# Parameter Estimation in Step Stress Partially Accelerated Life Testing under Different Types of Censored Data

Authors: Mustafa Kamal; Sabir Ali Siddiqui; Ahmadur Rahman; Hassan Alsuhabi; Ibrahim Alkhairy; Thierno Souleymane Barry
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

A long testing period is usually required for the life testing of high-reliability products or materials. It is possible to shorten the testing process by using ALTs (accelerated life tests). Due to the fact that ALTs test products in harsher settings than are typical use conditions, the life expectancy of the objects they evaluate is reduced. Censored data in which the specific failure timings of all units assigned to test are not known, or all units assigned to test have not failed, may arise in ALTs for a variety of reasons, including operational failure, device malfunction, expense, and time restrictions. In this paper, we have considered the step stress partially accelerated life test (SSPALT) under two different censoring schemes, namely the type-I progressive hybrid censoring scheme (type-I PHCS) and the type-II progressive censorship scheme (type-II PCS). The failure times of the items are assumed to follow NH distribution, while the tampered random variable (TRV) model is used to explain the effect of stress change. In order to obtain the estimates of the unknown parameters, the maximum likelihood estimation (MLE) approach is adopted. Furthermore, based on the asymptotic theory of MLEs, the approximate confidence intervals (ACIs) are also constructed. The point estimates under two censoring schemes are compared in terms of root mean squared errors (RMSEs) and relative absolute biases (RABs), while ACIs are compared in terms of their lengths and coverage probabilities (CPs). The performance of the estimators has been evaluated and compared under two censoring schemes with various sample sizes through a simulation study. Simulation results show that estimates with type-I PHCS outperform estimates with type-II PCS in terms of RMSEs, RABs, lengths, and CPs. Finally, a real-world numerical example of insulating fluid failure times is presented to show how the approaches will work in reality.


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## Body

## 1. Introduction

The customer’s proclivity to place greater trust and happiness in a product has been thoroughly tested to ensure that it will fulfil its intended function with high reliability. Moreover, with the widespread use of computers, automation, and simulation, the overall quality of the manufacturing process has improved significantly, making the goods more reliable than their previous versions. Scientists require failure data to produce an efficient forecast about the product’s expected life. Obtaining such failure data using ordinary life testing (OLT) is a time-consuming and expensive process for such highly reliable goods, and in such cases, OLTs are not suitable. As a solution, more sophisticated tests, such as ALTs and partial ALTs, are used to get rapid failures of goods by testing them under more severe conditions (such as temperature, voltage, humidity, and so on) than typical usage settings, resulting in decreased testing time, labor, and money.In ALTs, products or materials are evaluated only at stress levels that are greater than the stress level at which they function in regular usage, with the hypothesis that a product’s failure mechanisms and process will follow the same profile as when tested at normal stress. ALTs are often classified into three kinds depending on the stress loading modalities: constant stress ALT, step stress ALT, and progressive stress ALT (commonly abbreviated to CSALT, SSALT, and PSALT) [1, 2]. In CSALT, units are tested at more than one constant high stress level until all failures of all units are observed or the test is terminated for reasons such as a censoring scheme or an inexplicable failure cause. For further information, readers are directed to some good and relevant references, including [3–14] based on CSALT models. In SSALT, the testing units are initially subjected to a starting high level of stress; the failures are noted; and then the test items are removed at a prespecified time to test at the next level of stress, and so on. Many scholars have looked at the SSALT models, including [15–24]. PSALT, in which test units are exposed to gradually increasing stress over time to obtain failure data for testing, was initially proposed by [25]. In their study, they obtained estimates of the parameters of both the exponential and Weibull life distributions under PSALT. Ever since, numerous authors have looked at PSALT for various distributions and data kinds, including [26–30].Censoring in life testing experiments can occur at any time, either intentionally or unintentionally. In intentional censoring, testing is ended after a specific period or a number of failures owing to cost and time limitations, whereas unintentional censoring is generally caused by operational failure or equipment malfunction. The most frequent types of intentional censoring are type-I (time-constrained) and type-II (failure-restricted). Because of the specified test termination time in type-I, an experiment may have a very low failure rate or even no failures, but the experiment length in type-II censoring may be rather long, rendering it impractical in many situations. Reference [31] proposed type-I hybrid censoring, which is basically a combination of type-I and type-II censoring, to circumvent these restrictions. One of the most significant shortcomings of type-I, type-II, and type-I hybrid censorings is that test items cannot be withdrawn at any time throughout the test other than the test termination points. More comprehensive censoring, such as type-II PCS and type-I PHCS, must be used in life testing studies to solve this issue [32]. In type-II PCS, a randomly selected sample of n items is first placed on an experiment with a predefined number of failures m and a preset randomized removing strategy r1,r2…,rm. At the initial failure time y1,m,n, the experiment may proceed by removing r1 test items from the remaining n−1 survivors. Similarly, at the second failure time y2,m,n, r2 test items will be removed from the remaining n−1−r1 survivals and so on until the mth failure ym,m,n occurs. When the mth failure occurs, the test is stopped, and all the remaining rm=n−m−∑i=1mri surviving survivors are eliminated. References [33–35] provide further details on type-II PCS. However, because of the preset size of observable failures, the major issue with type-II PCS is that the test length may be quite long, potentially resulting in additional expenditures and resources. Reference [36] introduces type-I PHCS with random terminal time T0∗=minym,m,n,T0 as a solution to this issue, where T0 is a preset test stoppage time. The most significant benefit of this censoring is that the test is now time-limited. See [32, 35, 36] for further details and insights.In ALTs, the life of the test item at the stress level it will be used in real life is estimated by extrapolating the lifetime data obtained at high stress levels to the typical usage stress level. Although such life-stress links are very complicated or perhaps do not exist in some situations. To address this issue, PALTs, which may be thought of as a logical combination of OLTs and ALTs, are more suited for conducting life testing. In contrast to ALT, test units in PALT are allocated to both normal and accelerated circumstances in order to gather failure data. Furthermore, PALT does not require a life-stress relation to calculate the estimated life of a product under real-world conditions. CSPALT and SSPALT are two extensively utilized core PALT classifications. In CSPALT, product samples are tested under both regular and accelerated settings at the same time until the test is ended owing to a censoring scheme or an unforeseen malfunction. Readers can find more information about CSPALT models in [5, 37–40]. In SSPALT, products are tested up to prespecified time at normal use conditions, and then all products that still working are assigned to test on accelerated conditions until the test is ended owing to a censoring scheme or an unforeseen malfunction. To reflect the effect of stress change, [41] introduced the TRV model for SSPALT. More details on TRV models can be found in [41, 42].Many studies for SSPALT based on different censoring schemes so far have been carried out; see [43–53] for example. Reference [47] considered the SSPALT based on the TRV model under type-II PCS to obtain the Bayes and ML estimates of the parameters of the Lomax distribution. Reference [48] estimated the parameters of the Weibull exponential distribution using the MLE approach based on SSPALT with type-II PCS. Reference [49] discussed the estimation of the stress-strength reliability under the assumption that the strength variable belongs to SSPALT and the components of strength and stress follow exponential distributions. Reference [50] described a k-stage SSPALT and derived model parameter estimates using interval type-I PCS with equal lengths of inspection interval. Reference [51] discussed and compared the MLEs of Weibull distribution parameters and AF based on adaptive type-I PHCS and type-I PHCS for SSPALT using the TRV model. Reference [52] investigated and compared MLEs of Burr type-XII distribution and AF under SSPALT based on the TRV model with type-I and adaptive type-II PHCS. Reference [53] produced parameter inferences based on SSPALT based on progressive hybrid censored masked data for a three-component hybrid system employing power-linear hazard rate distribution as lifetime distribution.The NH distribution was proposed by [54] in 2011 as an extension of the exponential distribution. The NH distribution has some useful and appealing features, such as having an always zero mode and a closed-form HF that can explain increasing, decreasing, and constant hazard rates, making it an excellent choice in lifetime data analysis. As a specific case, particular probability distributions, such as the exponential distribution, may be generated. As a result, it is a feasible alternative to the Weibull, exponential, and gamma distributions. So far, several studies such as [55–63] considering the problem of estimation of the parameters of the NH distribution using MLE and BE techniques have been conducted. Assuming that the scale parameter of NH distribution has a log-linear relation with stress, [55] obtained MLEs of the parameters under CSALT and SSALT models. Under CSALT and PSALT for type-II PC data, [56, 57] considered the MLE and BE techniques to obtain the estimates of model parameters. Reference [59] explored optimum plans for k-level CSALT plans under complete data for NH distribution using D and C optimality. Recently, taking into account the MLE and BE techniques, [62] developed a CSPALT based on type-II PCS for estimating the parameters of the NH distribution. To the best of the authors’ knowledge, there is no study based on SSPALT that discussed the estimation of the parameters of the NH distribution and the AF for type-II PCS and type-I PHCS. Reference [63] used the NH distribution as a lifetime distribution to estimate unknown model parameters in SSPALT with adaptive type-II PHCS and proposed two feasible optimum test approaches based on the A and D optimality.This paper has two major goals: first, to present an SSPALT plan utilizing type-II PCS and type-I PHCS to estimate the parameters of the NH distribution and AF and, second, to compare the estimates using different sample combinations under the two mentioned censoring schemes. The remainder of the article is divided into the following sections: Section2 discusses test assumptions and methodologies. In Section 3, SSPALT with type-II PCS is formulated, and MLEs and associated ACIs are produced. SSPALT with type-I PHCS is formulated and MLEs and associated ACIs are obtained in Section 4. In Section 5, for illustration purposes, a simulation study is carried out, and the results for the suggested models are discussed. In Section 6, a numerical example of insulating fluid failure times is utilized to show the applicability of the proposed estimation approach under SSPALT based on type-II PCS and type-I PHCS. Finally, Section 7 concludes the study with some remarks and future research directions.


## 2. Test Assumptions and Procedure

In this article, for both type II-PCS and type I-PHCS data, we made the following assumptions under SSPALT:a1: The SSPALT is formulated using two stress levelsSu and SaSu<Sa, where Su represents use (normal) stress conditions and Sa represents severe (accelerated) stress conditions.a2: There aren items that are put on the life test, which are identical and independent in nature. At least one failure at each stress Su and Sa must be observed.a3: The failure timeT of each test item follows the NH distribution, with the probability density function (PDF), cumulative distribution function (CDF), survival function (SF), and hazard rate function (HRF) provided by(1)ft;α,θ=αθ1+θtα−1exp1−1+θtα,t>0,α>0,θ>0,(2)Ft;α,θ=1−exp1−1+θtα,t>0,α>0,θ>0,(3)Rt=exp1−1+θtα,t>0,α>0,θ>0,(4)ht=αθ1+θtα−1,t>0,α>0,θ>0,whereθ and α represents the scale and shape parameters of the distribution, respectively. Figure 1 displays various shapes of PDF and HRF generated with varying input values of parameters.a4: Alln units are initially tested under stress Su until a prespecified stress change time τ, at which point all surviving survivors are moved to be tested at stress Sa. The switching impact of stress on product life from Su to Sa can be determined by multiplying inverse of AF by the residual life of the product, and total life Y at Sa may theoretically be described by the TRV model as follows:(5)Y=TifT≤ττ+β−1T−τifT>τ,whereT is the lifespan of the test unit at condition Su and β>1 is the AF, which is in general depends on the Su and Sa. Now, we can describe the PDF and RF of Y based on a4 as follows:(6)fy=0ify≤0,f1yif0<y≤τ,f2yify>τ,(7)Ry=0ify≤0,R1yif0<y≤τ,R2yify>τ.Figure 1
The PDF and HRF curves with various combinations of the values of parameters.
(a)(b)Using equations (1), (2), (5)–(7), the following expressions can be obtained easily:(8)fy=0ify≤0,f1y;α,θ=αθ1+θyα−1exp1−1+θyαif0<y≤τ,f2y=αβθ1+θτ+βy−τα−1exp1−1+θτ+βy−ταify>τ,(9)Ry=0ify≤0,R1y;α,θ=exp1−1+θyαif0<y≤τ,R2y=exp1−1+θτ+βy−ταify>τ.


## 3. SSPALT Formulation and Parameter Inference with Type-II PCS

In this section, we determined the MLEs and ACIs of the parameters using the SSPALT model and type-II PCS. Assume that alln components/items are assigned to stress level Su to begin the testing process with some of the prespecified test restrictions τ,m, and r1,r2…,rmτ<m<n. Continue the experiment at stress Su until time τ, assuming that y1,m,n<y2,m,n<⋯<yn1,m,n are the failure data observed before τ and r1,r2…,rn1 are the total number of items eliminated at usage stress Su due to the type-II PCS. All components/items that have not failed/been removed by time τ are now allocated to test at Sa, and the experiment continues to run in the same manner as at Su until the occurrence of mth failure. The observed failure sample at Sa is yn1+1,m,n<yn1+2,m,n<⋯<ym,m,n, and the total removals are rn1+1,rn1+2…,rn1+m. Finally, when mth failure is observed, the test is stopped, and all remaining rm=n−m−∑i=1m−1ri test items that have not yet failed are removed from the experiment.The test description makes it clear that the total number of failures observed atSu prior to time τ is n1, and consequently, m−n1 is the total number of failures observed at Sa. Under SSPALT, the entire observed failure data with type-II PCS of size m will now be of the form y1,m,n<y2,m,n<⋯<yn1,m,n≤τ<yn1+1,m,n<⋯<ym,m,n, and therefore, the appropriate likelihood function may thus be expressed as follows:(10)Ly,α,θ,β=C∏i=1n1f1yiR1yiri∏i=n1+1mf2yiR2yiri,where C=nn−1−r1n−2−r1−r2…n−mn−m−∑i=1m−1ri and yi=yi,m,n,i=1,2,3…m. Now using equations (1), (3), (8), and (9), and equation (10) can be rewritten as follows:(11)Ly,α,θ,β=C∏i=1n1αθAyiα−1exp1−Ayiαexp1−Ayiαri∏i=n1+1mαβθByiα−1exp1−Byiαexp1−Byiαri,where Ayi=1+θyi, Byi=1+θτ+βyi−τ, and Byi−1=θτ+βyi−τ. Now, the log-likelihood Logy,α,θ,β=ℓ of equation (11) can be derived as follows:(12)ℓ=logC+mlogα+mlogθ+m−n1logβ+α−1∑i=1n1logAyi−∑i=1n1ri+1Ayiα+α−1∑i=n1+1mlogByi−∑i=n1+1mri+1Byiα.


### 3.1. Point Estimates

Now, by differentiating (12) with respect to α,βandθ, the following equations are obtained:(13)∂ℓ∂α=mα+∑i=1n1logAyi−α∑i=1n11+riAyiαlogAyi+∑i=n1+1mlogByi−∑i=n1+1n1+n21+riByiαlogByi=0,(14)∂ℓ∂β=m−n1β+α−1θ∑i=n1+1myi−τByi−αθ∑i=n1+1m1+riyi−τByiα−1=0,(15)∂ℓ∂θ=mθ+α−1∑i=1n1yiAyi−α∑i=1n11+riAyiα−1+α−1∑i=n1+1mτ+βyi−τByi−α∑i=n1+1m1+riτ+βyi−τByiα−1=0.The MLEsα^,θ^,β^ of the unknown parameters α,θ,β of the model discussed here can be obtained by solving equations (13)–(15) simultaneously. Unfortunately, the system of equations (13)–(15) is nonlinear; therefore, no closed-form solution can be obtained analytically. As a result of this problem, some iterative methods must be used to obtain estimates of unknown parameters α,θ,β of the model. There are several iterative approaches, such as the Newton–Raphson method for solving nonlinear equations. In this case, we implemented the optim () function of the R statistical software/language to solve our nonlinear equations.


### 3.2. Interval Estimates

In this subsection, using the asymptotic properties of MLEs, we determine the ACIs of model parameters. Given specific regularity requirements, asymptotic features indicate that MLEs are approximately distributed according to the normal distribution with mean zero and varianceF−1, which can be represented mathematically as follows:(16)α^−α,θ^−θ,β^−β∼N0,F−1,where F−1 is the inverse of the observed Fisher information matrix and is commonly referred to as a variance-covariance matrix for MLEs. It is possible to derive it as follows:(17)F−1=−−∂2ℓ∂α2−∂2ℓ∂α∂θ−∂2ℓ∂α∂β∂2ℓ∂θ∂α−∂2ℓ∂θ2−∂2ℓ∂β∂θ−∂2ℓ∂β∂α−∂2ℓ∂θ∂β−∂2ℓ∂β2α^,θ^,β^−1=varα^covarα^θcovarα^β^covarθ^α^varθ^covarβ^θ^covarβ^α^covarθ^β^varβ^.The elements ofF can be expressed by the following equations:(18)∂2ℓ∂α2=−mα2−∑i=1n11+riAyiαlogAyi2−∑i=n1+1m1+riByiαlogByi2∂2ℓ∂θ2=−mθ2−α−1∑i=1n1yi2Ayi2−αα−1∑i=1n11+riyi2Ayiα−2−α−1∑i=n1+1mτ+βyi−τ2Byi2−αα−1∑i=n1+1m1+riτ+βyi−τ2Byiα−2∂2ℓ∂β2=−m−n1β2−α−1θ2∑i=n1+1myi−τ2Byi2−αα−1θ2∑i=n1+1m1+riyi−τ2Byiα−2∂2ℓ∂θ∂α=∂2ℓ∂α∂θ=∑i=1n1yiAyi−∑i=1n11+riyiAyiα−11+αlogAyi+∑i=n1+1mτ+βyi−τByi−∑i=n1+1m1+riτ+βyi−τByiα−11+αlogByi∂2ℓ∂α∂β=∂2ℓ∂β∂α=θ∑i=n1+1myi−τByi−∑i=n1+1m1+riyi−τByiα−11+αlogByi∂2ℓ∂θ∂β=∂2ℓ∂β∂θ=α−1∑i=n1+1n1+n2yi−τByi2+α∑i=n1+1n1+n21+riyi−τByiα−2Byi.Now, two-sided1001−Δ% ACIs for the parameter α,θ, and β can be obtained as follows:(19)α^±ZΔ/2varα^;θ^±ZΔ/2varθ^;β^±ZΔ/2varβ^,where ±ZΔ/2 represents standard normal distribution’s upper and lower Δ/2th percentile. varα^,varθ^, and varβ^ are the diagonal entries of F−1.


## 3.1. Point Estimates

Now, by differentiating (12) with respect to α,βandθ, the following equations are obtained:(13)∂ℓ∂α=mα+∑i=1n1logAyi−α∑i=1n11+riAyiαlogAyi+∑i=n1+1mlogByi−∑i=n1+1n1+n21+riByiαlogByi=0,(14)∂ℓ∂β=m−n1β+α−1θ∑i=n1+1myi−τByi−αθ∑i=n1+1m1+riyi−τByiα−1=0,(15)∂ℓ∂θ=mθ+α−1∑i=1n1yiAyi−α∑i=1n11+riAyiα−1+α−1∑i=n1+1mτ+βyi−τByi−α∑i=n1+1m1+riτ+βyi−τByiα−1=0.The MLEsα^,θ^,β^ of the unknown parameters α,θ,β of the model discussed here can be obtained by solving equations (13)–(15) simultaneously. Unfortunately, the system of equations (13)–(15) is nonlinear; therefore, no closed-form solution can be obtained analytically. As a result of this problem, some iterative methods must be used to obtain estimates of unknown parameters α,θ,β of the model. There are several iterative approaches, such as the Newton–Raphson method for solving nonlinear equations. In this case, we implemented the optim () function of the R statistical software/language to solve our nonlinear equations.


## 3.2. Interval Estimates

In this subsection, using the asymptotic properties of MLEs, we determine the ACIs of model parameters. Given specific regularity requirements, asymptotic features indicate that MLEs are approximately distributed according to the normal distribution with mean zero and varianceF−1, which can be represented mathematically as follows:(16)α^−α,θ^−θ,β^−β∼N0,F−1,where F−1 is the inverse of the observed Fisher information matrix and is commonly referred to as a variance-covariance matrix for MLEs. It is possible to derive it as follows:(17)F−1=−−∂2ℓ∂α2−∂2ℓ∂α∂θ−∂2ℓ∂α∂β∂2ℓ∂θ∂α−∂2ℓ∂θ2−∂2ℓ∂β∂θ−∂2ℓ∂β∂α−∂2ℓ∂θ∂β−∂2ℓ∂β2α^,θ^,β^−1=varα^covarα^θcovarα^β^covarθ^α^varθ^covarβ^θ^covarβ^α^covarθ^β^varβ^.The elements ofF can be expressed by the following equations:(18)∂2ℓ∂α2=−mα2−∑i=1n11+riAyiαlogAyi2−∑i=n1+1m1+riByiαlogByi2∂2ℓ∂θ2=−mθ2−α−1∑i=1n1yi2Ayi2−αα−1∑i=1n11+riyi2Ayiα−2−α−1∑i=n1+1mτ+βyi−τ2Byi2−αα−1∑i=n1+1m1+riτ+βyi−τ2Byiα−2∂2ℓ∂β2=−m−n1β2−α−1θ2∑i=n1+1myi−τ2Byi2−αα−1θ2∑i=n1+1m1+riyi−τ2Byiα−2∂2ℓ∂θ∂α=∂2ℓ∂α∂θ=∑i=1n1yiAyi−∑i=1n11+riyiAyiα−11+αlogAyi+∑i=n1+1mτ+βyi−τByi−∑i=n1+1m1+riτ+βyi−τByiα−11+αlogByi∂2ℓ∂α∂β=∂2ℓ∂β∂α=θ∑i=n1+1myi−τByi−∑i=n1+1m1+riyi−τByiα−11+αlogByi∂2ℓ∂θ∂β=∂2ℓ∂β∂θ=α−1∑i=n1+1n1+n2yi−τByi2+α∑i=n1+1n1+n21+riyi−τByiα−2Byi.Now, two-sided1001−Δ% ACIs for the parameter α,θ, and β can be obtained as follows:(19)α^±ZΔ/2varα^;θ^±ZΔ/2varθ^;β^±ZΔ/2varβ^,where ±ZΔ/2 represents standard normal distribution’s upper and lower Δ/2th percentile. varα^,varθ^, and varβ^ are the diagonal entries of F−1.


## 4. SSPALT Formulation and Parameter Inference with Type-I PHCS

In this section, SSPALT with type-I PHCS will be formulated first, followed by MLEs and ACIs of model parameters. In SSPALT with type-I PHCS, a random sample ofn test items is randomly allocated for testing under stress Su with prefixed experimental restrictions τ,m,T0 and progressive removal pattern r1,r2…,rm. Now, ri,i=1,2,…n1 test items are removed from the test randomly at ith failure observation yi,m,n and the experiment continue to run until time ττ≥yn1. At time τ, all of the survivors n−n1−∑i=1n1−1ri at Su are removed and then assigned for testing at Sa until the random termination time T0∗=minym,m,n,T0 of the experiment. For ym,m,n≤T0, this means that the mth failure ym,m,n is observed before time T0, and the test is stopped at mth failure time ym,m,n by removing all the remaining survivals rm=n−m−∑i=1m−1ri. For ym,m,n>T0, this means mth failure is not observed before time T0, and only J failures are observed; then, at time T0, test will be terminated by removing all rm=n−J−∑i=1Jri remaining survivals. Hence, under SSPALT with type-I PHCS, we observed two types of data: (i) y1,m,n<y2,m,n<⋯<yn1,m,n≤τ<yn1+1,m,n<⋯<ym,m,nifym,m,n≤T0 and (ii) y1,m,n<y2,m,n<⋯<yn1+1,m,n≤τ<yn1+1,m,n<⋯<yJ,m,nifyJ,m,n≤T0<yn1+n2,m,n.Suppose thatn1 is the size of failure sample observed at Su before time τ and n2 is the size of the observed failure sample at Sa after time τ. Under SSPALT, the entire observed failure data with type-type-I PHCS will now be of the form y1,m,n<y2,m,n<⋯<yn1,m,n≤τ<yn1+1,m,n<⋯<ym,m,n≤T0, and therefore, the appropriate likelihood function may be expressed as follows:(20)Ly,α,θ,β=C∏i=1n1f1yiR1yiri∏i=n1+1n1+n2f2yiR2yiriR2T0r,where C=nn−1−r1n−2−r1−r2⋯n−J−∑i=1m−1ri; for case (i), n1+n2=m; and for case (ii), n1+n2=J. yn1+j,m,n,…,ym,m,n are not observed. Now using equations (1), (3), (8), (9), and (23) can be rewritten as follows:(21)Ly,α,θ,β=C∏i=1n1αθAyiα−1exp1−Ayiαexp1−Ayiαri∏i=n1+1n1+n2αβθByiα−1exp1−Byiαexp1−Byiαriexp1−BT0αr,where Ayi=1+θyi, Byi=1+θτ+βyi−τ, and BT0=1+θτ+βT0−τ. Now, log-likelihood Logy,α,θ,β=l of equation (24) can be derived as follows:(22)l=logC+n1+n2logα+logθ+n2logβ+n2r1−BT0α+α−1∑i=1n1logAyi+∑i=1n11+ri1−Ayiα+α−1∑i=n1+1n1+n2logByi+∑i=n1+1n1+n21+ri1−Byiα.


### 4.1. Point Estimates

Now, by differentiating (25) partially with respect to α,θandβ, the following equations are obtained:(23)∂l∂α=n1+n2α−n2rBT0αlogBT0+∑i=1n1logAyi−∑i=1n11+riAyiαlogAyi+∑i=n1+1n1+n2logByi−∑i=n1+1n1+n21+riByiαlogByi=0,(24)∂l∂θ=n1+n2θ−n2rατ+βT0−τBT0α−1+α−1∑i=1n1yiAyi−α∑i=1n11+riAyiα−1+∑i=n1+1n1+n2Byi−1θByi−α∑i=n1+1n1+n21+riByi−1θByiα−1=0,(25)∂l∂β=n2β−n2rθT0−τBT0α−1+α−1θ∑i=n1+1n1+n2yi−τByi−αθ∑i=n1+1n1+n21+riyi−τByiα−1=0.The MLEsα^,θ^,β^ of model parameters α,θ,β under type I-PHCS can be obtained by solving equations (26)–(28) simultaneously. Unfortunately, again the system of equations (26)–(28) is nonlinear; therefore, no closed-form solution can be obtained analytically. Again, the optim () function of the R statistical software/language is implemented to solve nonlinear equations.


### 4.2. Interval Estimates

The same approach outlined in Subsection3.2 can be used to produce ACIs. The entries in the Fisher information matrix are as follows:(26)∂2l∂α2=−n1+n2α2−n2rBT0αlogBT02−∑i=1n11+riAyiαlogAyi2−∑i=n1+1n1+n21+riByiαlogByi2∂2l∂θ2=−n1+n2θ2−n2rαα−1Byi−1θ2BT0α−1−α−1∑i=1n1yi2Ayi2−αα−1∑i=1n11+riyi2Ayiα−2−α−1∑i=n1+1n1+n2Byi−12θByi2−αα−1∑i=n1+1n1+n21+riByi−1θ2Byiα−2∂2l∂β2=−n2β2−n2rαα−1θ2T0−τ2BT0α−2−α−1θ2∑i=n1+1n1+n2yi−τ2Byi2−αα−1θ2∑i=n1+1n1+n21+riyi−τ2Byiα−2∂2l∂α∂θ=∂2l∂θ∂α=−n2rτ+βT0−τBT0α−11+αlog1+θτ+βT0−τ+∑i=1n1yiAyi−∑i=1n11+riyiAyiα−11+αlogAyi+∑i=n1+1n1+n2Byi−1θByi−∑i=n1+1n1+n21+riByi−1θByiα−11+αlogByi∂2l∂α∂β=∂2l∂β∂α=−n2rθT0−τBT0α−11+αlogBT0+θ∑i=n1+1n1+n2yi−τByi−∑i=n1+1n1+n21+riyi−τByiα−11+αlogByi∂2l∂θ∂β=∂2l∂β∂θ=−n2rαT0−τBT0α−21+αBT0−1+α−1∑i=n1+1n1+n2yi−τByi2+α∑i=n1+1n1+n21+riyi−τByiα−2Byi.


## 4.1. Point Estimates

Now, by differentiating (25) partially with respect to α,θandβ, the following equations are obtained:(23)∂l∂α=n1+n2α−n2rBT0αlogBT0+∑i=1n1logAyi−∑i=1n11+riAyiαlogAyi+∑i=n1+1n1+n2logByi−∑i=n1+1n1+n21+riByiαlogByi=0,(24)∂l∂θ=n1+n2θ−n2rατ+βT0−τBT0α−1+α−1∑i=1n1yiAyi−α∑i=1n11+riAyiα−1+∑i=n1+1n1+n2Byi−1θByi−α∑i=n1+1n1+n21+riByi−1θByiα−1=0,(25)∂l∂β=n2β−n2rθT0−τBT0α−1+α−1θ∑i=n1+1n1+n2yi−τByi−αθ∑i=n1+1n1+n21+riyi−τByiα−1=0.The MLEsα^,θ^,β^ of model parameters α,θ,β under type I-PHCS can be obtained by solving equations (26)–(28) simultaneously. Unfortunately, again the system of equations (26)–(28) is nonlinear; therefore, no closed-form solution can be obtained analytically. Again, the optim () function of the R statistical software/language is implemented to solve nonlinear equations.


## 4.2. Interval Estimates

The same approach outlined in Subsection3.2 can be used to produce ACIs. The entries in the Fisher information matrix are as follows:(26)∂2l∂α2=−n1+n2α2−n2rBT0αlogBT02−∑i=1n11+riAyiαlogAyi2−∑i=n1+1n1+n21+riByiαlogByi2∂2l∂θ2=−n1+n2θ2−n2rαα−1Byi−1θ2BT0α−1−α−1∑i=1n1yi2Ayi2−αα−1∑i=1n11+riyi2Ayiα−2−α−1∑i=n1+1n1+n2Byi−12θByi2−αα−1∑i=n1+1n1+n21+riByi−1θ2Byiα−2∂2l∂β2=−n2β2−n2rαα−1θ2T0−τ2BT0α−2−α−1θ2∑i=n1+1n1+n2yi−τ2Byi2−αα−1θ2∑i=n1+1n1+n21+riyi−τ2Byiα−2∂2l∂α∂θ=∂2l∂θ∂α=−n2rτ+βT0−τBT0α−11+αlog1+θτ+βT0−τ+∑i=1n1yiAyi−∑i=1n11+riyiAyiα−11+αlogAyi+∑i=n1+1n1+n2Byi−1θByi−∑i=n1+1n1+n21+riByi−1θByiα−11+αlogByi∂2l∂α∂β=∂2l∂β∂α=−n2rθT0−τBT0α−11+αlogBT0+θ∑i=n1+1n1+n2yi−τByi−∑i=n1+1n1+n21+riyi−τByiα−11+αlogByi∂2l∂θ∂β=∂2l∂β∂θ=−n2rαT0−τBT0α−21+αBT0−1+α−1∑i=n1+1n1+n2yi−τByi2+α∑i=n1+1n1+n21+riyi−τByiα−2Byi.


## 5. Simulation Study

In this section, Monte Carlo simulation techniques were used to determine the unknown parameters of the distribution and AF. MLEs in type-II PCS and type-I PHCS are compared using RMSEs and RABs, whereas ACIs are compared using lengths and CPs. The simulation was run for prefixed values ofn,m,τ,T0, and the removal scheme r1,r2…,rτ,…,rm. The parameters and AF are then estimated using different samples of type-II PC and type-I PHC data obtained through simulation under SSPALT. The estimation process is carried out in accordance with the following steps using numerical simulation:Step 1: Initializen,mτandT0.Step 2: Initializeα,β,θ.Step 3: Generate type-II PC sample from NH distribution as follows:(i)
Generate a random sample (u1,u2,…,un1) of size n1 from uniform distribution U (0, 1) with removals r1,r2,…,rn1 at stress Su. The failure data at stress Su from the NH distribution may then be derived using the inverse CDF technique by using the following equation:(27)yi=1θ1−log1−ui1/α−1ifyi<τ,i=1,2,…,n1.(ii)
Similar to step (i), the failure data at stressSa from the NH distribution may then be derived by using the following equation:(28)yi=1β1θ1−log1−ui1/α−1−τ+τ,yi>τ,i=1,2,…,n2,where n2=m−n1 and the removals are rn1+1,rn1+2,…,rm. At mth failure ym:m:n, stop the test by removing all rm=n−m−∑i=1m−1ri survivals.Step 4: Obtain type-I PHC data under SSPALT by repeating steps 1–3. Stop the test atT0∗=minym,m,n,T0. If ym:m:n≤T0, stop the test at time ym:m:n by removing all rm=n−m−∑i=1m−1ri survivals (case I). If ym:m:n>T0, stop the test at time T0 by removing all rm=n−j−∑i=1j−1ri survivals (case II).Step 5: Obtain the MLEs of the parametersΘ=α^,β^,θ^ using some numerical techniques from equations (13)–(15) simultaneously for type-II PCS and from equations (26)–(28) simultaneously for type-I PHCS using the data generated in steps 1–4.Step 6: Repeat steps 1–5 up to 10,000 times, obtain the average MLEs with their RMSEs and RABs.Step 7: Obtain ACIs with their lengths and CPs.Step 8: Adopt the following progressive censoring schemes for different specified sets of values ofn,m,τ,T0 and α,β,θ under SSPALT:(29)Scheme1:ri=n−m,i=m,0,otherwise,Scheme2:ri=n−1.5m+1,i=m,1,otherwise,Scheme3:ri=n−mm,i=1,2,…,m0,otherwise.Taking into account the above-mentioned algorithm, we set the initial values forτ,T0 = (0.40, 0.65), (0.50, 0.80), (0.60, 1.20) and the combinations of sample sizes n,m = (80, 50), (80, 60), (100, 60), (100, 70), (120, 70), (120, 80). Assuming that the true values of parameters α,β,θ = (1.7, 1.3, 1.5), the MLEs of the parameters with their respective RABs and RMSEs are obtained and given in Tables 1–3 under both types of censored data. Lengths and CPs of corresponding 95% ACIs are also computed and provided in Tables 4–6. Figure 2 depicts plots of 10000 repetitions of type-II PCS data based on SSALT. Comparative plots of RMSEs and RABs are given in Figures 3–5. Comparative plots of lengths and CPs of corresponding 95% ACIs are given in Figures 6–8.Figure 2
Plots of type-II PCS data based on SSALT.
(a)(b)(c)(d)(e)(f)Table 1
MLEs, RMSEs, and RABs with true values ofα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.40, T0 = 0.65).
(n, m)CSType-II PCSType-I PHCSαβθαβθMLERMSERABMLERMSERABMLERMSERABMLERMSERABMLERMSERABMLERMSERAB(80, 50)11.7250.4910.3121.6250.5060.3611.6930.4860.3271.8950.1290.0541.4620.4210.3121.6950.2730.213(80, 60)11.730.4590.3011.5420.5160.3581.6580.4470.3191.7970.1060.0861.1890.3110.2571.780.2680.213(100, 60)11.6880.4130.2841.4930.4510.3111.7430.4260.3121.8570.0980.0681.1990.2870.2181.7550.2550.199(100, 70)11.6750.4260.2751.5060.3340.3041.5670.3760.3021.6450.0980.0641.1930.2910.2071.6730.2510.206(120, 70)11.7140.3990.2621.4520.320.2431.5440.3290.2391.5740.0780.0131.3820.2690.1981.680.2460.201(120, 80)11.6180.3810.2271.4820.2850.2171.6060.3090.2131.4930.0180.0041.3180.2520.1931.6810.2290.184(80, 50)21.7130.5270.3081.4960.5190.3591.8750.5520.3871.7870.1140.0741.4980.3370.2791.7380.2640.215(80, 60)21.6940.4460.2991.5050.5180.3541.7230.4640.3261.5130.110.0721.1360.3230.2481.7160.2630.211(100, 60)21.730.4580.2911.4390.510.3171.7050.4260.2971.790.0990.0711.210.2970.2261.6690.2540.207(100, 70)21.7560.4330.2871.4070.4510.2541.5990.3670.3131.7520.0280.0091.4710.2910.2161.6920.2440.204(120, 70)21.6190.4040.2531.5350.3840.2541.8310.3620.2741.710.0180.0051.5090.2780.2111.7840.2320.199(120, 80)21.660.3760.2111.5030.3210.2411.5810.350.2641.6830.0110.0041.5980.2270.1831.6630.2150.186(80, 50)31.7160.4890.3031.5980.4470.3841.8020.5270.3581.7540.1280.0781.4860.4040.2941.7270.2590.207(80, 60)31.7580.4240.2961.5690.4120.3661.8060.4780.3521.4480.1190.0911.0570.3090.2551.7540.2540.209(100, 60)31.6910.4220.2931.5540.3870.3311.7560.3680.3281.7870.0970.0691.2640.2730.2311.7590.2530.209(100, 70)31.7270.420.2831.4970.4050.2951.6960.3560.2781.7290.0370.0131.1290.2660.2031.6680.2330.194(120, 70)31.6210.3980.2791.5130.3880.2631.7610.3950.2581.6630.0370.0091.2690.2660.2081.7240.2280.175(120, 80)31.670.3990.2261.5240.3010.2251.6060.3780.2571.8070.0170.0051.4730.2480.1891.6890.2150.172Table 2
MLEs, RMSEs, and RABs with true values ofα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.50, T0 = 0.80).
(n, m)CSType-II PCSType-I PHCSαβθαβθMLERMSERABMLERMSERABMLERMSERABMLERMSERABMLERMSERABMLERMSERAB(80, 50)11.7130.5250.4361.4980.4110.3411.6780.5290.4391.8850.580.4811.4860.560.4651.6870.2720.226(80, 60)11.7250.480.3981.6020.370.3071.5980.4340.361.8750.4150.3441.5190.3350.2781.6870.260.216(100, 60)11.6920.4620.3831.620.330.2741.7710.4060.3371.5780.330.2741.4970.330.2741.7630.2530.21(100, 70)11.620.4540.3661.6030.3090.2561.5540.4060.3371.7990.290.2411.4690.3050.2531.7370.2430.202(120, 70)11.6580.4340.3561.4960.2850.2371.680.3970.331.6410.2650.221.4360.3010.251.7380.2320.193(120, 80)11.7620.4240.3481.5930.2850.2371.5790.3780.3141.6940.0270.0221.5220.2740.2271.6810.2120.176(80, 50)21.7170.5340.4431.5830.3730.311.8720.4490.3731.7740.6180.5131.4930.6290.5221.7540.3360.279(80, 60)21.7290.4790.3981.560.3320.2761.6430.4350.3611.7730.4690.3891.3440.3320.2761.6880.2770.23(100, 60)21.6120.4210.3491.4920.3150.2611.9710.390.3241.6850.4350.3611.6260.3310.2751.7340.2570.213(100, 70)21.7590.4170.3461.4810.3150.2611.5340.3840.3191.6030.2370.1971.5110.3010.251.7210.2530.21(120, 70)21.6870.3970.331.5750.2830.2351.7140.3810.3161.5910.150.1251.6140.2660.2211.7090.2270.188(120, 80)21.6540.3560.2951.5970.2360.1961.620.3030.2511.5360.1490.1241.5260.2570.2131.7250.2130.177(80, 50)31.7380.4730.3931.6050.4070.3381.5020.4650.3861.8510.5890.4891.5290.4250.3531.7180.2560.212(80, 60)31.7120.4490.3731.4010.3750.3111.8920.440.3651.5570.4750.3941.6030.3040.2521.7170.2530.21(100, 60)31.7550.4460.371.3690.3270.2711.8320.4020.3341.8790.3440.2861.4980.3030.2511.7440.2510.208(100, 70)31.6920.4090.3391.6040.3190.2651.4980.3960.3291.7660.2670.2221.2570.2870.2381.6590.2350.195(120, 70)31.6610.3780.3141.5380.310.2571.8220.3790.3151.6270.2590.2151.5090.2780.2311.8130.2050.17(120, 80)31.6190.3560.2951.5420.3050.2531.5340.2920.2421.6770.1050.0871.3090.2730.2271.7120.2020.168Table 3
MLEs, RMSEs, and RABs with true values ofα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.60, T0 = 1.20).
(n, m)CSType-II PCSType-I PHCSαβθαβθMLERMSERABMLERMSERABMLERMSERABMLERMSERABMLERMSERABMLERMSERAB(80, 50)11.730.5280.4381.4970.4110.3411.7610.430.3571.5360.3760.3121.2610.4640.3851.6670.2740.227(80, 60)11.7140.4560.3781.4020.3470.2881.5810.4290.3561.8110.2850.2371.2610.3990.3311.7420.2650.22(100, 60)11.760.4350.3611.5690.3390.2811.8430.3630.3011.5280.1940.1611.4840.2950.2451.7530.2620.217(100, 70)11.6890.3970.331.5820.3140.2611.9220.3480.2891.6050.1650.1371.3370.2780.2311.7310.2450.203(120, 70)11.6210.3770.3131.5050.3090.2561.8420.3460.2871.7530.1330.111.4080.2580.2141.690.2190.182(120, 80)11.6680.3750.3111.5350.2250.1871.7770.320.2661.6980.0570.0471.5160.2340.1941.7140.2160.179(80, 50)21.7220.510.4231.4630.4730.3931.990.4460.3711.5610.5170.4291.2560.4380.3641.7010.280.232(80, 60)21.740.4360.3621.4520.4010.3331.8420.4430.3681.780.3530.2931.3240.3070.2551.730.2660.221(100, 60)21.6840.390.3241.4130.2950.2451.6640.3880.3221.5420.2470.2051.3060.3060.2541.8090.2260.188(100, 70)21.620.3890.3231.4370.2890.241.6940.3650.3031.8980.2070.1721.20.2810.2331.7150.2230.185(120, 70)21.660.3890.3231.4780.2870.2381.7150.3610.3051.9460.1910.1591.5230.2540.2111.6840.2130.177(120, 80)21.7550.3770.3131.530.2370.1971.8980.3340.2771.9630.1840.1531.4090.2460.2041.7230.210.174(80, 50)31.7120.4820.41.6090.3820.3171.7820.3950.3281.5980.6420.5331.3590.5410.4491.6940.2840.236(80, 60)31.7340.4630.3841.3980.3320.2761.6580.3840.3191.5180.3980.331.3720.3030.2511.7440.2680.222(100, 60)31.6880.4340.361.5810.3210.2661.7720.3330.2761.7250.2960.2461.2650.2890.241.7170.2640.219(100, 70)31.6150.4150.3441.5150.2740.2271.9240.3280.2721.6830.2580.2141.4050.280.2321.6930.2370.197(120, 70)31.7560.3810.3161.5130.2350.1951.6940.3260.2711.7010.1470.1221.3930.2440.2031.7750.2330.193(120, 80)31.660.3740.311.3950.2340.1941.840.3220.2671.6520.1390.1151.4520.2220.1841.7170.1810.15Table 4
Lengths and CPs of 95% ACIs withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.40, T0 = 0.65).
(n, m)CSType-II PCSType-I PHCSαβθαβθLengthCPLengthCPLengthCPLengthCPLengthCPLengthCP(80, 50)11.92494.871.04495.061.90694.910.50694.480.69495.011.0794.61(80, 60)11.894.960.79894.571.75294.670.41694.390.3894.331.0594.26(100, 60)11.6794.580.55894.851.6794.990.38494.070.33294.221.0195.01(100, 70)11.61894.930.43894.671.47494.160.30694.860.28694.860.98494.37(120, 70)11.56495.130.40294.641.2994.710.23594.610.28494.390.96494.33(120, 80)11.49494.790.31894.281.21295.040.17294.650.24894.690.86294.86(80, 50)22.06695.041.05695.042.16494.610.44694.240.44694.931.03494.76(80, 60)21.79694.921.05293.991.81894.860.43295.060.40894.561.0294.71(100, 60)21.74895.081.0295.031.6794.930.1195.040.34694.140.95694.63(100, 70)21.69895.070.79894.961.43894.560.07894.20.33294.610.9194.65(120, 70)21.58494.870.57894.491.4294.780.0794.820.30294.070.84294.14(120, 80)21.47494.690.40494.781.37294.440.04494.370.20294.670.78494.56(80, 50)31.91694.942.58494.892.06694.540.50294.930.6494.291.01695.04(80, 60)31.66294.970.78494.731.84294.650.46694.910.37494.460.99694.18(100, 60)31.65495.020.64295.011.6394.520.18894.970.29294.260.99294.69(100, 70)31.64694.630.5994.631.5694.840.14694.260.27894.410.91494.24(120, 70)31.56495.120.39894.711.39694.390.14694.590.27894.440.89494.48(120, 80)31.5694.760.35694.371.35294.890.06694.140.24294.710.84294.11Table 5
Lengths and CPs of 95% ACIs withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.50, T0 = 0.80).
(n, m)CSType-II PCSType-I PHCSαβθαβθLengthCPLengthCPLengthCPLengthCPLengthCPLengthCP(80, 50)12.05894.590.66293.992.07494.182.27494.871.2394.371.06693.99(80, 60)11.88295.170.53694.921.70294.041.62694.40.4495.171.0293.91(100, 60)11.81293.930.42694.871.59293.931.29494.430.42694.90.99294.7(100, 70)11.70294.650.37494.321.59294.131.13695.090.36494.290.95294.37(120, 70)11.70293.960.31894.571.55695.091.03895.060.35693.960.9194.73(120, 80)11.70294.020.31894.841.48294.810.10695.170.29494.260.83294.29(80, 50)22.09494.40.54694.371.7693.962.42295.011.5594.11.31894.15(80, 60)21.87894.260.43295.141.70694.921.83894.150.43294.181.08694.57(100, 60)21.6594.790.38895.091.52894.841.70694.950.4394.571.00894.18(100, 70)21.63494.540.38894.621.50694.460.9394.460.35694.730.99294.62(120, 70)21.55693.880.31495.171.49493.880.58894.370.27895.010.8994.81(120, 80)21.39694.290.21894.021.18894.650.58494.90.25894.870.83495.09(80, 50)31.85494.810.6594.541.82295.22.30894.510.70893.911.00495.03(80, 60)31.7694.350.55295.121.72494.871.86294.240.36294.210.99294.48(100, 60)31.74894.460.4295.011.57695.171.34893.880.3694.40.98494.02(100, 70)31.60494.870.39894.71.55294.241.04694.540.32294.070.92294.32(120, 70)31.48294.510.37694.261.48694.541.01694.320.30293.990.80494.95(120, 80)31.39694.370.36494.981.14495.060.41294.620.29294.240.79294.07Table 6
Lengths and CPs of 95% ACIs withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.60, T0 = 1.20).
(n, m)CSType-II PCSType-I PHCSαβθαβθLengthCPLengthCPLengthCPLengthCPLengthCPLengthCP(80, 50)12.0794.570.66294.541.68694.871.47493.790.84494.731.07494.79(80, 60)11.78894.90.47293.821.68294.981.11895.040.62493.791.03894.26(100, 60)11.70694.960.4594.91.42295.090.7694.650.34294.321.02895.07(100, 70)11.55694.870.38694.211.36493.870.64694.870.30294.930.9694.57(120, 70)11.47894.150.37494.291.35694.010.52294.150.2694.10.85893.85(120, 80)11.4793.820.19894.011.25494.290.22493.850.21494.90.84694.98(80, 50)22.00294.540.87893.871.74893.792.02694.210.75295.091.09894.62(80, 60)21.7194.510.6393.791.73694.511.38494.840.3793.981.04294.93(100, 60)21.52894.930.34294.351.5294.590.96894.510.36894.870.88694.65(100, 70)21.52494.230.32894.931.4393.960.81295.090.3195.040.87494.01(120, 70)21.52494.820.32295.091.41694.350.74893.960.25295.120.83494.37(120, 80)21.47893.870.2294.841.3194.710.72294.070.23894.840.82495.09(80, 50)31.8994.010.57294.151.54893.852.51694.231.14894.961.11494.71(80, 60)31.81494.260.43294.681.50693.981.5694.710.3694.621.0594.04(100, 60)31.70294.180.40494.231.30694.041.1694.10.32894.261.03494.54(100, 70)31.62694.590.29494.11.28694.791.01294.730.30894.40.9393.87(120, 70)31.49494.210.21694.761.27894.370.57693.930.23494.210.91494.21(120, 80)31.46694.730.21494.651.26294.680.54495.120.19494.820.7194.84Figure 3
Plots of RMSEs and RABs of the estimates withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.40, T0 = 0.65): (a) type-I and type-II RMSE α with t = 0.40, T0 = 0.65; (b) type-I and type-II RAB α with t = 0.40, T0 = 0.65; (c) type-I and type-II RMSE β with t = 0.40, T0 = 0.65; (d) type-I and type-II RAB β with t = 0.40, T0 = 0.65; (e) type-I and type-II RMSE θ with t = 0.40, T0 = 0.65; and (f) type-I and type-II RAB θ with t = 0.40, T0 = 0.65.
(a)(b)(c)(d)(e)(f)Figure 4
Plots of RMSEs and RABs of the estimates withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.50, T0 = 0.80): (a) type-I and type-II RMSE α with t = 0.50, T0 = 0.80; (b) type-I and type-II RAB α with t = 0.50, T0 = 0.80; (c) type-I and type-II RMSE β with t = 0.50, T0 = 0.80; (d) type-I and type-II RAB β with t = 0.50, T0 = 0.80; (e) type-I and type-II RMSE θ with t = 0.50, T0 = 0.80; and (f) type-I and type-II RAB θ with t = 0.50, T0 = 0.80.
(a)(b)(c)(d)(e)(f)Figure 5
Plots of RMSEs and RABs of the estimates withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.60, T0 = 1.20): (a) type-I and type-II RMSE α with t = 0.60, T0 = 1.20; (b) type-I and type-II RAB α with t = 0.60, T0 = 1.20; (c) type-I and type-II RMSE β with t = 0.60, T0 = 1.20; (d) type-I and type-II RAB β with t = 0.60, T0 = 1.20; (e) type-I and type-II RMSE θ with t = 0.60, T0 = 1.20; and (f) type-I and type-II RAB θ with t = 0.60, T0 = 1.20.
(a)(b)(c)(d)(e)(f)Figure 6
95% ACIs lengths and CPs of the estimates withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.40, T0 = 0.65): (a) type-I and type-II 4:95% ACIs of α with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.40, T0 = 0.65); (b) type-I and type-II 4:95% ACIs of α with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.40, T0 = 0.65); (c) type-I and type-II 4:95% ACIs of β with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.40, T0 = 0.65); (d) type-I and type-II 4:95% ACIs of β with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.40, T0 = 0.65); (e) type-I and type-II 95% ACIs of θ with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.50, T0 = 0.80); and (f) type-I and type-II 4:95% ACIs of θ with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.40, T0 = 0.65).
(a)(b)(c)(d)(e)(f)Figure 7
95% ACIs lengths and CPs of the estimates withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.50, T0 = 0.80): (a) type-I and type-II 95% ACIs of α with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.50, T0 = 0.80); (b) type-I and type-II 5:95% ACIs of α with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.50, T0 = 0.80); (c) type-I and type-II 95% ACIs of β with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.50, T0 = 0.80); (d) type-I and type-II 5:95% ACIs of β with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.50, T0 = 0.80); (e) type-I and type-II 95% ACIs of θ with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.50, T0 = 0.80); and (f) type-I and type-II 5:95% ACIs of θ with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.50, T0 = 0.80).
(a)(b)(c)(d)(e)(f)Figure 8
95% ACIs Lengths and CPs of the estimates withα = 1.7, β = 1.3, θ = 1.5, and (τ = 0.60, T0 = 1.20): (a) type-I and type-II 95% ACIs of α with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.60, T0 = 1.20); (b) type-I and type-II 95% ACIs of α with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.60, T0 = 1.20); (c) type-I and type-II 95% ACIs of β with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.60, T0 = 1.20); (d) type-I and type-II 95% ACIs of β with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.60, T0 = 1.20); (e) type-I and type-II 95% ACIs of θ with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.60, T0 = 1.20); and (f) type-I and type-II 95% ACIs of θ with true values of α = 1.7, β = 1.3, θ = 1.5, and (t = 0.60, T0 = 1.20).
(a)(b)(c)(d)(e)(f)From the results in Tables1–3 and Figures 3–5, it can easily be observed that the results are consistent and the estimates are quite close to their true values for both cases of censored data. Estimates based on type-I PHCS in most of the cases are with smaller RMSEs and RABs as compared to the estimates based on type-II PCS. RMSEs and RABs are decreasing as a result of an increase in values of n and m for fixed values of τ,T0 in all cases for both censoring schemes, and this is expected because the results are more accurate for large samples. The RMSEs and RABs are considerably smaller for type-I PHCS than that of type-II PCS in most of the cases. For fixed values m,τ, and T0, a decreasing pattern is observed in the values of the RMSEs and RABs with an increase in the values of n for type-I PHCS. The same pattern is also observed for type-II PCS but RMSEs and RABs are smaller for type-I PHCS in general for all cases. With an increase in the time of stress change τ, the RMSEs and RABs are decreasing for fixed values of n,T0, and m, and this is also quite obvious due to the fact that increasing the stress change time may result in more failures under normal use conditions.For fixed values ofn,m, and τ with an increasement in censoring time T0, RMSEs and RABs result in decreasing values for type-I PHCS, but this is not true for type-II PCS because the predetermined numbers of failures in type-II PCS, and when T0 increases, no additional failures are observed in type-II PCS. From Tables 4–6 and Figures 6–8, it is also observed that the lengths and CPs of 95% ACIs are reasonably precise for both censoring schemes in all cases but the ACIs are narrower for type-I PHCS.As the values ofn and m increase, the lengths of the ACIs decrease, and CPs are approaching 95%, and this is natural since the accuracy of the estimates depends on the size of the sample. For fixed values of τ and T0, with an increase in the values of n and m, the lengths of ACIs are getting narrower for all cases under the two considered censoring schemes. It is also noted that the width of ACIs for type-I PHCS is smaller than that of type-II PCS. ACIs are also getting narrower with an increasement in both stress change time τ and censoring time T0 for fixed values of n and m for type-I PHCS. The same pattern is true for type-II PCS, but ACIs for type-I PHCS are smaller than that of type-II PCS in general for all cases.


## 6. Real-Life Data Application

In this section, we will implement the models for the real data set of insulating fluid failures, which was initially reported in [64] (page 105). Table 7 displays 19 breakdown times (in minutes) for an insulating fluid placed between two electrodes exposed to a 34 kV voltage. The goal of the experiment was to see if the time to breakdown at this voltage follows an exponential distribution as predicted by theory. If necessary, the distribution can be utilized to estimate the likelihood of fluid breakdown during real-world applications. Reference [35] investigated the data further in the context of type-II PCS, whereas [65] investigated the data in the context of adaptive type-II PCS.Table 7
Insulating fluid data.
0.190.780.961.312.783.164.154.674.856.507.358.018.2712.0631.7532.5233.9136.7172.89We begin by verifying that the NH distribution may be utilized to examine the provided data set. The Kolmogorov–Smirnov (K–S) goodness-of-fit test is used to fit the NH distribution to real data as well as to compare the results of the NH distribution with other existing similar distributions such as the generalized exponential and Weibull generalized exponential distributions. The K–S test compares a real data set to a similar probability distribution. The test employs the K–S distance between the empirical distribution and the referenced cumulative distribution, as well as the associatedp-values for the goodness of fit. Table 8 shows the MLEs of unknown parameters, K–S distances, and p-values for all three competing distributions, including the NH, generalized exponential, and Weibull generalized exponential distributions for the considered data set. The R statistical software/language is utilized for the computation of MLEs, K–S distances, and p-values for each. Figure 9 exhibits a plot of the empirical CDF versus fitted CDF, as well as a histogram of data against fitted PDF of the NH, generalized exponential, and Weibull generalized exponential distributions. From Table 8 and Figure 9, we can see that the NH distribution, when compared to the other distributions, gives a very excellent fit to the provided data set. As a result, the given data can be used as an illustration for our models.Table 8
MLEs, K–S distances, andp-values based on complete insulating fluid data.
DistributionAlphaBetaThetaK–S testp-valueNH distribution0.4978590.276878—0.142380.7855Weibull generalized exponential15.1210280.0022180.7850.203290.3625Generalized exponential0.8034190.102165—0.229580.2309Figure 9
(a) Plot of the empirical CDF versus fitted CDF and (b) histogram of data against fitted PDF.
(a)(b)Now, in SSPALT, we set the value ofτ=7 and the total number of failures, m=12, which are chosen from a total of 19 =n observations, and the removal scheme, which is set to r1=r2=r3=r4=r5=r6=0,r7=r8=r9=1 at use stress level, while r10=r11=r12=0,r13=1,r14=r15=r16=0,r17=r18=r19=1 at accelerated stress level to generate the type-II PC data for illustration purpose. The generated type-II PC data is provided in Table 9. The MLEs of the parameters and their related MSEs under SSPALT with initial values of 0.497859 and 0.276878 computed for the generated type-II PC data are shown in Table 10. Similarly, we set the value of τ=6.5, T0=35 and the total number of failures, m=12, which are chosen from a total of 19 =n observations and the removal scheme, which is set to r1=r2=r3=r4=0,r5=r6=r7=1,r8=r9=r10=0 at normal stress levels and r11=0,r12=1,r13=r14=r15=0,r16=1,r17=0,r18=r19=1 at high stress levels to generate the type-I PHC data. The generated type-I PHC data is provided in Table 9. The MLEs of the parameters and their related MSEs under SSPALT with initial values of 0.497859 and 0.276878 computed for the generated type-I PHC data are shown in Table 10.Table 9
The generated type-II PC and type-I PHC data sets.
Type-II PC dataNormal stress: 0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 6.50Accelerated stress: 7.35, 8.01, 12.06, 31.75, 32.52Type-I PHC dataNormal stress: 0.19, 0.78, 0.96, 1.31, 4.67, 4.85, 6.50Accelerated stress: 7.35, 8.27, 12.06, 31.75, 33.91Table 10
MLEs with their related MSEs under SSPALT.
Censoringα^β^θ^MLEMSEMLEMSEMLEMSEType-II PCS0.8951650.002340.3587430.001981.2019320.000365Type-II PHCS0.9427670.0008230.4628850.0005191.2663810.000173


## 7. Conclusions

In this article, an SSPALT model with type-II progressively and type-I progressively hybrid censored data has been developed. Under the premise that the TRV model describes the life of the experimental units and the lifetimes of experimental units follow the NH distribution, MLEs of the unknown parameters and AF were derived. The Monte Carlo simulation study was used to compute point and interval estimates of the parameters numerically. As per the simulation results, the MLEs are fairly near to their real values and are consistent with small RMSEs and RABs for both censoring schemes. It is also noted that ACIs are relatively precise, and the predicted values for both censoring techniques fall within these ranges. As a result, it is possible to infer that the estimations are performed satisfactorily. Point estimates were compared based on their RMSEs and RABs, while interval estimates were compared based on their lengths and CPs. As a comparison between the two censoring schemes, it is found that the RMSEs and RABs for type-I PHCS are smaller than those for type-II PCS in all the sampling combinations. It is also noticed that the lengths of the ACIs of type-I PHCS are more precise than those of type-II PCS in all cases. The CPs of type-I PHCS are closer to 95% than type-I PCS. In general, it can therefore be concluded that type-I PHCS performs better than type-II PCS-based MLEs and ACs in terms of RMSEs, RABs, lengths, and CPs.To demonstrate the applicability of the suggested estimation technique under SSPALT based on type-II PCS and type-I PHCS, a real-life numerical example of insulating fluid failure times is employed. As per the K–S distance and thep-value, the data set shows a good match for the NH distribution. To ensure that the distribution is a good fit for this data, we plotted the empirical CDF vs the fitted CDF, as well as a data histogram versus the fitted PDF of the NH distribution. In the future, the Bayesian estimation approach might be used to estimate the parameters under SSPALT for the same censoring schemes. The optimum SSPALT design can also be established in terms of the time and cost constraints on the test.



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