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Akhila K Varghese, V. M. Chacko RT&A, No 3 (69) ESTIMATION OF STRESS-STRENGTH RELIABILITY_Volume 17, September 2022
ESTIMATION OF STRESS-STRENGTH RELIABILITY FOR AKASH DISTRIBUTION
Akhila K Varghese, V. M. Chacko
Department of Statistics St. Thomas' College (Autonomous) Thrissur, Kerala, 680 001, India [email protected] [email protected]
Abstract
In this paper, we consider the estimation of the stress-strength parameter R = P[Y < X], when X and Y are following one-parameter Akash distributions with parameter 61 and d2 respectively. It is assumed that they are independently distributed. The maximum likelihood estimator (MLE) of R and its asymptotic distribution are obtained. Asymptotic distributions of the maximum likelihood estimator is useful for constructing confidence interval of P[Y < X]. The Bootstrap confidence interval of P[Y < X] is also computed. The illustrative part consists of the analysis of two real data sets, (i) simulated and (ii) real.
Keywords: stress-strength model; maximum-likelihood estimator; bootstrap confidence intervals; asymptotic distributions
I. Introduction
In reliability analysis, estimation of stress-strength reliability is one of the important and difficult but tractable problem, while using various distributions. In the statistical literature, estimating the stress-strength parameter, R, is quite useful. For example, if X is a measure of a system's strength when it is subjected to a stress Y, then R is a measure of system performance that naturally occurs in a system's mechanical dependability. The system fails if and only if the applied stress exceeds its strength at any point. In reliability analysis, a variety of lifespan distributions are used. In dependability analysis, terms like exponential, Weibull, log-Normal, and their generalizations are frequently used. A number of academics have recently proposed several distributions, with the new ones demonstrating a superior fit than current well-known distributions. While using better fitted models in stress-strength analysis, one may have to inspect its estimation procedure, since, if the estimation procedure fails with available techniques, one may not be able to solve the problem with new models. So estimation of various reliability parameters is vital and researchers have to give more concentration of estimation while using better fitted models.
The estimation of reliability or survival probability of a stress-strength model when X and Y have specified distributions has been discussed in literature. The survival probabilities of a single component stress-strength (SSS) model have been considered by several authors for different
Akhila K Varghese, V. M. Chacko RT&A, No 3 (69) ESTIMATION OF STRESS-STRENGTH RELIABILITY_Volume 17, September 2022
distributions, see Raqab and Kundu [13], Kundu and Gupta [9,10], Constantine and Karson [6]
and Downtown [7]. Several authors have studied the problem of estimating R. Church and Harris
[5] derived the MLE of R when X and Y are independently Normally distributed. The MLE of R,
when X and Y have bivariate exponential distributions has been considered by Awad et. al. [2].
Awad and Gharraf [3] provided a simulation study to compare three estimates of R when X and Y
are independent but not identically distributed Burr random variables. Ahmad et. al. [1} and Surles
and Padgett [15,14] provided estimates for R when X and Y are having Burr Type X distribution.
In this paper, we consider the problem of estimating the stress-strength reliability parameter R = P(Y < X), when X and Y be independent strength and stress random variables having Akash distribution with parameters 01 and 62 respectively. Rama Shanker [12] introduced Akash distribution by considering a two-component mixture of an Exponential distribution having scale parameter 0 and a Gamma distribution having shape parameter 3 and scale parameter 0. The probability density function (pdf) of Akash distribution can be defined as
#3
f(x; 9) = (1 + x2)e-#x; x>0,9>0.
The corresponding cumulative distribution function (cdf) is given by
6x(6x + 2)
F(x) = 1-
1+
92 + 2
e-#x;x> 0,9 > 0.
The estimation of the stress-strength parameter R = P[Y < X], when X and Y are having one-parameter Akash distributions with parameter 01 and 62 respectively, is an unsolved problem. Statistical inference on stress-strength parameters is important in reliability analysis. It is observed that the maximum likelihood estimators can be obtained implicitly by solving two nonlinear equations, but they cannot be obtained in closed form. So, MLE's of parameters are derived numerically. It is not possible to compute the exact distributions of the maximum likelihood estimators, and we used the asymptotic distribution and we constructed approximate confidence intervals of the unknown parameters.
The rest of the paper is organized as follows. In Section 2, the MLE of R is computed. The asymptotic distribution of the MLEs are provided in Section 3. Bootstrap confidence interval is presented in Section 4. In Section 5, simulation study is given. Theoretical results are verified by analyzing one data set in Section 6 and conclusions are given in Section 7.
II. Maximum Likelihood Estimator of R
In this section, the procedure of estimating the reliability of P[Y<X] models using Akash distributions, is considered. It is clear that
+
R = P(Y<X) = jx<) f(x,y)dxdy = f* f(x;91)F(x;e2)dx
where f (x,y), is the joint pdf of random variables X and Y, having Akash distributions. If the r.v's X and Y are independent, then f (x, y) = f (x) g(y), where f (x) and g(y) are the marginal pdfs of X and Y, so that
JOT 0
9"
+2
(1+x 2)e-##x1 -
1 + 92x(92x + 2)
9"2 + 2
,-67x
dx.
9?
On simplification we get.
A _ 1 e^[ef6+4e1e2+6e1^e^+4e'^e'2++22e1e'2+e4ie'^+22e'1e'^+4e'1+2e1+20e1e2+i0e'2e2+40e'2+8e^]
~ ' (e2+2)(2+e2)(ei+e2)5
If we have two ordered random samples representing strength (X1,X2,...,Xn) and stress (Y1, Y2,...,Ym) of sizes n and m respectively, following Akash distribution with parameters 01 and 62, respectively. Then the likelihood function of the combined random sample can be obtained as follows
93 93
L = nr=1 efPl(1 + Xl2)e"#lXi nr=1 W^ (1 + y'2)e"e2y&■
The log-likelihood function is
I = logL = 3n log Q1 - (9" + 2) - e1Jirn=ixi + £"=1 log (1 + xi 2) + 3m log 02 -
mlog (022 + 2)- ^ZE^i + Y6=i^og @1 + yj).-----------------(1)
The solution of the following non-linear equations yield the MLE of the parameters 01 and 02. Differentiating (1) with respect to parameters 01 and 02, we get
dl 3n n291 n M¡ = ~9¡ - (9f + 2) - S1=1Xi
and
dl 3m
d9¡ = ~9~' (92 + 2) The second partial derivatives of (1) with respect to parameters d1 and 02, are
d2l -3 n 2n(2-9f)
2 _i_ .
and
д92 el (в1 + 2)2 d2l -3m 2m(2 - 922)
d9l 91 (91 + 2)2
MLE of R is obtained as
n _ i g3[026+4ê1ê2+6ê?êl+4ê?^2++22ê1ê3+^lê2+22ê2ê2+4ê?+2êl+2oê1ê2+loê3ê2+4oê2+8êl]
ti - 1----^-r-
(Эl+2)(2 + 02)(б?l+02)
This can be used in estimation of stress-strength for the given data.
III. Asymptotic Distribution and Confidence Intervals
In this section, the asymptotic distribution and confidence interval of the MLE of R are obtained. To find an asymptotic variance of the MLE R , let us denote the Fisher information
Akhila K Varghese, V. M. Chacko RT&A, No 3 (69) ESTIMATION OF STRESS-STRENGTH RELIABILITY_Volume 17, September 2022
matrix of 9 = (6V 82 ) as 1(8) = [li7 (8); i,j = 1,2], i.e.,
1(8) = E
—d2l -d2l 381" dd1dd2 -d2l -d2l
ddndd-, 362
To establish the asymptotic Normality, we define
fdR dft Y -a*)
where
as
03
(40| + 120+0/ + 120,00 + 2200 + 4000, + 440+0,
(0+ + 2)(00 + 2)(0+ + 0,)1 + 300, + 200, + 800 + 80+)
(0, + 2)(0f + 2)(0+ + 02)|30, - 00(0, + 2)[(0, + 2)5(0+ + 02)4 + 20+]
+
(0, + 2^ (00 + 2),(01 + 0,)+1
(0| + 40+0
+ 60, + 0/ + 80/ + 4000, + 220+0, + 0/0, + 220,0, + 400, + 10000, + 200+0, + 20/ + 40,
as 50,
00
(60, + 200+0/ + 240,0, + 320, + 12000, + 660+0,
(0, + 2)(0, + 2)(0+ + 0,)1
+ 20/0, + 440, 0, + 800, + 1000 + 200+)
00(0, + 2)[(0, + 2)5(0+ + 0,)/ + (0+ + 0,)20,]
+
(0+ + 2),(0, + 2)2(0l + 02)1o
(0| + 40+ + 0, + 60,
+ 220, + 80, + 0, + +000, + 220+0, + 0, + 100! 0, + 200+0, + 20/ +40,:
We obtain the asymptotic distribution of R" as
Vn + m(R~ML -ft) N(O,d'(0)/-1 (0)d(0)).
»ML .
The asymptotic variance of fi is obtained as
av@r-ml a =
n + m
d'(6)-1I(6)d(d).
i.e.,AV(R~ML) = V(8\)d! + V(8\)d2 + 2d1d2Xd\
Asymptotic 100(1 — y)% confidence interval for R can be obtained as
rML ± z'jav@r"ml a .
IV. Bootstrap Confidence Intervals
In this section, we use confidence intervals based on the parametric percentile bootstrap methods (we call it from now on as Boot-p), Kundu et. al. [11]. Bootstrapping is a statistical approach that resamples a single dataset in order to build up a huge proportion of simulated samples. To estimate confidence intervals of R in this methods, the following steps are used.
1.Estimate 9 and ,say 9, from the sample using maximum likelihood estimate method
2. Generate a bootstrap sample (x1:n,x2:n,, x2:n,,... xn:n) using 9, where x3:n represents the ith observation when there are n observations in the experiment. Obtain the bootstrap estimate of 9, say 9 * using the bootstrap sample.
3. Repeat Step [2] NBOOT times.
4. Let CD~F(x) = P(§" < x), be the cumulative distribution function of A"*. Define f"Boot-p (x) = CDF-1(x) for a given x. The approximate 100(1 — a)% confidence interval for 6 is given by
f Boot-p , @ Boot-p Xl — 2)j
V. SIMULATION STUDY
In this section, we present some results based on inversion method to assess the performance of estimators of R. For this purpose, we have generated 1000 samples from independent Akash (01) and Akash (02) distributions. We considered sets of parameter values 1.25 and 1.75 which correspond to the R values 0.6491261. The bias and the mean square error (MSE) of the parameter estimates are calculated. In Tables 1, Maximum likelihood estimate of R (R(ML)), the average biase, MSE, asymptotic confidence Intervals (AS(CI)) and Bootstrap confidence interval (BT(CI)) corresponding to different (n,m) values are calculated by the method explained in section 3.
Table 1
(n,m) R(ML) BIAS MSE AS(Cl) BT(CI)
(7,7) 0.712703 0.06357 0.007773 0.5929103,0-8324957 0.6775518,0.8844710
(15,15) 0.714086 0.06495 0.006078 0.6295406,0.7986314 0.6777669,0.8394092
(30, 30) 0.713589 0.06446 0.005118 0.652743,0.7744354 0.64231680.7997702
(30,35) 0.713068 0.06399 0.004900 0.6571613,0.7689664 0.5521229,0.7371387
(40,40) 0.713401 0.06426 0.004835 0.6614061,0.7654101 0.6363909,0.7394242
(50, 50) 0.713477 0.06341 0.004699 0.667147,0.7598043 0.6572927,0.7705276
From the simulation results, it is observed that as the sample size (n,m) increases the biases and the MSEs decrease. Thus the consistency properties of all the methods are verified.
VI. DATA ANALYSIS
In this section, we consider two real data sets of the breaking strengths of jute fiber at two different gauge lengths (see Xia et. al. [16]). Two sets of real data are shown as follows.
Data set I: Breaking strength of jute fiber length 10 mm (variable X) 693.73, 704.66, 323.83, 778.17, 123.06, 637.66, 383.43, 151.48, 108.94, 50.16, 671.49, 183.16, 257.44, 727.23, 291.27, 101.15, 376.42, 163.40, 141.38, 700.74, 262.90, 353.24, 422.11, 43.93, 590.48, 212.13, 303.90, 506.60, 530.55, 177.25.
Data set II: Breaking strength of jute fiber length 20 mm (variable Y) 71.46, 419.02, 284.64, 585.57, 456.60, 113.85, 187.85, 688.16, 662.66, 45.58, 578.62, 756.70, 594.29, 166.49, 99.72, 707.36, 765.14, 187.13, 145.96, 350.70, 547.44, 116.99, 375.81, 581.60, 119.86, 48.01, 200.16, 36.75, 244.53, 83.55
These data were first used by Xia et al. [16] and later by Saracoglu et. al. [4]. Shahsanaei and Daneshkhah [8] used the data to study the estimation of stress-strength parameter for generalized linear failure rate distribution (GLFRD) under progressive type-II censoring and studied the validity of GLFRD for both data sets.
The table 2 gives the result of goodness of fit test. The test used to check whether the considered distribution is a good fit to the data.
Table 2
PLANE MLEs K-S Statistic P-value
length 10 mm(X) 0.00831404 0.13641 0.5847
length 20 mm(F) 0.00880360 0.20925 0.1248
MLEs of parameters of Akash (01) and Akash (02) distributions are 0.00831404 and 0.00880360. Reliability P(Y<X) value for the data is 0.526798. The 95% asymptotic interval of R is (0.36446, 0.7335) and 95% bootstrap confidence interval is (0.376799,0.676798).
VII. CONCLUSION
In this paper, we considered the problem of estimation of P(Y<X) using Akash distribution. The MLE of SSS reliability, R is obtained. Also, asymptotic 100(1 — v)% CI for the reliability parameter is computed. Bootstrap confidence interval is also obtained. When the sample size is increased, MSE caused by the estimates comes nearer to zero by extensive simulation. Finally, real data sets are analyzed.
Acknowledgement
The authors are thankful for the comments of referees and editors which helped to improve the paper.
References
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Akhila K Varghese, V. M. Chacko RT&A, No 3 (69)
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