24E-4 | 1.60E-4 | 1.56E-4 | 1.65E-4 | -0.0208 | 0.0013 | 2.01E-4 | 0.0019 |
Q2, R1 | 0.0052 | 0.0082 | 0.0078 | 0.0083 | 0.0708 | 0.0893 | 0.0872 | 0.0899 |
Q2, R2 | 0.0065 | 0.0096 | 0.0092 | 0.0097 | 0.0793 | 0.0972 | 0.0950 | 0.0977 |
Q2, R3 | 0.0016 | 0.0038 | 0.0035 | 0.0038 | 0.0373 | 0.0597 | 0.0579 | 0.0601 |
Q3, R1 | 0.0024 | 0.0046 | 0.0040 | 0.0046 | 0.0421 | 0.0638 | 0.0620 | 0.0641 |
Q3, R2 | 0.0039 | 0.0066 | 0.0063 | 0.0067 | 0.0574 | 0.0775 | 0.0757 | 0.0780 |
Q3, R3 | 9.39E-4 | 9.27E-4 | 8.63 E-4 | 9.09E-4 | -0.0144 | 0.0205 | 0.0192 | 0.0193 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0165 | 0.0179 | 0.0129 | 0.0175 | 0.0385 | 0.1173 | 0.0967 | 0.1177 |
Q1, R2 | 0.0064 | 0.0061 | 0.0059 | 0.0059 | 0.0047 | -0.0063 | -0.0235 | -0.0015 |
Q1, R3 | 0.0060 | 0.0066 | 0.0044 | 0.0066 | 0.0331 | 0.0447 | 0.0240 | 0.0483 |
Q2, R1 | 0.0008 | 0.0009 | 0.0010 | 0.0007 | 0.1113 | -0.0162 | -0.0204 | -0.0100 |
Q2, R2 | 0.0032 | 0.0035 | 0.0038 | 0.0027 | 0.0530 | -0.0504 | -0.0538 | -0.0427 |
Q2, R3 | 0.0021 | 0.0023 | 0.0025 | 0.0017 | 0.1008 | -0.0397 | -0.0434 | -0.0327 |
Q3, R1 | 0.0032 | 0.0032 | 0.0026 | 0.0037 | 0.1068 | 0.0424 | 0.0353 | 0.0480 |
Q3, R2 | 0.0027 | 0.0028 | 0.0027 | 0.0027 | 0.0377 | -0.0019 | -0.0082 | 0.0059 |
Q3, R3 | 0.0021 | 0.0022 | 0.0019 | 0.0023 | 0.0941 | 0.0134 | 0.0063 | 0.0201 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 0.0021 | 0.0021 | 0.0018 | 0.0025 | 0.0385 | 0.0386 | 0.0342 | 0.0433 |
Q1, R2 | 0.0008 | 0.0008 | 0.0007 | 0.0009 | 0.0047 | 0.0048 | 0.0009 | 0.0101 |
Q1, R3 | 0.0017 | 0.0018 | 0.0015 | 0.0021 | 0.0331 | 0.0340 | 0.0295 | 0.0380 |
Q2, R1 | 0.0136 | 0.0141 | 0.0118 | 0.0145 | 0.1113 | 0.1139 | 0.1036 | 0.1161 |
Q2, R2 | 0.0029 | 0.0049 | 0.0037 | 0.0053 | 0.0530 | 0.0565 | 0.0465 | 0.0605 |
Q2, R3 | 0.0113 | 0.0118 | 0.0097 | 0.0123 | 0.1008 | 0.1026 | 0.0923 | 0.1052 |
Q3, R1 | 0.0129 | 0.0134 | 0.0110 | 0.0136 | 0.1068 | 0.1097 | 0.0987 | 0.1111 |
Q3, R2 | 0.0035 | 0.0037 | 0.0029 | 0.0040 | 0.0377 | 0.0408 | 0.0313 | 0.0451 |
Q3, R3 | 0.0102 | 0.0108 | 0.0087 | 0.0111 | 0.0941 | 0.0964 | 0.0855 | 0.0988 |
Censored scheme | MSE | AB | ||||||
ˆδAML | ˆδS | ˆδE | ˆδAML | ˆδS | ˆδE | |||
d=3 | d=−3 | d=3 | d=−3 | |||||
Q1, R1 | 3.61E-4 | 3.50E-4 | 3.11E-4 | 3.95E-4 | 0.0137 | 0.0132 | 0.0117 | 0.0149 |
Q1, R2 | 7.16E-4 | 6.92E-4 | 6.24E-4 | 7.66E-4 | 0.0229 | 0.0223 | 0.0208 | 0.0239 |
Q1, R3 | 6.24E-4 | 6.54E-4 | 7.08E-4 | 5.76E-4 | -0.0208 | -0.0215 | -0.0228 | -0.0196 |
Q2, R1 | 0.0052 | 0.0053 | 0.0049 | 0.0055 | 0.0708 | 0.0710 | 0.0688 | 0.0724 |
Q2, R2 | 0.0065 | 0.0066 | 0.0062 | 0.0068 | 0.0793 | 0.0798 | 0.0775 | 0.0811 |
Q2, R3 | 0.0016 | 0.0016 | 0.0015 | 0.0017 | 0.0373 | 0.0371 | 0.0350 | 0.0387 |
Q3, R1 | 0.0024 | 0.0024 | 0.0022 | 0.0025 | 0.0421 | 0.0426 | 0.0407 | 0.0442 |
Q3, R2 | 0.0039 | 0.0039 | 0.0037 | 0.0041 | 0.0574 | 0.0574 | 0.0554 | 0.0589 |
Q3, R3 | 9.39E-4 | 9.52E-4 | 9.92E-4 | 8.85E-4 | -0.0144 | -0.0151 | -0.0168 | -0.0131 |
Censored scheme | CP | AW | ||||
Case 1 | Case 2 | Case 3 | Case 1 | Case 2 | Case 3 | |
Q1, R1 | 0.8838 | 0.9564 | 0.9993 | 0.3178 | 0.2518 | 0.1287 |
Q1, R2 | 0.8384 | 0.9075 | 0.9951 | 0.3723 | 0.2999 | 0.1288 |
Q1, R3 | 0.8855 | 0.9289 | 0.9976 | 0.3001 | 0.2714 | 0.1315 |
Q2, R1 | 0.9681 | 0.9456 | 0.9869 | 0.2493 | 0.2657 | 0.1341 |
Q2, R2 | 0.8468 | 0.9702 | 0.9840 | 0.3669 | 0.2500 | 0.1339 |
Q2, R3 | 0.9238 | 0.9396 | 0.9707 | 0.2783 | 0.2672 | 0.1374 |
Q3, R1 | 0.9059 | 0.9422 | 0.9766 | 0.2456 | 0.2599 | 0.1238 |
Q3, R2 | 0.8831 | 0.9747 | 0.9873 | 0.2831 | 0.2588 | 0.1287 |
Q3, R3 | 0.8298 | 0.9657 | 0.9634 | 0.2730 | 0.2588 | 0.1283 |
Data sets | A-D | p-values | ||||||||
T | 0.0137 | 0.0272 | 0.0295 | 0.0308 | 0.0315 | 0.0829 | 0.1209 | 0.1248 | 0.6006 | 0.1132 |
0.1361 | 0.1538 | 0.2062 | 0.2141 | 0.2410 | 0.3165 | 0.3597 | 0.7634 | |||
1.0417 | 1.2821 | 5.2632 | ||||||||
Z | 0.0392 | 0.0726 | 0.1869 | 0.2506 | 0.2725 | 0.3448 | 0.3690 | 0.3876 | 0.3121 | 0.5858 |
0.4831 | 0.5076 | 0.5917 | 1.0101 | 1.0417 | 1.6949 | 2.8571 |
Censored scheme | Q | R |
1 | (2*5, 0*2, 1*1) | (1*5, 0*5) |
2 | (0*7, 11*1) | (0*9, 5*1) |
3 | (0*3, 5*1, 0*3, 6*1) | (0*4, 5*1, 0*3) |
Censored scheme | ˆδAML | ˆδS | ˆδE | ACI | |
d=3 | d=−3 | ||||
1 | 0.5179 | 0.5228 | 0.5080 | 0.5335 | (0.3815, 0.6579) |
2 | 0.5055 | 0.5131 | 0.4912 | 0.5244 | (0.2808, 0.7303) |
3 | 0.4425 | 0.4507 | 0.4315 | 0.4634 | (0.2852, 0.5998) |