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Parameter estimation for progressive censored data under accelerated life test with k levels of constant stress
Mustafa Kamal •
Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, KSA e-mail: m.kamal@seu.edu.sa
Abstract
Accelerated life testing (ALT) is a time-saving technique that has been used in a variety of sectors to get failure time data for test units in a relatively short time it takes to test them under regular operating circumstances. One of the primary goals of ALT is to estimate failure time functions and reliability under typical use. In this article, an ALT with k increasing stress levels that is stopped by a type II progressive censoring (TIIPC) scheme is considered. At each stress level, it is assumed that the failure times of test units follow a generalized Pareto (GnP) distribution. The link between the life characteristic and stress level is considered to be log-linear. The maximum likelihood estimation (MLE) method is used to obtain inferences about unknown parameters of the model. Furthermore, the asymptotic confidence intervals (ACIs) are obtained by utilizing the inverse of the fisher information matrix. Finally, a simulation exercise is presented to show how well the developed inferential approaches performed. The performance of MLEs is assessed in terms of relative mean square error (RMSE) and relative absolute bias (RAB), whereas the performance of ACIs is assessed in terms of their length and coverage probability (CP).
Keywords: Simulation, type-II progressive censoring, multiple constant stress accelerated life test
I. Introduction
Traditional reliability tests are designed to examine failure time data acquired under normal operating circumstances. However, due to restricted testing time and highly reliable products like as electronics systems, insulating materials, engines, and so on, such life data is not easy to obtain. As a result, the use of traditional reliability tests is inappropriate, time consuming and expensive. Therefore, ALT is widely used in the manufacturing and production industries due to its capacity to provide timely and adequate failure data for product reliability and design assessment. Furthermore, the growing competitiveness of innovation, as well as the desire to shorten product development time, have underlined the use and significance of ALT techniques. It is especially difficult to detect faults in highly reliable items or systems under typical operating conditions in a short period of time. Thus, in the manufacturing business, ALT has become an essential element of the product design and development process. In ALT, samples of test items are exposed to more intense levels of stress, such as temperature, voltage, humidity etc. to cause early failures, and the resultant failure times and the used censoring schemes are recorded. The data is then utilized to create an ALT model for extrapolating the product's reliability under normal operation conditions.
The stress loading in ALT may be applied in a various different way, however the most often utilized stress loadings are constant, step, and progressive stress loadings [1] (abbreviated as
Mustafa Kamal RT&A, No 3 (63)
PARAMETER ESTIMATION FOR PROGRESSIVE CENSORED DATA... Volume 16, September 2021 CSALT, SSALT & PSALT). In CSALT, units are tested at two or more constant levels of high stress until they all fail or the test is stopped owing to some censoring scheme or other factors. Many researchers have investigated CSALT models, including [2-9]. In SSALT, the testing units are initially subjected to a starting high level of stress and the failures are noted and then the test items are removed at prespecified time to test at the next level of stress, and so on. Many scholars have looked at the SSALT models, including [10-19]. Progressive stress loading, often known as ramp stress loading, is a method of exposing test units to gradually increasing stress over time. PSALT designs were initially proposed by [20], who took into account both exponential and Weibull life distributions. Since then, several writers have explored PSALT for different distributions with different types of data, including [21-25].
Most items are subjected to constant stress when they are in use in general. The CSALT is straightforward in most tests, making it easier to maintain a consistent stress level and the CSALT mimics actual use of the product. For some materials and products, CSALT models are better designed and empirically validated. Furthermore, data analysis for estimating reliability is well established and automated. However, CSALT is the most commonly utilized ALT method because of its above-mentioned benefits, but the majority of ALT research has concentrated on statistical inference with only two or three levels of constant stress.
So far, there are only a few studies which have considered multi-stress CSALT. [26] discussed the reliability analysis of type-I censored Weibull failure data obtained from a system of multiple components connected in series under CSALT assuming a log-linear relation between scale parameter and the stress variable. [27] used TIIPC data to develop MLEs and Bayes estimates (BEs) of the parameters of the extended exponential distribution under CSALT. [28] examined several estimating methods for the parameters of the exponentiated distributions family, with the exponentiated inverted Weibull regarded as a particular example under CSALT. [29] estimated the parameters of a lower-truncated family of distributions using the MLE approach for simple CSALT under TIIPCS. Assuming mean life as a linear function of the stress, [30] considered hybrid type-I censored data from a CSALT and obtained the MLEs and approximation MLEs of the parameters of a generalized log-location-scale distribution. [31] investigated a multiple stress CSALT and utilized MLE and BE approaches to construct point and interval estimates of Weibull distribution parameters based on TIIP adaptive hybrid censored data. Assuming that failure under arithmetically increasing stress levels of CSALT, [32] employed MLE methods for estimating the Burr-X life distribution parameters. [33] studied CSALT and estimated the doubly truncated Burr-XII parameters using MLE and BE methods. [34] address the problem of statistical inference using MLE and BE approaches for TIIPC data under multiple stress CSALT, assuming that failure times follow the modified Kies exponential distribution and removal follows a binomial distribution.
In this paper, an ALT with k constant stress levels which is stopped by a TIIPC scheme is considered. The following is how the paper is structured. Section 2 provides fundamental terminology, failure distribution, and basic multi-stress CSALT assumptions. In Section 3, the MLE technique is employed to derive estimates of the parameters using TIIPC data. In Section 4, a simulation study with different test setting is conducted to compare the performance of the proposed model. Section 5 concludes the paper with some remarks.
II. Assumptions and procedure for fc-stress level ALT
A k levels of CSALT is considered. Let Q0 be the normal stress level and Qt, i = 1,2, ...k are the k levels of applied higher constant stress levels. The following assumptions are used in this paper: 1. Suppose -n.", i = 1,2,...,k are samples containing independent and identical items put on test at the same time at stress levels Q" ,i = 1,2,... k in such way that M = -n^, where N total number of items assigned on all stress levels to test.
Mustafa Kamal RT&A, No 3 (63)
PARAMETER ESTIMATION FOR PROGRESSIVE CENSORED DATA... Volume 16, September 2021 2. The product's life has a GnP distribution under normal stress Q0 and accelerated stress Qt, i = 1,2,...k. The density function, cumulative distribution function, the reliability function and the hazard rate function of GnP distribution are as follows:
m ,rpi) = rp" (1 + rp"ti)-()i+1), ti ,xPi > 0 T(tt , , Vi) = 1-(1+ ipiti)-)i, ti, , vi >0 R(ti ) = (1+ vi, > 0
h(ti ,<Pi,^d =
(1+ Viti)'
ti, Vi>0
(1) (2)
(3)
(4)
where ^ is the shape parameter and the scale parameter is rpt at stress level Qi, i = 1,2, ...k.
3. At each increased stress, the product's failure mechanism stays unchanged. Because specifies the failure mechanism, it follows that
<Po = <P& = <P2 =... = <Pk = <P (5)
4. The parameter ^ has a log-linear relationship with the stress variable Qi and may be described as follows:
log^i = a + pZi, i = 0,1,2,...,k (6)
where a and /3(> 0) are the unknown parameters of the relationship and their values usually depend on true nature of the test items. And Zi = Z(Qt) is an increasing function of stress Qi. Eq. (6) depends on the type of stress used for testing, e. g., if stress is temperature, then, the Arrhenius model is used and can written as log ^ = a + p /Qi, where Qi is temperature stress. If stress is voltage, then the inverse power model is appropriate to be used and can written as log ^ = a + P(log(Qi)), where Qi is voltage stress. For weather conditions, exponential model is used and can written as log^ = a + PQi. Above defined three well-known models may be converted into the linear form as in eq. (6) by transforming the stress with Z(Qt) = 1/Qit log(Qi) and Qi respectively.
Let ti-, i = 1,2, ...k; j = 1,2,...,-n-i are observed ordered failures with progressive censoring scheme = (Wi &, ^ 2,..., m.) , i = 1,2, ...k; j = 1,2,..., m^ at each Qt stress level. Based on k stress ALT with TIIPC scheme , -n^ units are put under accelerated testing condition Qi and the experiment will be run until m^ failures at each stress level and the number of failures is prefixed. Now, the TIIPC can be implemented as follows: at each Qt, i = 1,2, ...k, at first failure time tt&, & items are omitted from the remaining (-n^ — 1) survivals randomly. Similarly at time ^2, Wj2 is the number of removed items from (-n^ — 2 — &) remaining survivals and so on until the desired number of failures m0 i = 0,1,2, ...,k at each stress level obtained and then the test is terminated by removing all the remaining survivals = ^ — m^ — J/^i & from the test.
III. Parameter estimation
In statistics, MLEs and BE techniques are two of the most significant and commonly used approaches. The MLEs are asymptotically normal and consistent. The BE technique necessitates the selection of previous knowledge of unknown parameters, although this is generally a challenging task in reality. Furthermore, BE method frequently necessitates the use of complicated integral procedures. As a result, in a CSALT, this paper uses an MLE method utilizing TIIPC data.
Let the obtained observed failure TIIPC samples at ith stress level in the considered ALT t^& < ti2 < ••• < t^m , i = 0,1,2,... ,k, then the likelihood for the observed data under TIIPC scheme can
Mustafa Kamal RT&A, No 3 (63)
PARAMETER ESTIMATION FOR PROGRESSIVE CENSORED DATA. Volume 16, September 2021 be obtained in the following form
k i / ^ ) = °fTij fa- ){1-^rii M)"11 J (7)
where, Q = /ni(ni — 1 — wi1)(-ni — 2— wi1 — wi2)... & wij. Now, the log likelihood I = L(y,, ^,a,S ) corresponding to Eq. (7) after substituting the values of fTi(¿¡j) & TT..{ti-) and taking log on both sides is obtained as follows:
k /
1 = Z + loaWi) — ij $ + $ + 1)l09(1 + Wij )\ (8)
i%1 j%1
Now, from equation (6), we can drive
^ = ^0e(42i~4Zo) =rp0d6i, i = 0,1,2,..., k (9)
Where, = a + represents the GnP distribution's scale parameter at stress Q0 and = $ = e4(Z$~Zo) denotes the acceleration factor from Q1 to Q0, and
tti = (Zi —Z0)/(Z1 —z0), i = 0,1,2.....k (10)
Because the transformation from (a,p,rf>) to is one-to-one, we can immediately
calculate the product's life at Q0 using the new transformed parameters. As a result, the likelihood function (8) may be rewritten as follows:
k mi
l = Z Z109^) + + l0gid) -(1 + $ + <P)l09(l + Tpo^tij(11)
i=i j=i
By solving the following likelihood equations, the MLEs of the parameters can now be calculated:
= 0 (12)
dl _ 1 d6(1 + (1+-wij)<t>)ti}
% 1 + t"j
=0 (13)
di _ ni (1 + (1 + wij)^)tij
d-d -8 1 + \p0dai tij ^ = ■^-(1+ Wij )l°9(1 + *ij ) = 0 (14)
We now have a system of three nonlinear equations with three unknowns, making it difficult to find closed-form solutions manually. Hence, numerical solution to equations is obtained using Newton Raphson iterative approach, the R programming language is used to get the solutions.
By using asymptotic characteristics of the MLEs, the ACIs of the parameters may now be estimated using TIIPC by mathematically inverting Fisher's information matrix. As a result, we can compute the estimates of 95% two-sided ACIs for and 0 as follows:
d ± 1.96 Jvar(ip0) ; â ± 1.96 fvar(â); <p ± 1.96 fvar(<p) (15)
Where, var(xp0), var(tf) and var(<fi) are main diagonal entries of inverted Fisher matrix and the elements of the matrix are given by following equations:
d2l d^2
1
(16)
d2l dß2
Ui(1 + ip0ß6tjj(1 - 4(1 + Wij)(1 + rpoß6tjj) + U(1 + $ + Wjj$)))
ß2(1 + ^0ß6i tij)2
(17)
" _ + V V 1 ™ijjyj"ij
dtâ (1 + №6tij)2
d2l 1 ß26 (1 + (1 + wij
(18)
d2l _ d2l _ rpoUßai-1(1 + wij)tij d$dß _ dßd$ _ 1 + tij
(19)
d2l d2l ß6 (1+wij)tij
(20)
d^dipo dip0d$ 1 + ip0ß6i tij
d2l d2l ßai-1(1 + $ + wij$)tij
(21)
dip0dß dßdip0 (1 + xp0ß6i tij )2
IV. Simulation Study
In this section, the performance of the considered methodology for estimating the parameters of the GnP distribution based on the CSALT with k stresses for TIIPC data utilizing the log liner association between stress and life characteristic is examined through a Monte-Carlo simulation using the R-package. Two set of initial values = 1.2,d = 0.75,$ = 0.5), = 1.5,$ = 0.5,$ = 0.8) of parameter with various sample combinations (■ni ,mi )= (20, 10), (20, 15), (30, 15), (30, 20), (30, 25), (40, 20), (40, 25), (40, 30), (50, 25), (50, 30), (50, 35), (60, 35), (60, 40), (60, 45) are selected for simulation. Apart from the normal stress level Q0 = 110, three levels of constant stress are assumed as; Q& = 150, Q2 = 220 and Q3 = 250 under TIIPC scheme. Additionally, two distinct removal schemes i.wil,wi2,... ,wi(jn-l) = (*ni —mi) & wim. = 0; H.wil,wi2,... ,wi(jn-r) = 1 & wim = -ni —ms + 1 are used to generate TIIPC samples with various combinations of (■ni,mi) under three different constant-stress levels. For each test scheme, average RABs and RMSEs for point estimates, as well as lower and upper ACI limits (LACIL, UACIL) and ACIs lengths (ACIsL) of 95% ACIs with corresponding CPs are computed. Following steps are used to perform the simulation study:
Step 1: Set the initial values of the parameters and Step 2: Set the values of stress levels Qi, i = 0,1,2,..., k. Step 3: Set the values of (-ni, mi), i = 0,1,2,...,k at each stress levels Qi.
Step 4: Now using the defined values in step 1-3, generate i = 1,2,...,k random samples of size mt of TIIPC data from Uniform (0, 1) distribution according to the steps outlined by [35]. Step 5: Using inverse CDF method, for each sample size mt obtained in step 4, generate TIIPC data from GnP distribution using (exp(— ln(1 — u)/$) — 1)/^.
Step 6: For each stress levels along with removal scheme, repeat the steps 1-5 for 10000 times. Step 7: Compute the average of MLEs of d and $ with their respective RABs and RMSEs. Step 8: Compute LACIL, UACIL, ACIsL of 95% ACIs with corresponding CPs of rp0, d and Table 1-6 displays the numerical findings of RMSEs and RABs of MLEs and LACIL, UACIL, and ACIsL, as well as the corresponding CPs of ACIs. Figures 1-3 represent the behavior of the RABs and RMSEs with respect to the sample size.
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PARAMETER ESTIMATION FOR PROGRESSIVE CENSORED DATA. Volume 16, September 2021
Table 1: MLEs with RMSEs, RABs of ip0 & LACH, UACIL, ACIsL, CPs of ACIs with = 1.2, fl = 0.75,0 = 0.5)
('ni,m,- ) CS MT,F, RMSF, RAB TACTT, UAC.TT, ACTsT, CP
20, 10 1 1.379815 0.960779 0.439453 -0.503310 3.262942 3.766253 0.950000
20, 15 1 1.439942 0.797518 0.433950 -0.123190 3.003076 3.126269 0.940000
30, 15 1 1.370340 0.705081 0.408689 -0.011620 2.752298 2.763916 0.960000
30, 20 1 1.486785 0.666933 0.357396 0.179597 2.793973 2.614376 0.950000
30, 25 1 1.312451 0.664671 0.392446 0.009696 2.615206 2.605510 0.950000
40, 20 1 1.291834 0.644747 0.383245 0.028131 2.555537 2.527407 0.950000
40, 25 1 1.370246 0.639035 0.389515 0.117737 2.622754 2.505017 0.940000
40, 30 1 1.365194 0.611388 0.358700 0.166874 2.563514 2.396640 0.980000
50, 25 1 1.371536 0.588579 0.350251 0.217920 2.525151 2.307231 0.970000
50, 30 1 1.053091 0.533587 0.382436 0.007259 2.098922 2.091663 0.940000
50, 35 1 1.244513 0.521590 0.330190 0.222196 2.266830 2.044634 0.950000
60, 35 1 1.275332 0.504480 0.344008 0.286551 2.264113 1.977562 0.940000
60, 40 1 1.294818 0.451734 0.283193 0.409419 2.180216 1.770797 0.950000
60, 45 1 1.244503 0.451180 0.266636 0.360191 2.128815 1.768624 0.950000
20, 10 2 1.359424 0.946580 0.432959 -0.495874 3.214721 3.710595 0.980000
20, 15 2 1.418662 0.785732 0.427537 -0.121372 2.958696 3.080068 0.940000
30, 15 2 1.350089 0.694661 0.402649 -0.011446 2.711624 2.723070 0.940000
30, 20 2 1.464813 0.657076 0.352115 0.176943 2.752682 2.575739 0.950000
30, 25 2 1.293055 0.654848 0.386646 0.009553 2.576557 2.567005 0.950000
40, 20 2 1.272743 0.635218 0.377582 0.027715 2.517771 2.490056 0.950000
40, 25 2 1.349996 0.629591 0.383759 0.115997 2.583995 2.467997 0.950000
40, 30 2 1.345019 0.602353 0.353399 0.164408 2.525630 2.361222 0.940000
50, 25 2 1.351267 0.579881 0.345075 0.214700 2.487834 2.273134 0.970000
50, 30 2 1.037528 0.525702 0.376785 0.007152 2.067903 2.060751 0.940000
50, 35 2 1.226121 0.513882 0.325310 0.218912 2.233330 2.014417 0.950000
60, 35 2 1.256485 0.497025 0.338925 0.282316 2.230653 1.948337 0.970000
60, 40 2 1.275682 0.445058 0.279008 0.403369 2.147996 1.744628 0.950000
60, 45 2 1.226111 0.444512 0.262695 0.354868 2.097355 1.742486 0.970000
Table 2: MLEs with RMSEs, RABs of ip0 & LACIL, UACIL, ACIsL, CPs of ACIs with (ip0 = 1.5, fl = 0.5,0 = 0.8)
(n^m,) CS MT,F, RMSF RAB TACTT, UAC.TT, ACTsT, CP
20, 10 1 1.594526 0.948091 0.458727 -0.263732 3.452785 3.716517 0.930000
20, 15 1 1.556388 0.844726 0.385720 -0.099275 3.212051 3.311326 0.950000
30, 15 1 1.582766 0.812384 0.367365 -0.009506 3.175038 3.184544 0.930000
30, 20 1 1.546824 0.807062 0.381692 -0.035018 3.128666 3.163684 0.970000
30, 25 1 1.189059 0.807045 0.483961 -0.392749 2.770867 3.163616 0.960000
40, 20 1 1.557939 0.806578 0.406983 -0.022955 3.138833 3.161788 0.940000
40, 25 1 1.589465 0.763197 0.372843 0.093600 3.085331 2.991731 0.920000
40, 30 1 1.520138 0.754312 0.365067 0.041687 2.998588 2.956901 0.950000
50, 25 1 1.560208 0.736059 0.399957 0.117533 3.002883 2.885350 0.970000
50, 30 1 1.147427 0.687020 0.415694 -0.199131 2.493985 2.693117 0.970000
50, 35 1 1.581000 0.661877 0.319240 0.283721 2.878278 2.594557 0.950000
60, 35 1 1.466991 0.645759 0.352575 0.201303 2.732678 2.531375 0.940000
60, 40 1 1.582213 0.589180 0.280259 0.427421 2.737005 2.309584 0.950000
60, 45 1 1.350998 0.574095 0.326686 0.225772 2.476224 2.250452 0.970000
20, 10 2 1.496527 0.812236 0.370885 -0.095460 3.088511 3.183967 0.940000
20, 15 2 1.521890 0.781138 0.353235 -0.009140 3.052921 3.062062 0.940000
30, 15 2 1.487331 0.776021 0.367011 -0.033670 3.008333 3.042004 0.950000
30, 20 2 1.143326 0.776005 0.465347 -0.377640 2.664295 3.041939 0.930000
30, 25 2 1.498018 0.775556 0.391330 -0.022070 3.018108 3.040180 0.940000
40, 20 2 1.528332 0.733843 0.358503 0.091020 2.966664 2.876664 0.950000
40, 25 2 1.461671 0.725300 0.351026 0.040084 2.883258 2.843174 0.950000
40, 30 2 1.500200 0.707749 0.384574 0.113013 2.887387 2.774375 0.960000
50, 25 2 1.103295 0.660596 0.399705 -0.191470 2.398063 2.589535 0.960000
50, 30 2 1.520192 0.636420 0.306962 0.272809 2.767575 2.494766 0.960000
50, 35 2 1.410568 0.620922 0.339014 0.193561 2.627575 2.434014 0.950000
60, 35 2 1.521359 0.566519 0.269480 0.410982 2.631736 2.220754 0.950000
60, 40 2 1.299036 0.552014 0.314121 0.217088 2.380985 2.163896 0.960000
60, 45 2 1.429823 0.520777 0.260784 0.409101 2.450546 2.041445 0.970000
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PARAMETER ESTIMATION FOR PROGRESSIVE CENSORED DATA... Volume 16, September 2021
Table 3: MLEs with RMSEs, RABs of fl & LACH, UACIL, ACIsL, CPs of ACIs with = 1.2, fl = 0.75,0 = 0.5)
('ni,m,-) CS MTF RMSF, RAB T.ACTT, UACTT, ACTsT, CP
20, 10 1 0.956331 0.146834 0.116788 0.668536 1.244127 0.575591 0.960000
20, 15 1 0.970048 0.144884 0.116523 0.686076 1.254020 0.567944 0.950000
30, 15 1 0.964952 0.142178 0.112322 0.686283 1.243622 0.557339 0.970000
30, 20 1 0.938985 0.138562 0.116477 0.667403 1.210567 0.543164 0.950000
30, 25 1 0.964784 0.127289 0.102683 0.715298 1.214270 0.498972 0.940000
40, 20 1 0.937890 0.120433 0.104640 0.701840 1.173940 0.472099 0.950000
40, 25 1 0.948488 0.119736 0.097341 0.713805 1.183171 0.469366 0.950000
40, 30 1 0.951535 0.117819 0.098871 0.720610 1.182461 0.461852 0.960000
50, 25 1 0.936677 0.115799 0.101152 0.709711 1.163643 0.453932 0.960000
50, 30 1 0.879029 0.112284 0.099044 0.658952 1.099106 0.440154 0.950000
50, 35 1 0.884003 0.110710 0.103224 0.667011 1.100995 0.433983 0.960000
60, 35 1 0.951515 0.102543 0.079229 0.750531 1.152500 0.401969 0.950000
60, 40 1 0.962086 0.094635 0.077911 0.776601 1.147571 0.370970 0.940000
60, 45 1 0.948595 0.092008 0.079129 0.768259 1.128932 0.360673 0.960000
20, 10 2 1.004047 0.169615 0.129299 0.671601 1.336493 0.664892 0.910000
20, 15 2 0.993732 0.160594 0.128606 0.678968 1.308495 0.629528 0.940000
30, 15 2 0.919236 0.159483 0.138557 0.606649 1.231823 0.625174 0.970000
30, 20 2 0.975282 0.152032 0.125374 0.677299 1.273265 0.595966 0.960000
30, 25 2 0.987848 0.149513 0.121256 0.694803 1.280893 0.586090 0.960000
40, 20 2 0.945983 0.138707 0.117505 0.674117 1.217849 0.543732 0.960000
40, 25 2 0.967796 0.136775 0.109025 0.699717 1.235876 0.536159 0.970000
40, 30 2 0.908435 0.136498 0.121973 0.640898 1.175972 0.535073 0.940000
50, 25 2 0.854772 0.134755 0.124194 0.590652 1.118892 0.528240 0.930000
50, 30 2 0.963932 0.132995 0.111667 0.703263 1.224602 0.521339 0.940000
50, 35 2 0.970456 0.129270 0.109791 0.717087 1.223824 0.506737 0.960000
60, 35 2 0.950931 0.121757 0.106660 0.712287 1.189574 0.477287 0.960000
60, 40 2 0.977545 0.116375 0.096155 0.749450 1.205640 0.456190 0.940000
60, 45 2 1.003948 0.101925 0.080405 0.804175 1.203721 0.399546 0.970000
Table 4: MLEs with RMSEs, RABs of ti & LACIL, UACIL, ACIsL, CPs of ACIs with ($0 = 1.5, fl = 0.5,0 = 0.8)
(/ni,mi) CS MTF RMSF RAB T.ACTT, UACTT ACTsT, CP
20, 10 1 0.729710 0.156055 0.171920 0.423842 1.035579 0.611737 0.970000
20, 15 1 0.635912 0.147363 0.168628 0.347080 0.924744 0.577663 0.970000
30, 15 1 0.752896 0.146134 0.156156 0.466473 1.039318 0.572845 0.950000
30, 20 1 0.761058 0.141137 0.146475 0.484429 1.037687 0.553258 0.980000
30, 25 1 0.659879 0.139685 0.167380 0.386096 0.933663 0.547567 0.940000
40, 20 1 0.755815 0.138839 0.148709 0.483691 1.027939 0.544249 0.950000
40, 25 1 0.737196 0.134817 0.147778 0.472956 1.001437 0.528481 0.970000
40, 30 1 0.744293 0.128940 0.144283 0.491570 0.997016 0.505445 0.960000
50, 25 1 0.743150 0.122515 0.130935 0.503021 0.983279 0.480258 0.960000
50, 30 1 0.747007 0.120722 0.130088 0.510392 0.983622 0.473230 0.960000
50, 35 1 0.616800 0.118942 0.144958 0.383673 0.849927 0.466253 0.950000
60, 35 1 0.761865 0.113971 0.122099 0.538481 0.985249 0.446768 0.960000
60, 40 1 0.756613 0.105303 0.110844 0.550219 0.963008 0.412788 0.960000
60, 45 1 0.741342 0.098291 0.106389 0.548692 0.933991 0.385299 0.950000
20, 10 2 0.635545 0.121039 0.186381 0.398309 0.872781 0.474472 0.950000
20, 15 2 0.757236 0.120128 0.156847 0.521786 0.992686 0.470901 0.970000
30, 15 2 0.617029 0.110272 0.174321 0.400897 0.833161 0.432264 0.940000
30, 20 2 0.705092 0.106901 0.148531 0.495566 0.914619 0.419054 0.960000
30, 25 2 0.614613 0.099448 0.154738 0.419694 0.809531 0.389836 0.940000
40, 20 2 0.605555 0.099098 0.157103 0.411322 0.799788 0.388466 0.960000
40, 25 2 0.712096 0.098604 0.106531 0.518833 0.905359 0.386527 0.970000
40, 30 2 0.621327 0.097804 0.154010 0.429632 0.813023 0.383391 0.950000
50, 25 2 0.614977 0.083327 0.128538 0.451657 0.778298 0.326641 0.950000
50, 30 2 0.669440 0.074306 0.107218 0.523801 0.815080 0.291279 0.950000
50, 35 2 0.617426 0.067618 0.102392 0.484895 0.749956 0.265061 0.940000
60, 35 2 0.613535 0.064974 0.104036 0.486187 0.740883 0.254696 0.940000
60, 40 2 0.612077 0.064608 0.106285 0.485446 0.738708 0.253262 0.930000
60, 45 2 0.613641 0.063932 0.104994 0.488334 0.738948 0.250613 0.930000
Table 5: MLEs with RMSEs, RABs of 0 & LACIL, UACIL, ACIsL, CPs of ACIs with = 1.2, fl = 0.75,0 = 0.5)
(n.-.m.) CS
MTF
RMSF,
RAB
T.ACTT,
UACTT,
ACTsT,
XP_
20, 10 20, 15 30, 15 30, 20 30, 25 40, 20 40, 25 40, 30 50, 25 50, 30 50, 35 60, 35 60, 40 60, 45
.356076 .423038 .442701 .385605 .408111 .322006 .365803 .335572 .448864 .420924 .469790 .411541 .326189 .333336
0.217232 0.170116 0.103119 0.096882 0.095668 0.092630 0.080635 0.078809 0.077859 0.064415 0.062875 0.053268 0.046967 0.035991
0.331312 0.307633 0.181128 0.187231 0.188336 0.225161 0.180866 0.191962 0.139534 0.122822 0.101525 0.103181 0.110134 0.087729
-0.069698 0.089610 0.240587 0.195716 0.220601 0.140452 0.207758 0.181107 0.296260 0.294671 0.346555 0.307136 0.234134 0.262794
0.781851 0.756466 0.644815 0.575493 0.595620 0.503560 0.523847 0.490037 0.601468 0.547176 0.593024 0.515946 0.418244 0.403879
0.851548 0.666856 0.404228 0.379777 0.375019 0.363108 0.316089 0.308930 0.305208 0.252505 0.246469 0.208809 0.184110 0.141085
0.980000 0.940000 0.960000 0.950000 0.950000 0.980000 0.950000 0.990000 0.960000 0.950000 0.940000 0.940000 0.950000 0.980000
20, 10 20, 15 30, 15 30, 20 30, 25 40, 20 40, 25 40, 30 50, 25 50, 30 50, 35 60, 35 60, 40 60, 45
.387373 .360833 .465610 .404602 .402596 .340660 .447136 .403542 .450346 .390654 .416645 .388899 .341503 .450346
0.124545 0.105094 0.104133 0.101346 0.094031 0.093892 0.078380 0.077751 0.073391 0.061132 0.059222 0.050428 0.037570 0.030832
0.226787 0.239976 0.167382 0.199987 0.194124 0.218835 0.134325 0.163517 0.134952 0.122158 0.115242 0.104176 0.081691 0.095481
0.143265 0.154850 0.261510 0.205964 0.218295 0.156631 0.293511 0.251151 0.306500 0.270836 0.300570 0.290060 0.267867 0.195635
0.631482 0.566817 0.669710 0.603241 0.586897 0.524689 0.600761 0.555934 0.594192 0.510473 0.532720 0.487738 0.415140 0.316495
0.488217 0.411967 0.408200 0.397277 0.368602 0.368058 0.307250 0.304783 0.287693 0.239636 0.232150 0.197678 0.147273 0.120860
0.970000 0.970000 0.920000 0.950000 0.970000 0.960000 0.930000 0.980000 0.960000 0.950000 0.960000 0.940000 0.950000 0.950000
Table 6: MLEs with RMSEs, RABs of 0 & LACIL, UACIL, ACIsL, CPs of ACIs with Q/>0 = 1.5, fl = 0.5,0 = 0.8)
(n.-.m.) CS
MTF
RMSF,
RAB
T.ACTT, UACTT
ACTsT,
XP_
20, 10 20, 15 30, 15 30, 20 30, 25 40, 20 40, 25 40, 30 50, 25 50, 30 50, 35 60, 35 60, 40 60, 45
0.534378 0.628356 0.716714 0.568052 0.788660 0.567105 0.763205 0.677318 0.552460 0.541943 0.622779 0.603203 0.628403 0.578427
0.248217 0.226171 0.218887 0.204952 0.188148 0.180587 0.171400 0.165821 0.158731 0.131083 0.130376 0.126617 0.101000 0.095940
0.339334 0.291756 0.253214 0.291313 0.175430 0.266770 0.166184 0.198586 0.226367 0.189980 0.170692 0.150484 0.126772 0.128139
0.047871 0.185060 0.287696 0.166346 0.419890 0.213155 0.427261 0.352309 0.241347 0.285021 0.367242 0.355034 0.430444 0.390385
1.020884 1.071652 1.145733 0.969758 1.157431 0.921055 1.099149 1.002327 0.863573 0.798864 0.878316 0.851372 0.826363 0.766470
0.973012 0.886592 0.858036 0.803412 0.737541 0.707900 0.671888 0.650018 0.622227 0.513844 0.511074 0.496338 0.395920 0.376086
0.940000 0.970000 0.960000 0.950000 0.940000 0.970000 0.950000 0.950000 0.940000 0.970000 0.960000 0.970000 0.950000 0.960000
20, 10 20, 15 30, 15 30, 20 30, 25 40, 20 40, 25 40, 30 50, 25 50, 30 50, 35 60, 35 60, 40 60, 45
0.562503 0.661427 0.754436 0.597950 0.830169 0.596952 0.803374 0.712967 0.581537 0.519055 0.570466 0.655557 0.634950 0.661477
0.261281 0.238075 0.230407 0.215739 0.198051 0.190091 0.180421 0.174548 0.167086 0.156485 0.137982 0.137238 0.133281 0.106316
0.357193 0.307111 0.266541 0.306645 0.184663 0.280810 0.174930 0.209037 0.238281 0.257407 0.199979 0.179676 0.158404 0.133444
0.050391 0.194800 0.302838 0.175101 0.441990 0.224373 0.449749 0.370852 0.254049 0.212344 0.300022 0.386571 0.373720 0.453098
1.074614 1.128055 1.206034 1.020798 1.218348 0.969531 1.156999 1.055081 0.909024 0.825767 0.840910 0.924543 0.896181 0.869856
1.024223 0.933255 0.903196 0.845697 0.776359 0.745158 0.707250 0.684230 0.654975 0.613423 0.540888 0.537972 0.522461 0.416758
0.940000 0.950000 0.940000 0.960000 0.950000 0.940000 0.970000 0.970000 0.950000 0.940000 0.940000 0.960000 0.940000 0.940000
m m
Figure 1: RMSEs and RABs of the estimates of ip0 with (ip0 = 1.2,-d = 0.75, cp = 0.5) & 0/>o = 1.5, tf = 0.5, cp = 0.8)
m m
Figure 3: RMSEs and RABs of the estimates of cp with (ip0 = 1.2, tf = 0.75, cp = 0.5) & 0/>o = 1.5, tf = 0.5, cp = 0.8)
It is evident from the results in tables 1-6 and Figures 1-3 that the results are consistent and that the estimates are quite closer to the real values of the parameters. The following observations can be made in general:
1. In all situations, the RMSEs and RABs decrease as the values of -n^ and mt increase, which is to be expected because greater samples produce more accurate results.
2. The lengths and CPs of 95% of ACIs are relatively precise in all situations, as shown in table 1-6. However, the ACIs are narrower for parameter setting 1 and removal scheme 2.
3. With increasing values of -n^ and mt, it can also be seen that the lengths of 95% percent ACIs are getting smaller and the CPs are getting larger.
As an outcome of the above observations, it is reasonable to infer that the proposed model and estimation method in this paper performed well, and that all statistical assumptions for fitting the model and estimation are suitable.
V. Conclusions
In this paper, the CSALT model has been considered with k levels of constant stress. The observed TIIPC failure data was assumed to come from a GnP distribution. The distribution's shape parameter has been assumed to be independent of the stress, whereas the scale parameter was assumed to have a log linear relationship with the stress variable. Model parameters are estimated using the MLE approach, and their performance is evaluated using the corresponding RABs and MSEs. The performance of MLEs has been found to be satisfactory, as the estimated values approaching real values as the sample size increases. The ACIs have also been constructed
based on the asymptotic properties of the MLEs. The performance of ACIs was evaluated in terms of their corresponding CPs and ACIIL. An alternative lifetime distribution can be considered in the future research, and the corresponding inferences under various censoring methods can be developed using the BE technique.
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