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ESTIMATION OF FRECHET PARAMETERS WITH TIME-CENSORED DATA IN ACCELERATED LIFE TESTING UTILISING THE GEOMETRIC PROCESS
Abdul Kalam , Cheng Weihu2, Ahmadur Rahman3,Mohammad Ahmad4
•
1,2,4Faculty of Science, Beijing University of Technology, China, 3 Department of Statistics and O.R., Aligarh Muslim University, Aligarh-202002, India [email protected], [email protected], [email protected],
[email protected] Corresponding Author
Abstract
The geometric process (GP) has been applied to estimate constant stress accelerated life testing for the Frechet failure item with time-censored data. A geometric process (GP) is developed by the failure time of tested items when stress levels are constantly rising. The estimates of the various parameters are calculated using the maximum likelihood estimation procedure. The asymptotic variance of estimates is obtained using a Fisher information matrix. The asymptotic variance is then used to calculate the distribution parameter asymptotic interval values. The statistical properties and confidence intervals of the required parameters are then illustrated using a simulation technique.
Keywords: Frechet Distribution, Geometric Process, Maximum Liklihood Estimate, Asymptotic Confidence Interval Estimate, Simulation Study.
I. Introduction
The most common approach to product evaluation is accelerated life testing (ALT), which gives the necessary details on the product's life under normal usage. It's widely employed in the manufacturing sector for the purpose of enhancing product quality. In order to gather data swiftly than under normal conditions, it enables the researcher to enhance the stresses on the life distribution parameters. Because working under normal conditions would be time consuming, it is impractical to test items under greater stress than is normal in order to induce early failure. The life distribution of a product, as well as any related characteristics under normal stress, must be extrapolated from test data using accelerated life analysis. Comparing such a test to tests conducted under normal conditions, time and money are saved.
Making the decision as to what stress should be imposed and how is the most challenging job in Accelerated Life Testing. The ALT contains a variety of stress loading types, such as constant stress , progressive stress, step stress, random stress and cyclic stress. ALT has mainly two types of data that is complete (every failure time is available) and censored (some failure time is unavailable).
Numerous authors have given their perspective on accelerated life testing (for constant stress), references includes [1, 2, 3, 4]. Yang[5], introduced optimal design using a four-level ALT and compared it with three-level ALT for different censoring schemes. ALT utilised Lam's[6] GP concept to investigate the problem of repair replacement. ALT plans for Generalized Exponential
Distribution under GP was analyse by Lone[7]. A great number of literature on ALT under GP model are (see,[8, 11, 9, 10]). Zhou[12] showed ALT with progressive hybrid censoring under geometric process for Rayleigh distribution. And the same time Huang[13] preseneted GP for exponential failure model in respect of complete as well censored observation. Fan [14] explored the constant ALT design for the generalised gamma model. For life distributions like exponential and lognormal distributions, Chen[15] discovered Bayesian approximations of the parameters in a generalised linear model (GP). So many works has done on GP in ALT, see Lone SA[16], Kamal M[17], Lone SA[18], Kamal M[19], Zarrin S[20], Lone SA[22], Lone SA[23], Ismail[24], Lone SA[25], Aly H[26], Alam I[27]. Using Informative and Noninformative Priors, Sindhu[28] performed a Bayesian Study for Censored Shifted Gompertz Mixture Distributions. Nassr, SG[29] extended Weibull distribution under adaptive type II progressive hybrid censoring. Hemmati, F[30] provided the log-normal distribution under type-II progressive hybrid censoring . For the Modified Kies exponential distribution, Hussam E[31] investigated simple and multiple ramp-stres ALT design for type-II censored data and Binomial Removal.
II. Model Description and Testing Procedure
I. Geometric Process (GP)
A sequence of stochastic variable {Xn}, n = 1,2,3... is referred to as a Geometric Process (GP) if {An-1 Xn,n = 1,2,3...} established a renewal process. Where, ratio of GP A(> 0) is a real valued. It may be demonstrated that if {Xn}, n = 1,2,3... develops a GP and there exist a random variable having pdf f (x) with mean 7 and variance a2 then the subsequent pdf of Xn will be given as An-1 f (An-1 x) with mean E(Xn) = j/An-1 and variance V(Xn) = a2/A2(n-1).
II. Frechet Failure Model
Probability density function of the Frechet variable is :
f (x) = apax—a— e( P) , x > 0, a > 0, p > 0 (1)
where, a(shape) and p (scale) are parameters of the life distribution:
0 1 2 3 4 5
Figure 1: PDF at different shapes and fixed scale (p = 2)
Figure 2: CDF at different shapes and fixed scale (fi = 2) CDF of Frechet variable takes the following expression:
_( x)-a
F(x) = e( ß) , x > 0, a > 0, ß > 0 (2)
Survival function for the Frechet variable is given as:
The hazard function (HF) is
_j x \-a
S(x) = 1 - e (ß) , x > 0 (3)
_( x)-a
,, . aßax-a-1 e (ß) ,A.
h(x) = —-_( x )-a (4)
1- e (ß)
III. Assumptions
1. The lifetime of failed items follows Frechet distribution at each stress.
2. Supoose a life test is organized with s number of stresses(increasing order). A random sample of n items are place on each stress and begin to operate simultaneously. Let the failure time of ith (ranges from 1 to n) item in kth (ranges from 1 to s) stress is denoted by x^. Now, failed items are removed and the test will run till the complete sample is exhausted at a predetermined t(censoring time)at each stress.
3. Stress is a log-linear function of the scale parameter fi i.e., log(fik) = a + bSk, where a and b are unknown parameters, values dependind on the nature of products and test method.
4. Let say the lifespan of items on each stress is represented by random variables X0, X\, X2,..., Xs, where X0 is the lifespan of the items under normal stress and sequence Xk, k = 1,2, ...s formulates a GP with ratio parameter A > 0.
All of the preceding assumptions, except for the last one assumption (4th assumption), are generally accepted in the ALT. Last one is based on the notion of a geometric procedure that is better than the traditional one without making computation more difficult. The following
theorem, which presupposes a log linear function between life and stress is demonstrates as:
Theorem 1. If stresses in an ALT increases constantly, then the lifespan of products at each stress develops a GP. i.e., if the difference (Sk+1 - Sk) = (AS )constant, for k = 1,2,...,s - 1, then {Xk}, k = 1,2,..., s develops a GP.
Proof.Last assumption (4th assumption) states that,
log(^f) = b(S+ - Sk) = b (AS )
(5)
This demonstrates that the excessive stresses comprise an arithmetic series with a difference AS. (constant ) Now, the previous expression can be rewritten as:
from (6), we have
pk+1 = bAS = 1
Pk A '
1 1
pk = A^k-1 = A2 fik-2
(6)
A, p
The lifetime pdf of an object has the following structure at the kth stress level
f(x) = apax-a-1 e
1*-(p)
And the cdf is written as
This shows that
fXk(x) = A-akrk arkpxrkxkle\ p
FXk (x) = e
fXk (x) = AkfX0 (Ax)
(7)
(8)
(9)
Hence, from the theory of GP and using the equation(9) it is obvious that if the pdf of lifespan of the X0 (normal stress) is fX0(x), then the pdf of the lifespan Xk (kth stress) is given by AkfXo (Akx). Finally , it is evident that lifespan of a arithmetically increasing stresses results a GP. ■
IV. Maximum Likelihood Estimation
The most significant and frequently applied estimation technique is the maximum likelihood (ML) approach. The applicability of other methods is restricted, whereas it can be used with any probability distribution. ML estimation achievement in ALT is more complicated, and closed-form estimates of parameters are typically unavailable. Consequently, to calculate them, an arithmetic technique Newton Raphson (NR) method has been used.
Let's say the test is stopped at time t for each stress level, and only xjki(< t) failures are recorded. Suppose that rk < n failures at the kth stress levels are obtained prior to suspending the test, and remaing (n — rk) items are survived till the entire test without any failure. The likelihood function of a particular stress is provided for time-censored Frechet failure data under GP with s number of stress:
(n - rk)!
i -akrk
irkrk fjxue
i=1
' xki P
1e
Consequently, the likelihood function for overall stresses is
n-rk
(10)
a
k
A x
Akx,.,
P
k
A" t
n
P
L
k
Ls (a, fi, A) = n Lk
s=1
n n! k=i (n - rk)!
A-akrk ark fiark n xkiA fi i=1
1- e
n-rk
Taking log both sides and the likelihood function for the above equation is
(11)
l = ln Ls (a, fi, A)
E
k=1
ln
(n - rk)!
Ak\ -a rk
+ rk (ln a + ln fi - ak ln A) - (a + 1) E ln xu - ( fi J E
i=1
i=1
(12)
+ (n - rk) ln < 1 - e
Ah ~ fi
MLE's of different parameters a, fi and A has found after solving the these normal equations dL dL anH dL = 0
da, dfi and dA = a
= e
Ta E
- k rk ln A + a \ - E ln xki +( fij In i fi) Exki - (n - rk) A 1
i=1
i=1
e
- A? - A?
(13)
di = e dfi h
fi-1rk - ( fi) In fi Exa + (n - rk) a-1 < e
- f) fi
(14)
= ^ dA = ¿1
rk
- a k rkA-1 - fiaa k A-(ak+1) E xki - (n - rk)A-1 \ a k A-(ak+1) -t e
i=1
where A = 1 e
Akt ~ fi
(15)
As we can see,expressions(13), (14) and (15) are not linear. Consequently, it is challenging to find a closed-form answer. Therefore, the estimate of a, fi and A is obtained by concurrently solving these expressions using the NR method.
V. Asymptotic Confidence Interval
Under some specific regularity restrictions, large sample theory assures the consistency and normality of ML estimators. Because the estimate of parameters are not forming closed form, exact confidence intervals of the parameters cannot be determined . As a consequence, asymptotic confidence intervals rather than exact confidence intervals develop using the asymptotic property of MLE's.
k
a
Akx
k
A" t
s
fi
s
n
a
x
a
a
k
At
fi
a
k
At
fi
a
k
a A" t
fi
a
Providing the The Fisher Information Matrix (FIM) as:
r d2l d2l d2l -
da2 dadß dadA
F = d21 d2l d2l
dßda dß2 dßdA
d2l d2l
__ . dAda dAdß dA2 J
The components of FIM can be obtained as follows:
dÜ - f
da2 = f o« ^=1
1- GO Mr 1
f xki - (n- rk) A-i=1
x < e
XA ( 2
-11- e
Akt ~ ß
Ah"
T
(16)
= f dß2 = ¿I
f\k\-a rk
ß-2 rk - i j\ (Inß)2 f Xki. + (n - rk)A-2
e
Akt ~ ß
- Aßf\ In ß ( A (ß* In ß - e
Akt ~ ß
- £) - in ß) - 1
(17)
= f
dA2 = ¿1
rk
- a k rkA-2 + ßa a (a + 1) k A-(ak+2) f xki + (n - rk) A
i=1
-2
i \ -a i —àb xl a k A-(«k+2)l ß) A - ß
Ai a kA-^l t) + ( -n V" /- ß
d2l = f dadß = ¡f
'AkS
ß \ß
Ak
-M « 1+aln M fxki + (n-rk)A
2
ß
i=1
e
Akt ~ ß
-9) ~ (ï + - Af
+ ß Mn ß - e
Akt ~ ß
(18)
- f) ]n ß
(19)
d2l dadA
f
k=1
- ^ +(£) J ( ß + a 'n(ß) " ) +(n - rk)A-2
x^« k A-("'-') I>
akx-wi -;)- + (ß
-a Akt
A- ß
2
a
a
k
a
a
A t
ß
a
a
a
a
a
a
a
a
a
d2' M 1u (Xk\
m=£ -A
Ah V
j)
The var-covariance matrix is:
s E - A
k—1
f ( \kt \
e
rk
!E-
i—1
\ all , ln J - + a ln —
J
J
- e
- f) J
(21)
r d2l d21 __'£]_ 1
da2 dadJ BadX
1 — d2l d21 d21
dJda dJ 2 BJB\
d2l a21 d21
. dXdlx BXBJ 9A2 J
F-
The diagonal elements in the above matrix are representing variances terms, and the non-diagonal are indicating covariances .
Estimates of asymptotic confidence interval for the proposed parameters a, J and A are written
as:
a ± Z1- v (SE(ol)) , J ± Z1- v (SE(J)) and til ± Z1-v (SE(A)) respectively.
III. Simulation study
Simulation study is a computational approach to analyze the behavior of the function. The uniform distribution has used to better understand the characteristics of the parameters. The
proposed simulation method going through these steps.
1. Generating a pseudo random sample using the distribution u[0,1].
2. Inverse-cdf method applied to transform the equation (8) in terms of u. Expression of
Xki (< t) is:
(6
xki —
Ak (- logu)
1/a
3. 5000 random samples of size 25, 50, 80, 120 and 150 have been produced from the Frechet Distribution.
4. Opted a fixed censoring time t — 3.5 at normal condition.
5. Taken values of number of failed items Tk — (0.8 x n), where n is sample size.
6. Chosen stress levels are s — 4,6 and 8 along with parameter values a — 1.4, A — 0.5, J — 1.1.
7. Finally, optim () function in R-Programming Software has been used to calculate the ML estimates of mean along with various statistical measurements such as root mean squared error (RMSE), relative absolute bias (RAB), and lower and upper limit of 95% confidence intervals for different samples of different sizes.
2
n
a
k
A* t
i
1
Table 1: Simulation findings for a = 1.4, A = 0.5, ß = 1.1 ands = 4
Sample Failed Estimates Mean RMSE RAB Lower Upper
size(n) items(rk) limit limit
a 1.3478 0.0721 0.0373 1.2065 1.4891
25 20 A 0.4010 0.0523 0.1980 0.2985 0.5035
ß 1.0501 0.1230 0.0454 0.8090 1.2912
a 1.3731 0.0515 0.0192 1.2722 1.4740
50 40 A 0.3044 0.0415 0.3912 0.2231 0.3857
ß 1.0623 0.0871 0.0343 0.8916 1.2330
a 1.3836 0.0409 0.0117 1.3034 1.4638
80 64 A 0.7133 0.0328 0.4266 0.6490 0.7776
ß 1.0603 0.0688 0.0361 0.9255 1.1951
a 1.3926 0.0335 0.0053 1.3269 1.4583
120 96 A 0.5040 0.0456 0.0080 0.4146 0.5934
ß 1.0847 0.0561 0.0139 0.9747 1.1946
a 1.3963 0.0300 0.0026 1.3375 1.4551
150 120 A 0.4659 0.0431 0.0688 0.3814 0.5504
ß 1.0847 0.0501 0.0139 0.9865 1.1829
Table 2: Simulation findings for a = 1.4, A = 0.5, ß = 1.1 ands = 6
Sample Failed Estimates Mean RMSE RAB Lower Upper
size(n) items(rk) limit limit
a 1.3218 0.0508 0.0558 1.2222 1.4214
25 20 A 0.8050 0.0714 0.6100 0.6651 0.9449
ß 1.1030 0.1230 0.0027 0.8619 1.3441
a 1.3429 0.0363 0.0407 1.2718 1.4140
50 50 A 0.8024 0.0513 0.6048 0.7019 0.9029
ß 1.1260 0.0710 0.0236 0.9868 1.2652
a 1.3571 0.0288 0.0306 1.3006 1.4135
80 64 A 0.3097 0.0363 0.3806 0.2386 0.3808
ß 1.0979 0.0561 0.0019 0.9879 1.2079
a 1.3648 0.0236 0.0251 1.3185 1.4110
120 96 A 0.1058 0.0213 0.7884 0.0641 0.1475
ß 1.0921 0.0459 0.0072 1.0021 1.1821
a 1.3683 0.0212 0.0226 1.3267 1.4098
150 120 A 0.0504 0.0113 0.8992 0.0283 0.0725
ß 1.0900 0.0411 0.0091 1.0094 1.1706
Table 3: Simulation findings for a = 1.4, A = 0.5, ß = 1.1 ands — 8
Sample Failed Estimates Mean RMSE RAB Lower Upper
size(n) items(rk) limit limit
a 1.3812 0.0454 0.0134 1.2922 1.4702
25 20 A 0.2087 0.0583 0.5826 0.0944 0.3229
ß 1.1030 0.0977 0.0027 0.9115 1.2945
a 1.3690 0.0285 0.0221 1.3131 1.4249
50 40 A 0.8024 0.0367 0.6048 0.7305 0.8743
ß 1.1505 0.0614 0.0459 1.0301 1.2708
a 1.3692 0.0226 0.0220 1.3249 1.4135
80 64 A 0.7041 0.0513 0.4082 0.6035 0.8046
ß 1.1447 0.0484 0.0406 1.0498 1.2396
a 1.3742 0.0184 0.0184 1.3381 1.4103
120 96 A 0.7084 0.0273 0.4168 0.6549 0.7619
ß 1.1943 0.0395 0.0857 1.1169 1.2717
a 1.3767 0.0165 0.0166 1.3443 1.4090
150 120 A 0.6099 0.0127 0.2198 0.5850 0.6348
ß 1.1681 0.0355 0.0619 1.0985 1.2376
In this study, various measures such as average mean values, RMSE and RAB are calculated using 5000 replications of different samples to avoid randomness. The results presented in Table 1-3 are based on different sample sizes with parameter values a — 1.4, A — 0.5, J — 1.1 and stresses s — 4, 6 and 8 to analyse the performance of the MLEs of the Frechet parameters. Table 1-3 shows that, nearly all of the parameter estimates in Table 3 result in lesser RMSEs and RABs compared with the estimates in Table 1-2. In every situation, the RMSEs of the MLEs of the parameters in Table 1-3 decrease as sample size increases.
IV. Conclusion
In the current study, a Frechet failure item accelerated life testing (ALT) design with time-censored data has been taken into consideration. The likelihood equation for the Frechet parameter is built using a geometric process, which is produced by the failure time of tested objects under constantly increasing stress leveles. Since the likelihood equation does not have the closed-form, the Newton-Raphson technique is used to calculate the mean, root mean square error (RMSE), and relative absolute bias (RAB) for the parameters. The results provided in Table 1-3 show that the estimates are reasonably near to their true values with low RMSEs. A larger sample number results in lower RMSE values and a narrower confidence interval. This work can be extended for various censoring schemes such progressive censoring.
References
[1] Kundu, D. and Gupta, R.D., (1999). Generalized exponential distribution. Aust NZJ Stat, 41, pp.173-188.
[2] Ahmad, K.E., Fakhry, M.E. and Jaheen, Z.F., (1997). Empirical Bayes estimation of P (Y< X) and characterizations of Burr-type X model. Journal of Statistical Planning and Inference, 64(2), pp.297-308.
Rahman, A. and Lone, S.A., (2019). Mathematical Model of Accelerated Life Testing Plan Using Geometric Process. Bayesian Analysis and Reliability Estimation of Generalized Probability Distributions, p.30.
Ahmad, N., (2010). Designing accelerated life tests for generalised exponential distribution with log-linear model. International Journal of Reliability and Safety, 4(2-3), pp.238-264. Yang, G.B., (1994). Optimum constant-stress accelerated life-test plans. IEEE transactions on reliability, 43(4), pp.575-581.
Lin, Y.L.Y., (1988). Geometric processes and replacement problem. Acta Mathematicae
Applicatae Sinica, 4, pp.366-377.
Lone, S.A., Alam, I. and Rahman, A., (2022). Statistical analysis under geometric process in accelerated life testing plans for generalized exponential distribution. Annals of Data Science, pp.1-13.
Lone, S.A. and Rahman, A., (2016). Arif-Ul-Islam, Estimation in Step-Stress Partially Accelerated Life Tests for the Mukherjee-Islam Distribution Using Time Constrain. International Journal of Modern Mathematical Sciences, 14(3), pp.227-238.
Rahman, A., Sindhu, T.N., Lone, S.A. and Kamal, M., (2020). Statistical inference for Burr Type X distribution using geometric process in accelerated life testing design for time censored data. Pakistan journal of statistics and operation research, pp.577-586. Mohamed, A.E.R., Abu-Youssef, S.E., Ali, N.S. and Abd El-Raheem, A.M., (2018). Inference on constant-stress accelerated life testing based on geometric process for extension of the exponential distribution under type-II progressive censoring. Pakistan Journal of Statistics and Operation Research, pp.233-251.
Watkins, A.J. and John, A.M., (2008). On constant stress accelerated life tests terminated by Type II censoring at one of the stress levels. Journal of statistical Planning and Inference, 138(3), pp.768-786.
Zhou, K., Shi, Y.M. and Sun, T.Y., (2012). Reliability analysis for accelerated life-test with
progressive hybrid censored data using geometric process.
Huang, S., (2011). Statistical inference in accelerated life testing with geometric process model (Doctoral dissertation, Sciences).
Fan, T.H. and Yu, C.H., (2013). Statistical inference on constant stress accelerated life tests under generalized gamma lifetime distributions. Quality and Reliability Engineering International, 29(5), pp.631-638.
Chen, J., Li, K.H. and Lam, Y., (2010). Bayesian computation for geometric process in maintenance problems. Mathematics and computers in simulation, 81(4), pp.771-781. Lone, S.A. and Ahmed, A., (2021). Design and Analysis of Accelerated Life Testing and its Application Under Rebate Warranty: Accelerated Life Testing. Sankhya A, 83, pp.393-407. Kamal, M., Khan, S., Rahman, A., Aldallal, R.A., Abd El-Raouf, M.M., Muse, A.H. and Rabie, A., (2022). Reliability Analysis of Hybrid System Using Geometric Process in Multiple Level of Constant Stress Accelerated Life Test through Simulation Study for Type-II Progressive Censored Masked Data. Mathematical Problems in Engineering, 2022. Lone, S.A. and Rahman, A., (2017). Step Stress Partially Accelerated Life Testing Plan For Competing Risk Using Adaptive Type-I Progressive Hybrid Censoring. Pakistan Journal of Statistics, 33(4).
Kamal, M., (2013). Design Of Accelerated Life Testing Using Geometric Process For Pareto Distribution With Type-I Censoring. Journal of Global Research in Mathematical Archives (JGRMA), 1(8), pp.59-66.
Zarrin, S., (2013). Designs Of Accelerated Life Tests (Doctoral Dissertation, Aligarh Muslim University Aligarh).
Kamal, M., Rahman, A., Zarrin, S. and Kausar, H., (2021). Statistical inference under step stress partially accelerated life testing for adaptive type-II progressive hybrid censored data. Journal of Reliability and Statistical Studies, pp.585-614.
[22] Lone, S.A., Panahi, H. and Shah, I., (2021). Bayesian prediction interval for a constant-stress partially accelerated life test model under censored data. Journal of Taibah University for Science, 15(1), pp.1178-1187.
[23] Lone, S.A., Rahman, A. and Tarray, T.A., (2021). Inference for step-stress partially accelerated life test model with an adaptive type-I progressively hybrid censored data. Journal of Modern Applied Statistical Methods, 19(1), p.13.
[24] Ismail, A.A., (2014). Inference for a step-stress partially accelerated life test model with an adaptive Type-II progressively hybrid censored data from Weibull distribution. Journal of Computational and Applied Mathematics, 260, pp.533-542.
[25] Lone, S.A. and Rahman, A., (2019). Designing Accelerated Life Testing for Product Reliability Under Warranty Prospective. Bayesian Analysis and Reliability Estimation of Generalized Probability Distributions, p.68.
[26] Aly, H.M., Bleed, S.O. and Muhammed, H.Z., (2020). Inference and optimal design of accelerated life test using geometric process for generalized half-logistic distribution under progressive type-II censoring. Journal of Data Science, pp.361-379.
[27] Alam, I., Ahmad, H.H., Ahmed, A. and Ali, I., (2022). Inference on adaptive progressively hybrid censoring schemes under partially accelerated life test for OLiHL distribution. Quality and Reliability Engineering International.
[28] Sindhu, T.N., Aslam, M. and Hussain, Z., (2016). A simulation study of parameters for the censored shifted Gompertz mixture distribution: A Bayesian approach. Journal of statistics and management systems, 19(3), pp.423-450.
[29] Nassr, S.G., Almetwally, E.M. and El Azm, W.S.A., (2021). Statistical inference for the extended weibull distribution based on adaptive type-II progressive hybrid censored competing risks data. Thailand Statistician, 19(3), pp.547-564.
[30] Hemmati, F. and Khorram, E., (2013). Statistical analysis of the log-normal distribution under type-II progressive hybrid censoring schemes. Communications in Statistics-simulation and Computation, 42(1), pp.52-75.
[31] Hussam, E., Alharbi, R., Almetwally, E.M., Alruwaili, B., Gemeay, A.M. and Riad, F.H., (2022). Single and multiple ramp progressive stress with binomial removal: practical application for industry. Mathematical Problems in Engineering, 2022.
