MUlTI-COMPONENT CONDITIONAL STRESS-STRENGTH PARAMETER Текст научной статьи по специальности «Математика»

RT&A, No 2 (73) Volume 18, June 2023

MUlTI-COMPONENT CONDITIONAL STRESS-STRENGTH PARAMETER

Kavoos Khorshidian1, Morteza Taheri Saif Abad 2

1

Department of Statistics, Faculty of Science, Shiraz University, Shiraz, Iran

khorshidian@shirazu.ac.ir Department of Statistics, Faculty of Science, Shiraz University, Shiraz, Iran taherisaifmorteza@gmail.com

2

Abstract

There are situations in which the experimenter has some information about the components of the operating system and he/she wants to use this information for better assessment or operating of the underlying system. In such cases the notion of conditional probability may help the operator to use that information and improve his/her task. In the present study this notion has been examined, and some conditional stress-strength parameters have been introduced for s of k systems. The multi-component conditional stress-strength parameter (MCCSSP) and its maximum likelihood estimator have been calculated when the strength and stress random variables are exponentially distributed. In the case of having extra information about the parameters, a closed form has been derived for the Bayes estimator of MCCSSP and has been calculated by using an algorithm together with Monte Carlo method. For the case of non-exponential stress or strengths, the nonparametric estimator of the defined parameter has also been derived. Finally, some simulation study on the MLE and Bayes estimator, as well as real data analysis for nonparametric estimators have been done to verify the analytic results.

Keywords: Conditional Reliability, Exponential Distribution, Maximum Likelihood Estimator,

Multi-Component Systems, Stress-Strength Parameter

The effects of resistance and shocks which enter to a system are usually studied via a stress-strength model. The term stress-strength was first introduced by [1]. Since then the stress-strength models have been inspected by many researchers due to their applicability in different fields, such as engineering, economics, psychology, medicine and so on. In such models, when the stress that experienced by the system have been represented by a random variable (RV) X and the strength of system by a RV Y, the stress-strength parameter is denoted by R = P(X > Y), it measures the chance that the system fails. It should be mentioned that 1 — R is the chance that the considered system operates well and is known as the reliability function or parameter of the system. For the majority of the well-known distributions, including Normal, Exponential, Pareto, Uniform, Weibull, Gamma, Beta, logistic, and Laplace, R has been studied by [2]. Some of the recent studies about R can be seen in [3], [4], [5], [6] and [7]

There are situations that one have some information about the stress and strength RV's and knows that they are greater than some pre-specified values, or one wants to know how much a system can be reliable when stress and strength increase or decrease. Considering conditions like these, the conditional stress-strength parameter was introduced by [8] as:

1. Introduction

R|a,b = P(X > Y | X > a, Y > b).

(1)

Nowadays in the real life and industries most of the operating systems have become complex with more than one active component, i.e., a lot of working systems are multi-component rather than simple and uni-component. The reliability of a multi-component stress-strength model was first developed by [9]. Afterwards, applications and studies on different characteristics of multi-component stress-strength models grow up rapidly. Some of the recent studies can be seen in [10],[11], [12], [13], [14] and [15].

By developments in most technologies, in many situations there are a lot of information about the working mechanisms which will be precise and helpful, if they have been employed corrected, e.g. in the case of second hand and used devices. For example, consider a large drilling machine in a mine. This machine uses several gears or drills simultaneously for drilling, which are the most important parts of this machine and are often iteratively replaced by another one. Therefore, a lot of information about the amount of stress and strength experienced by this part of the machine can be collected . In this article, we have focussed on the notion of conditional stress-strength parameter to extend, generalize and employ such information in multi-component systems. In order to prepare a complete pack about MCCSSP, it has been calculated and estimated by using different methods for employing it in different real situations of practice. For exponential distribution as the first and most exploited candidate of the lifetimes of components in operating systems, the MCCSSP has been calculated, its MLE has been estimated through samples and its asymptotic behaviors has been studied, as well. For the circumstances that we have extra information about the varying structure of exponentially distributed stress and strengths random variables, the Bayes estimators of MCCSSP has been also derived based on the information included in samples of stress and strength. For the case of non-exponential or unknown life time distributions the non-parametric estimators have been also derived.

The structure of this article is as follows: A general formula for computing MCCSSP will have been provided in Section 2. In Section 3, the MCCSSP has been computed in the case of exponential distributions as well as its maximum likelihood estimator and asymptotic distribution of the later. The Bayes estimator of this parameter has been obtained in Section 4, by adopting an algorithm and using the Monte Carlo method. The corresponding nonparametric estimator of this parameter has been obtained in Section 5. Section 6 is devoted to the presentation of some simulation studies on the MLE, Bayesian and nonparametric estimators and their comparison. Some numerical results for a real data-set have been presented in Section 7. Finally in Section 8, some concluding remarks have been given.

2. The MCCSSP

In this section, the MCCSSP will have been introduced and a general formula have been presented to compute it.

Definition 1. Consider the independent RV's X1,..., Xk with common continuous distribution function F(-), independent of continuous RV Y with distribution function G(-). The MCCSSP is defined as:

R? = p(at least s of X1,..., Xk exceed Y | X1 > a,..., Xk > a, Y > b). (2)

The particular cases s = 1 and s = k correspond to parallel and series systems, respectively. Note that a special case of this quantity for a = b = — to is

k /k\ rto

Rs,k = P( at least s of Xx.....Xfc exceed Y) = £ ^ J ^(1 — F(y))(F(y))k—*dG(y) (3)

which is introduced by [9] as the multi-component stress-strength parameter. Suppose that there a lot of information about one of the stress RV's Xz, some specified z, 1 < z < k,. For example, in some systems, one of the parts wears out more and is replaced more often, such as drilling machines, where the drill bit is very important and is replaced a lot, and

the other parts are replaced less often. Therefore, there are more information about the lifetime of a specified part than the other parts. For this case R^g(^,a,b as the MCCSSP when Xz > a is defined as:

Definition 2.

R\(z),a,b = p(at least s of X1,...,Xk exceed Y | Xz > a,Y > b)

(4)

Note that (3) is again a special case of (4). A formula for computing (2) has been presented in the following theorem.

Theorem 1. If R^bk is defined by (2), then

R

|a,b s,k

Tk=s (k ).!T [1-F(y)]i[F(y)-F(b)]k-idG(y)

[1-F(a)]k [1-G(b)] Lk=s (k)(J7 [G(x)-G(b)]dF(x))i(J™[1-G(x)]dF(x))k

[1-F(a)]k [1-G(b)]

a < b a > b

(5)

Proof. First, we write (2) as follows:

|a,b _ P(at least s of X1,..., Xk exceed Y, X1 > a,..., Xk > a, Y > b)

R

s,k

P(X1 > a,...,Xk > a,Y > b)

Since Xi,..., Xk and Y are independent, the dominator is (1 - F(a))k(1 - G(b)). To compute the numerator, first we write it as follows:

P(at least s of Xi exceed Y, X1 > a,..., Xk > a, Y > b)

P((X1.....Xk, Y) G A)

/ ... dF(x1 )...dF(xk)dG(y),

where A = {(x1,...,xk,y) | at least s of x1,...,xk exceed y, x1 > a,...,xk > a,y > b}. To compute this integral, partition A into two regions A1 and A2 for the cases a < b and a > b, where:

Aj = {(xj,..., xk, y) | at least s of xj,..., xk exceed y, xj > a,..., xk > a, y > b, a < b}

= {(x\,..., xk,y) | at least s of x1,..., xk exceed y, a < x\ < b,..., a < xk < b,y > b,a < b} |^J{(x1,..., xk,y) | at least s of x1,..., xk exceed y, x1 > b,...,xk > b,y > b,a < b}

= B1 U B2,

and

A2 = {(x1,..., xk, y) | at least s of x1,..., xk exceed y, y > b, x\ > a,..., xk > a, a > b}

= {(x\,..., xk,y) | at least s of (b, ),..., (b, xk) contain y,y > b,> a,...,xk > a, a > b}.

where

B1 = {(x1,...,xk,y) | at least s of x1,...,xk exceed y, a < x1 < b,...,a < xk < b,y > b,a < b}, B2 = {(x1,...,xk,y) | at least s of x1,...,xk exceed y, x1 > b,...,xk > b,y > b,a < b}.

Let

and

then

R1

R2

Mi

dF(x1 )...dF(xk )dG(y),

dG(y)dF(x-i )...dF(xk )

(6) (7)

R1

/ dF(x1)...dF(xk)dG(y) + dF(x1)...dF(xk)dG(y)

'B\ JB2

JB2 k fk

dF(x1)...dF(xk)dG(y)

[1 - F(y)]i[F(y) - F(b)]k-idG(y)

A

2

CO

l=s

The first integral becomes zero because P(X* > Y, a < X, < b, Y > b) = 0 for i = 1,...,k, and R2 = P(at least s of (b,X1),...,(b,Xk) contain Y, Y > b,X1 > a,...,Xk > a,a > b)

g (J) [/J[G(x) - G(b)]dF(x)]'[ f [1 - G(x)]dF(x)]k-i,

This completes the proof. ■

Remark 1. Consider an s of k multi-component system, which their strengths are denoted by iid RV's X1, X2,..., Xj with common continuous distribution function F( ). Also suppose that each component experiences a random stress Y with continuous distribution function G( ), independent of the strengths. Note that the system stays alive only if at least s of k strengths be greater than the stress. Then the conditional reliability of the multi-component system has the following form:

= P(at least s of X1,..., Xk exceed Y | X1 > a,..., Xk > a, Y > b). (8)

In this model, the conditional reliability of the system is represented by (5).

Remark 2. In practice the information in hand and given condition may not have exactly the form {x1 > a,...,Xk > a, Y > b}, but be as {X1 e A1,...,Xk e Ak, Y e B} where A1,..., Ak and B are linear Borel sets on (0, j). In this case, by applying some procedure similar to the approach of Theorem 1, one can compute this generalized MCCSSP. Based on the structure of A1,..., Aj and B, it is expected that the analytic derivations may be complicated. In this situation and more general case some non-parametric method similar to that given in section 5 as well as Monte Carlo simulation may be applied.

Remark 3. By formula (5), one may show that for the case a < b, the MCCSSP is an increasing function of a, which is expected trivially. Note that in this case:

= kf (a)(1 - F(a))k-1 Ek=s (k) r[1 - F(y)]i[F(y) - F(b)]k-idG(y) da [1 - F(a)]2k[1 - G(b)]

= < ffo > o.

According to the calculations resulting in the formula (5), it can be seen that if X1,..., Xj have different distributions, it is not easy to calculate the analogous of this formula. In what follows, the formula (5) has been calculated when X1,..., Xj and Y have the same distributions.

Corollary 1. Suppose that the continuous RV's X1,..., Xj and Y are independent and identically distributed with probability density function(pdf) f (.) and cumulative distribution function(cdf) F(.). Then,

f gk=s (k) jf(b) [1-y]'[y-F(b)]k~'dy b

R|a,b = J [1-F(a)]k[1-F(b)] , a < b (9)

S,J | (1 )kEj=s (k)[1-2F(b)+F(a)]i[1-F(a)]k-i > b W

2 ) [1-F(b)] , a > b.

Remark 4. Put rI^1^ = P(at least s of Xi,...,Xk exceed Y, Xz > Y | Xz > a, Y > b) and

Rikz)2,a,b = P(at least s of X1,...,Xk exceed Y, Xz < Y | Xz > a, Y > b) for some z, 1 < z < k. According to the approach of the proof for Theorem 1, after some computation, we have:

f E?=s (J) ¡T [1-F(y)]'F(y)J-'dG(y) b

R|(z)1,a,b I [1-F(«)][1-G(b)] . 1 k . a < b ,10.

s'k I gJ=s (.-¿-,)[¡T[G(x)-G(b)]dF(x)] * [r[G(*)-G(b)]riF(x)] [¡J [1-G(x)]dF(x)] * > V '

I [1-F(a)] [1-G(b)] a > b,

' (,.j-J,-1) ¡T[1-F(y)]i [F(y)-F(b)]F(y)J-(i+1)dG(y) b

R|(z)2,a,b I [1-F(a)][1-G(b)] a < b f11.

S'J I g-s1 (,,J-Ji-1)[XT[G(x)-G(b)]dF(x)]* [¡J[1-G(x)]dF(x)] [¡o°°[1-G(*)]dF(*)p'+1) ^ >

[1-F(a)] [1-G(b)] a > b.

Therefore

R

(z),a,b _ s,k _

R

\(z)i,a,b

Rl§2'a'b Xz < Y.

Xz>Y

(12)

3. Estimation for Exponential Distribution

In this section, the measure (5) has been evaluated for the Exponentially distributed stresses and strength RV's with different parameters. The probability density and cumulative distribution functions of a random variable X ~ E(a) are denoted by: f (x) = ae—ax, and F(x) = 1 — e—ax where x > 0, a > 0. Suppose that Xj ~ E(A1) for i = 1,..., k and Y ~ E(A2) are independent, we have:

k k-i

Ri _ À2e~b(Àik+*2) y E fk) ,.(_i\ , 1 2 iE^VI,l) Ài(l + j) + À2'

and

R2 _ e

_ e-ak(Ài +À2)

Ài

Ài + À 2

kk E

Ài + À2 e-À2 (b-a) _ i Ài

by dividing the above equations by [i _ F(a)]k[i _ G(b)] _ e_(akÀi +bÀ2) we have:

* I À2e_Àik(b_a) yk yk_i ( k ) (-i) a,b J À2e U_s Vl_0 (i,l) Ài (l+j)+À2

>,k 1 .-Ào (ak-b)\ Ài iksr-k fk\ rÀi +À2„-\

R

a<b

e_À2(ak_b)[àÀt2]" Vk_s (k)[^e_À2(b-a) _ i]1 a > b.

(i3)

Remark 5. From (13), we conclude that R^ for a < b in Exponential distribution depends only

on the difference between a and b. In other words , if b1 — a1 = b2 — a2 then R^l^1 a1 < b1 and a2 < b2.

RSaf2 for

Figure i show the effect of changes in the values a and b in (i3). These figures show what happens when the values a and b increase or decrease, in all Figures (s, k) _ (i,3).

(a) b=600, (Ài, À2 ) (0.0049,0.0005)

(b) a _ 0.i, (Ài,À2) _ (c) b _ 0.i, (Ài,À2) _ (d) a _ 2, (Ài,À2) (0.0049,0.0005) (i.4, i.7) (i.4,i.7)

Figure 1: MCCSSP

By assuming the Exponential distributions for stresses and strength, from (10) and (12), after some calculation it follows that for the case Xz > Y, we have:

R

\ (z),a,b s,k

ab

Ek Ek_l ( k ) À2(_i)le_Ài(b(k_l)-a) yl_s yi_0 (l,j) à1 (k_j)+À2

e_À2(a_b)[ÀÀÀVk_s (k)[ÀiÀÀe_À2b _ i]l-i[^e_À2(b-a) _ i] a > b,

and for the case Xz < Y :

R

[fb_ f À2e_Ài(b_a) [Eki E^ (k)(k_l)(_i)le_Ài

\(z),a,b _ s,k

e_Ài (b_a) ,-,

àie+i+i)+à2]] a ^b

e_À2(a_b)[ààà]k[Et; (k)[^ÀÀe_À2b _ i]1

Ài (l+j) + À2

a > b.

(i4)

(i5)

k

l_s

Indeed, as in (13) to (15), the stress-strength parameters and Ri(z),a,b are functions of A1 and A2. Therefore, it is rational that for evaluating the maximum likelihood estimators of MCCSSP, the first step to be calculating the MLE's of A1 and A2.

3.1. Maximum Likelihood Estimation

Suppose that X1,.... Xn and Y1,.... Ym are two independent random samples from E(A1) and E(A2). Then the likelihood function is

L(Ai, A2) = A? Ame-A1 S^2

(16)

and the MLE's of the parameters A1 and A2 are A1 = X and A2 = =, respectively. Therefore, by using the invariance property for MLE's, and substituting A1 and A2 instead of A1 and A2 in (14) and (15), one may write the MLE of (13) by:

R

ÎA2e-Alk(b-a) (j

(-i)j

s,k

A1(i+j)+A2

« < b

e-A2(ak-b) []k Ek=s (k) [Al+ke-A2(b-a) - 1p « > b.

(17)

3.2. Asymptotic Distribution

«,b

In this subsection the asymptotic distribution of R i k will have been obtained by using the asymptotic normality of the MLE's and the multivariate delta method. By the fact that X ^ N2(X,E) as n,m tend to infinity, m ^ d for some 0 < d < œ, where X = (A^ A2)T, X = (A1, A2)t and E is the inverse of Fisher's information matrix I(X), it is easy to see that

I(X)

— 0

A2 0

0 m

0 a2j

and so E

A 0 n

0 ^

m

The well-known delta method enables us to derive the asymptotic behaviour of functions of an estimator, whenever the estimator is itself asymptotically normal. The delta method have been present and applied in different forms, we have used the following presentation.

Proposition 1. Let g(.) be a mapping g(.) : Rd ^ R, such that g(.) is continuous in a neighborhood of p e Rd. If Xn is a sequence of d-dimensional random vectors such that Xn ^ Nd (p, E) in distribution, then g(Xn)-g{p) ^ N(0,1) in distribution, where t2 = VTEV > 0 and V = .

We will apply Proposition 1 to Xn = X and

g(x1 x2 )

r,e-*1 k(b-a) Ek Ek-i ( k )_(

(-1)j

s^;=0 Vi,;Vx1 (¿+j)+X2

« < b

e-x2(ak-b) []k Ek=s (k)[x1+2e-x2(b-a) - 1p « > b.

The asymptotic distribution of R^ may be obtained as below:

(R if - Rif) ^ N (0, VT EV),

(18)

where

V = ( dg(A1. A2) dg(A1. A2) )T

3A1 3A2

VTEV = [dg(A1. A2) ]2 Ai + [dg(A1. A2) ]2A2

[ 3A1 ] n dA2 ] m .

For the cases a < b and a > b, denote (19) by of and of respectively. Put $ = dg(A1'A2) |a<b, V = ^^AA1 la>b, 0 = la<b, k = ^^AA! |a>b, we arrive at: 1

o2 = M + 02 M,

1 n m

(19)

A2 A2

o2 = v2 ^ + K2 A2. (21)

2 n m

Therefore, the asymptotic normalized distribution of Rfor different values of a and b are as follow:

R|a,b _ R|a,b

—^-^ ^ N(0,1) i = 1,2, (22)

°i

where oi and 02 stands for the cases a < b and a > b respectively The above statistics can be used for constructing confidence intervals for R^. By employing a similar approach and performing some steps like the above, using lemma 1 and asymptotic normality of RS(jz),a,b, one may arrive at the asymptotic distribution of Rj(jZ),a,b.

4. Bayes Estimation

In this section, the Bayesian estimation of the reliability parameter (13) has been considered. Suppose that the parameters A1 and A2 are RV's, and have independent Gamma prior distributions with parameters (a, ft), i = 1,2 respectively. The pdf of a random variable X ~ Gamma(ai, ft) is denoted by

Bxi

n(x) = rayx"'-1 x > 0, ai > 0, ft > 0. (23)

The joint posterior density function of the parameters based on this prior density and the likelihood function can be written as follows:

n* (A1, A2 I x, y) = f„ f„ "A' A2' y)dA dA (24)

J0 J0 n(A1, A2, x, y)dA1 UA2

where

n(A1, A2,x,y) = n(A1 )n(A2)L(A1, A2) a Aa11+n_1e_A1(P1+£n=1 xi)Ap^V^2+£f=1y). It is easily seen that the posterior density functions of A1 and A2 are respectively

n

n* (Ai |A2, x, y) «r(«i + n, ft + £ X'), (25)

i=1

m

n*(A2IA1,x,y) «r(«2 + m,+ £ y). (26)

j=1

The Bayes estimator of R^ under the squared error loss (SEL) is obtained as

R^ = E(R^b|x, y) = Jo" Jo" R^n* (A1, A2 | x, y)dA1 dA2. (27)

It is not possible to calculate equation (27) analytically. Therefore, to compute the Bayes estimate of reliability parameter R^, a Monte Carlo (MC) method has been adopted as follows: Step 1: Set l=1. ,

Step 2: Generate X1,..., Xn from Exp(A1). Step 3: Generate Y1,..., Ym from Exp(A2)

Step 4: Generate A1 from Gamma(a1 + n, ft + X'). Step 5: Generate a2 from Gamma(a2 + m,+ £m=1 y).

Step 6: Compute Rs 7b at (A1, A2). Step 7: l=l+1. ,

Step 8: Repeat Steps 2 to 7, M times and obtain the posterior sample Rfc^ for l = 1,•••,M. Now the Bayes estimate of R^ with respect to SEL will be obtained as follows:

I h 1 M n u

Rff = M MR1!;*. (28)

M 1=1

5. NONPARAMETRIC ESTIMATION

In this section a nonparametric method for estimating R^ has been presented. In many situations,

we may have no information about the distribution of data or computing R^ via 1 may require complex computations, or even may not have a definite answer. Therefore, employing the nonparametric method, in which the structure of the model may have been determined from data, can lead us to better results or at least be more applicable. Let n(.) be the counting measure. For the sample space S and the event D as a subset of S the nonparametric estimator of P(D) is defined as P(D) = n^p). To obtain the nonparametric estimator of MCCSSP, one may write (2) in the form:

R| a,b = P(at least s of X1,..., Xk exceed Y, X1 > a,..., Xk > a, Y > b) s,k = P(X1 > a,...,Xk > a,Y > b) ( )

where P(X1 > a,..., Xk > a, Y > b) > 0. Since X1,..., Xk and Y are independent, equation (29) can be written as follows:

R|a,b = P(at least s of X1,..., Xk exceed Y, X1 > a,..., Xk > a, Y > b)

Rsk = P(X1 > a.....Xk > a)P(Y > b) , ( )

where P(X1 > a,..., Xk > a)P(Y > b) > 0.

Let A = {(x1,..., xk, y) | at least s of x1,..., xk exceed y, x1 > a,..., xk > a, y > b}, B = {(x1,...,xk) | x1 > a,...,xk > a} and C = {y | y > b}. The nonparametric estimator of (30) can be written as follows:

RNpla,b = n(A)

Rs,k n(B)n(C). (31)

Let X1j,..., Xki ~ X for i = 1,..., n and Y1,..., Ym ~ Y be independent random samples. Also, let 1(E) be the indicator function of the event E, that is a RV that takes value 1 when the event E happens and 0 when the event does not happen. By assuming n(.) as the counting measure, we

have:

n

n(B) = £ I(X1i > a.....Xki > a), (32)

i=1

m

n(C) = £ I(Yj > b), (33)

j=1

and by the properties of the indicator function:

nm

n(A) = £ £ I(s of X1i,..., Xki exceed Yj )I(X1i > a,..., Xki > a)I(Yj > b) + ... i=1 j=1

nm

+ £ £ I(k of Xu.....Xkl exceed Yj)I(Xu > a.....Xki > a)I(Yj > b).

i=1 j=1

(34)

Let Xi = (X1i,..., Xki) for i = 1,..., n. Those observations Xi and Yj for them both Xi < a and Yj < b simultaneously, have been removed in calculating n(A) , since in details of calculating

P(A) or RNPla,b = "((A) , the numerator is an strict subset of denominator. Note that in this

v ' s,k n(B)n(C)'

case the values of the second and third indicators will automatically equal one in n(A), (34). It is worth noting that the number of reminded samples of Xi and Yj are n(B) and n(C), so n(A) can be written as follows:

n(B) n(C) n(B) n(C)

n(A) = £ £ I(s of X1i,..., Xfci exceed Yj) + ■ ■ ■ + £ £ I(k of X1i,..., Xfci exceed Yj)

i=1 j=1 i=1 j=1

In the case of n = m, the formula (31) may have simpler form and computations, since we only keep those (X1i,...,X^, Yi) i = 1,...,n which for them (X1i > a,...,Xfci > a, Yi > b) and remove the rest and also, n(B) = n(C). In what follows, we introduce a definition and representation for non-parametric estimator of multi-component stress-strength parameter. To the best of our knowledge, interestingly this estimator has not been defined till now.

Definition 3. The nonparametric estimator of Rs,fc is defined as follows:

nNP _ n(A) /or\

Rs,fc = n(B)n(C) (35)

where n(B) = n, n(C) = m and

n(A) = £ £I(s of X1i,...,Xfci exceed Yj) + ■ ■ ■ + £ £I(k of X1i,...,Xfci exceed Yj).

i=1 j=1 i=1 j=1

Note that (35) can be obtained from (31) by assuming a = b = 0. Remark 6. (i): By (31), and according to the definitions of n(A),n(B) and n(C), it can be concluded that for fixed values of a, a < b, the estimator RNfcP|a,b is a decreasing function of b. (ii): By (31), and according to the definitions of n(A),n(B) and n(C), it can be concluded that for fixed values of b, a > b, the estimator RNp|a,b is an increasing function of a.

In applications, the data observed for different stresses may differ greatly in their values. Therefore, selecting a minimum value of a, w.r.t. it all stresses in MCCSSP through definition 1,

satisfy the corresponding condition Xi > a, may be not useful. So, in what follows, the MCCSSP has been defined in some general way to be more realistic and applicable.

Definition 4. The generalized conditional multi-component stress-strength parameter is defined as follows:

R^.....ak,b = P(at least s of X1,..., Xfc exceed Y | X1 > a1,..., Xfc > afc, Y > b) (36)

where the RV's Y,X1,..., Xfc are independent, G( ) is the continuous distribution function of Y and F(-) is the common continuous distribution function of X1,...,Xfc.

Theorem 2. If Xri > max(a1,..., ak) for r = 1,...,k; i = 1,..., n and Yj > b for j = 1,..., m then

RNp|a1,...,ak,b _ RNP Rs,k = Rs,k .

Proof. Replace I(X1i > a1,...,Xfci > afc) with I(X1i > a,...,Xfci > a) in (32) and (34). Since I(X1i > a1,...,Xki > ak) = 1 and I(Yj > b) = 1 we have n(B) = n, n(C) = m and

n(A) = £n=1 £m=11(s of X1i.....Xfci exceed Yj) + ■ ■ ■ + £=1 £m=11(k of Xu.....Xfci exceed Yj). ■

Of course, a special case of (36) is (2). In parametric case (MLE method) when a1,..., afc are closed in values, a can be considered as the minimum or maximum of a1,..., afc and approximate (36) through (4). In some situations, a1,..., afc are very different, and using (36) is not very helpful or may not be accurate. In these cases, the non-parametric method is more practical and it is enough to consider A = {(x1,..., Xfc, y) | at least s of x1,..., Xfc exceed y, x1 > a1,..., Xfc > afc,y > b}, and B = {(x1,..., Xfc) | x1 > a1,..., Xfc > afc} in (31). It is easy to see that the results of nonparametric estimation of (29) can also be used for nonparametric estimation of (36), where ai is substituted instead of a for i = 1,..., k. Note that in this case, one advantage of the nonparametric method is that the assumption of common distribution for stress RV's may be relaxed. The later makes this method much more practical. If B = {(x1,..., xk, y) | x1 > a1,..., xk > ak, y > b}, then the nonparametric estimator of the generalized MCCSSP where stresses and strength RV's are not independent, can also be easily computed through the same method.

6. Simulation

In this section, a simulation study has been done to assess the quality and the efficiency of performance of R^, its MLE, Bayes and nonparametric estimators. The performances of the MLE, Bayes and nonparametric estimators have been studied by using their biases. The performances of the confidence intervals for MLE are studied by using average confidence lengths (ACL's) and coverage probabilities (CP's). It would be mentioned that, the proportion of the times that the intervals contain the true value of interest is called the coverage probability of a confidence interval. The simulations have been only done for a = b since for a = b the conditional and unconditional cases have the same results.

The results for R^, MLE's, Biases, MSE's, ACL's and CP's and different values of m and n where the other parameters are fixed, have been shown in the Tables 1 for A1 = 1, A2 = 2 and shown in 2 for A1 = 1.5, A2 = 0.7. According to these tables larger sample sizes have more reliable

results. A comparison among MLE, R^|a,b and R1a3b assuming a1 = 2, = 3, a2 = 5, j82 = 4 for different values of a and b, n = m = 100, A1 = 0.0003 and A2 = 0.0005 has been done and the results presented in Tables 3 and 4. A comparison among ^2"4b and ^2"4b assuming A1 = 3, A2 = 2, a1 = 5, = 0.8, a2 = 4, j82 = 0.2 for different sample sizes has been done and the results presented in Table 5. A nonparametric simulation for different values of a1, a2, a3, A1 = 0.004 and A2 = 0.002 has been done and the results are presented in Table 6.

Table 1: Comparison of estimators, r!0^'1 = 0.3659, R^7'1 = 0.2409

n 15 20 35 50 85 100

(s,k) m 15 25 35 50 75 100

R|0.7,1 R1,3 0.3556 0.3577 0.3612 0.3628 0.3632 0.3646

MSE 0.0439 0.0256 0.0180 0.0125 0.0083 0.0062

(1,3) Bias -0.0102 -0.0082 -0.0046 -0.0030 -0.0026 -0.0013

ACL 0.6889 0.5202 0.4425 0.3674 0.2990 0.2588

CP 0.9384 0.9498 0.9736 0.9802 0.9818 0.9844

Rl0.7,1 r2,4 0.2372 0.2357 0.2390 0.2393 0.2400 0.2400

MSE 0.0243 0.0139 0.0100 0.0069 0.0046 0.0034

(2,4) Bias -0.0036 -0.0052 -0.0018 -0.0015 -0.0008 -0.0008

ACL 0.5145 0.3879 0.3296 0.2741 0.2228 0.1921

CP 0.8988 0.9012 0.938 0.9518 0.9640 0.9690

Table 2: Comparison of estimators, Rl^2'0'5 = 0.4603, R^2'05 = 0.2798

n 15 20 35 50 75 100

(s,k) m 15 25 35 50 75 100

R 11.2,0.5 R1,3 0.4404 0.4481 0.4551 0.4537 0.4562 0.4567

MSE 0.0391 0.0225 0.0097 0.0096 0.0060 0.0046

(1,3) Bias -0.0199 -0.0122 -0.0052 -0.0065 -0.0041 -0.0035

ACL 0.5655 0.4581 0.3110 0.3085 0.2481 0.2183

CP 0.9374 0.9696 0.9826 0.9838 0.9876 0.9932

R 11.2,0.5 R2,4 0.2608 0.2679 0.2718 0.2742 0.2758 0.2771

MSE 0060 0.0045 0.0025 0.0017 0.0010 0.0008

(2,4) Bias -0.0197 -0.0118 -0.0079 -0.0056 -0.0040 -0.0027

ACl 0.2349 0.2079 0.1600 0.1351 0.0.0104 0.0964

CP 0.9156 0.9548 0.9620 0.9690 0.9712 0.9837

Table 3: Comparison o/JJ JR[83!'/or a < b

a 10 25 70 78 170 215 300

b 20 40 74 120 190 260 310

J1,3 0.8607 0.8568 0.8653 0.8362 0.8530 0.8340 0.0.8607

J |a'b 0.8555 0.8516 0.8602 0.8311 0.8478 0.8327 0.8555

J |a'b 0.8598 0.8559 0.8646 0.8349 0.8519 0.8321 0.8598

RNP|a,b J1,3 0.8657 0.8655 0.8697 0.8681 0.8691 0.8646 0.8668

Bias(JJ 1a3b) -0.0051 -0.0051 -0.0051 -0.0051 -0.0051 -0.0013 -0.0051

Bias(JJ -0.0008 -0.0009 -0.0007 -0.0013 -0.0010 -0.0019 -0.0008

tt /„NPla'b,. Bias(R131 ) 0.0049 0.0087 0.0043 0.0618 0.0161 0.0305 0.0061

Table 4: Comparison o/JJ1"^, R^'^18'6, J^/or a > b

22 45 67 100 120 240

11 38 65 90 70 230

0.9377 0.9124 0.8877 0.8664 0.8867 0.7532

0.9338 0.9088 0.8844 0.8633 0.8836 0.7514

0.9313 0.90559 0.8812 0.8598 0.8805 0.7466

0.8658 0.8658 0.8653 0.8674 0.8686 0.8680

-0.0038 -0.0036 -0.0033 -0.0030 -0.0030 -0.0017

-0.0063 -0.0064 -0.0065 -0.0065 -0.0062 -0.0065

-0.0719 -0.0466 -0.0224 0.0097 -0.0181 0.1147

a b

7 4

J1,3

jjhb

J |a'b

RNP|a,b J1,3

Bias(JJ l1^6) Bias(JJ

TT /,,NP|a,b..

Bias(R131 )

0.9437 0.9397 0.9372 0.8654 -0.0039 -0.0064 -0.0782

Table 5: Comparison o/ JJ24, JJ2a4b, exact va/wes 4°46'°'8 = 0.0430, J1^9,0'6 = 0.0243

n 10 20 30 60 95 100 150

m 10 22 30 58 100 120 150

jj 10.6,0.8 J 2,4 0.0505 0.0475 0.0457 0.0443 0.0437 0.0436 0.0434

jj 10.6,0.8 J 2,4 0.0360 0.0407 0.0405 0.0418 0.0419 0.0421 0.0424

Bias(JJ i0fa8) -0.0075 -0.0045 -0.0027 -0.0013 -0.0007 -0.0006 -0.0004

Bias(JJ2°46,0.8) 0.0069 0.0022 0.0024 0.0011 0.0010 0.0008 0.0005

JJ |0.S>,,0.6 J2,4 0.0269 0.0260 0.0254 0.0249 0.0246 0.0246 0.0245

JJ 10.9,0.6 J 2,4 0.0342 0.0301 0.0290 0.0271 0.0261 0.0257 0.0252

Bias(JJ 2°49,0.6) -0.0026 -0.0016 -0.0010 -0.0006 -0.0003 -0.0002 -0.0001

Bias(iJ 2°49,06) -0.0098 -0.0057 -0.0046 -0.0027 -0.0017 -0.0013 -0.0008

Table 6: Values of r^1"1'"2'"3'6 for A1 = 0.004 and A2 = 0.002

"i 1 1 8 27 40 40 95 100 100

a2 3 5 14 60 40 42 98 100 100

a3 7 7 28 90 40 47 100 100 100

b 5 3 19 43 30 30 110 110 180

RI^P\a1,a2,g3,b q.532 0.535 q.532 q.54Q q.546 0.549 q.5Jq q.5Q9 q.468

7. Real Data Analysis

In this section the numerical results of the parameters estimation for a real data set with Exponential distribution have been presented. This data set was used for the first time by [16] and can be find in it. Also, it have been used by many other authors, e.g., [17], [18] and [19].These data present the tensile properties of the jute fibres at different gauge lengths 5,10,15 and 20 mm which measured in MPa. The data sets corresponding to the breaking strength of jute fibres with 10mm and 15mm gauge lengths have been considered as the stresses measurement and 20mm in gauge lengths, which represents the strength measurement.

Each data has been separately fitted to the some Exponential distribution and examined by using the Kolmogorov-Smirnov goodness-of-fit test, the results have been reported in Table 7. The Kolmogorov-Smirnov statistics and the corresponding P-values indicate that the Exponential distribution fits the data sets. The estimation of MCCSSP for different values of a and b by MLE, nonparametric methods and Bayesian approach assuming a1 = 2, fii = 3, a2 = 5, j82 = 4 for parameters of prior distributions have been presented in Table 8. The estimation of MCCSSP for different values of a1, a2 and b by nonparametric methods have been presented in Table 9. The estimation of (4) for ai = 0 or a2 = 0 by nonparametric methods have been presented in Table 10. The data set consisting of the breaking strength of jute fiber 5 mm in gauge length have been fitted with the Normal distribution with mean 384.37 and standard deviation 188.77 using the Kolmogorov-Smirnov goodness-of-fit test. For this data, the Lilliforce significance correction criteria (modified Kolmogorov-Smirnov test to check the normality of the data) and the P-value are 0.143 and 0.122. Note that by adding this length to the model, the assumption of exponentially for all stresses fails and the MLE method may not be employed. The nonparametric estimators of MCCSSP for real data and different values of a1, a2, a3 and b have been presented in Table 11 where X1 has Normal distribution, X2 and X3 have Exponential distribution.

Table 7: Estimate of parameters, K-S test for strength of jute fiber data

data Mean A K-S p-value

10 mm 365.72 0.0027 0.958 0.317

15 mm 367.87 0.0027 0.999 0.271

20 mm 340.74 0.0029 0.727 0.666

Table 8: Values of estimates of MCCSSP for real data

a 30 45 45 78 85 100 220

b 25 50 40 90 75 80 245

R \a'b 0.7200 0.6680 0.6893 0.6432 0.6280 0.6288 0.5996

R \a'b 0.7431 0.6737 0.6458 0.6207 0.6477 0.5970 0.6151

RNP\a,b R1 ,2 0.6744 0.6760 0.6886 0.6462 0.6485 0.6485 0.6944

Table 9: Va/wes ofR^181^^/^ real data

81 20 30 90 180 202 200 300

82 40 100 70 170 200 250 280

b 30 70 80 175 201 225 290

jNP|a1,a2,b J 1,2 0.674 0.640 0.654 0.755 0.755 0.694 0.760

Table 10: Va/wes ofR^1^''^/^ real data

81 0 0 90 160 0 190 0

82 30 50 0 0 150 0 280

b 45 35 40 145 160 255 290

jNP|a1,a2,b J 1,2 0.663 0.670 0.679 0.608 0.663 0.617 0.640

Table 11: Va/wes o/.RNp|al'a2'aз'b/or real data

81 10 42 80 111 150 215 300

82 30 58 90 121 160 221 400

83 60 71 100 171 170 240 100

b 34 54 85 154 165 220 340

„NP|a1,a2,a3,b J1,3 0.730 0.736 0.705 0.750 0.859 0.625 0.750

8. Conclusion

The MCCSSP (Jakb) as an appropriate extension of multi-component stress-strength parameter has been introduced. A general formula for computing J8^ in the case of continuous RV's has

been presented. The maximum likelihood estimator of j8^ for Exponential distribution has been estimated. The asymptotic distribution of maximum likelihood estimator has been obtained and been used to obtain asymptotic confidence intervals of j8^ . A Formula for estimating the MCCSSP by nonparametric method has also been presented. Some numerical computation and simulation studies have been done for illustrating the inferential procedures. In the past decades, a lot of researches have been done for studying the behavior of reliability function in multi-component stress-strength models, many of similar works can be done for the conditional case. As an specific idea, j8^ can be obtained and estimated for other distributions. As another idea, one may interested in the amounts of information which are measurable, lost, unpredictable, etc.

Declarations

• Authors Contribution: All parts of this study has been done jointly by both authors, unless the simulation and graphical study which has been compiled by M.Taheri and the final edition which has been done by K.Khorshidian.

• Competing Interest: There is no conflict of interest between authors.

• Availability of Data and Materials: Please contact M.Taheri, taherisaifmorteza@gmail.com in order to request any data corresponding to the simulation or graphical subsection.

• Funding: No funding was obtained for this study.

References

[1] Church, J. D. and Harris, B. (1970). The estimation of reliability from stress-strength relationships. Technometrics, 12(1): 49-54.

[2] Kotz, S., Lumelski, Y. and Pensky, M. The Stress-Strength Model and its Generalizations: Theory and Applications, World Scientific, Singapore, 2003.

[3] Jose, J. K. (2022). Estimation of stress-strength reliability using discrete phase type distribution. Communications in Statistics-Theory and Methods, 51(2): 368-386.

[4] Khan, A. H., Jan, T. R. (2020). Estimation of stress-strength reliability model using finite mixture of M-transformed Exponential distributions. Reliability: Theory and Applications, 15(2): 90-103.

[5] Varghese, A. K., Chacko, V. M. (2022). Estimation of stress-strength reliability for Akash distribution. Reliability: Theory and Applications, 17(3 ): 52-58.

[6] Chauhan, K. S. (2022). Estimation and testing procedures of P (Y< X) for the inverse distributions family under type-II censoring. Reliability: Theory and Applications, 17(3 ): 328-339.

[7] Xavier, T., Jose, J. K. (2021). A study of stress-strength reliability using a generalization of power transformed half-logistic distribution. Communications in Statistics-Theory and Methods, 50(18): 4335-4351.

[8] Saber, M. M. and Khorshidian, K. (2021). Introduction to reliability for conditional stress-strength parametter. Journal of sciencees, Islamic Republic of Iran, 32(4):349-357.

[9] Bhattacharya, G. k. and Johnson, R. A. (1974). Estimation of reliability in a multicomponent stress-strngth model. Journal of American statistical association, 69(348): 966-970.

[10] Cetinkaya C. (2021). Reliability estimation of the stress-strength model with non-identical jointly type-II censored Weibull component strengths. Statistical Computation and Simulations, 91(14): 2917-2936.

[11] Goswami, A., Seal, B. (2022). Stress-strength Reliability for Equi-correlated Multivariate Normal and its estimation. Reliability: Theory and Applications, 17(4 ): 249-267.

[12] Awodutire, P. O., Xavier, T., Jose, J. K. (2022). Inferences on stress-strength reliability in multi-component system for type I generalized half-Logistic distribution. Reliability: Theory and Applications, 17(1 ): 18-32.

[13] Jana, N. and Bera, N. (2022). Interval estimation of multi-component stress-strength reliability based on inverse Weibull distribution. Mathematics and Computers in Simulation, 92(4):667-704.

[14] Kotb, M. S. and Raqab, M. Z. (2021). Estimation of multi-component stress-strength model based on modified Weibull distribution. Statistical Papers, 62(6): 2763-2797.

[15] Saini, S., Tomer, S. and Garg, R. (2022). On the reliability estimation of multi-component stress-strength model for Burr XII distribution using progressively first-failure censored samples. Statistical Computation and Simulation, 191: 95-119.

[16] Xia, Z. P., Yu, J. Y., Cheng, L. D., Liu, L. F. and Wang, W. M. (2009). Study on the breaking strength of Jute Fibers using modified Weibull distribution. Compus-A, 40:54-59.

[17] Chaturvedi, A., Kumar, N. and Kumar, K. (2018). Statistical inference for the reliability functions of a family of lifetime distributions based on progressive type II right censoring. Statistica, 78(1):81-101.

[18] Hassan, A. S., Naqy, H. F., Muhammad, H. Z. and Saad, M. S. (2020). Estimation of multi-component stress-strengt reliability following weibull distribution based on upper record values. Journal ofTaibah University for science, 1(14):244-253.

[19] Rao, G. S., Mbwambo, S. and Pak, A. (2018). Estimation of multi-component stress-strength reliability from exponentiated inverse Rayleigh distribution. Journal of Statistics and Management Systems, 1-20.