Stress-strength Reliability for Equi-correlated Multivariate Normal and its estimation Текст научной статьи по специальности «Медицинские технологии»

Stress-strength Reliability for Equi-correlated Multivariate

Normal and its estimation

Anirban Goswami*1 and Babulal Seal2

Regional Research Institute of Unani Medicine, Patna, Bihar 2Department of Mathematics and Statistics, Aliah University, Kolkata, West Bengal [email protected], [email protected] * Corresponding author

Abstract

In this article it is mainly focused on discussion about estimation of stress-strength reliability under equi-correlated multivariate setup. It is seen in some situations that the components of a system are equi-correlated. Generally, the form of the equi-correlation structure within the components of a system is known for a given situation, however parameters that are involved in the equi-correlation structure always unknown. In this article, we propose a procedure to compute and estimate the stress-strength reliability R= Pr(a'x > b'y) when x and y are distributed non-independently equi-correlated multivariate normal distribution, where a and b are two known vectors. Here we have proposed the method of moments estimator to estimate these unknown parameters. Actually, we want to find out overall strength is larger than overall stress. In order to do that we take a'x and b'y as their representatives e.g. principal components of the respective vectors do the job approximately. An asymptotic distribution used to obtain confidence intervals for the stress-strength reliability. The performance of these intervals checked through the simulation study. Finally, we provide a real data analysis.

Keywords: Equi-correaled; Principal Component, Method of Moments Estimator (MOM); Asymptotic.

1. Introduction

The strength-stress model measured by R=Pr(X>Y), the lifetime of a component has a random strength X and it's subjected to random stress Y. In stress-strength model, at any time, the system fails if and only if, the stress is greater than its strength. First introduced to this model by Birnbaum [1] and then developed by Birnbaum and McCarty [2]. There has been a huge number of works as regards estimation of the reliability R= P(X>Y) in the field of stress-strength models. It has several applications particularly in engineering ideas, like structures, deterioration of rocket motors, static fatigue of ceramic parts, fatigue failure of craft structures, and also in mechanical, civil engineering. The R=Pr(X>Y) has been formulated for the huge majority of the popular statistical distributions when X and Y are independent random variables belonging to the same univariate family and also (X,Y) follows the bivariate distribution with dependence between X and Y. This form of R has been established for the bulk of popular statistical distributions, including Normal, uniform, exponential, gamma, beta, extreme value, Weibull, Laplace, logistic and the Pareto distributions...etc [3-7]. This model may be applied in clinical trial to comparing two treatment effects, it may be more useful to draw conclusions regarding the unit's free measure, rather than comparing the means [8]. Simonoff, Hochberg and Reiser [9] also used this model to find the effect of the treatment, if Y is the response

A. Goswami, B. Seal RT&A, No 4 (71)

STRESS-STRENGTH RELIABILITY FOR EQUI-CORRELATED_Volume 17, December 2022

for a control group, and X refers to a treatment group.

A numerical procedure obtained by Birnbaum and McCarty [2] based on the asymptotic distribution to find the sample size needed for setting up an upper confidence bound with the defined width and confidence coefficient. Sen [10] obtained the non-parametric confidence bounds for P(X<Y) based on independent samples. Govindarazulu [11] obtained two-sided confidence limits for R when X and Y are independent and also dependent normal variates. Church and Harris [12] obtained confidence intervals for R in case of independent normal varieties.

All these above existing works were done under the univariate or bivariate setup, Gupta and Gupta [13] first introduced the concept of estimating stress-strength reliability under multivariate normal setup. They considered the forms of R= Pr(a'x > b'y), when (xpiXl, yP2Xl) follows multivariate normal distribution with non-independent vector between xpiXl and yP2Xl, a' and b' are two known vectors. This problem arises when a system in the energy is supplied to the system by p$ sources and is consumed through p2 sources and the sources of energy supplied and consumed are linearly related with known vector a' and b'. Under this set up Gupta and Gupta [13] considered only special cases of a' and b' and compared the MVUE and MLE estimates of R using given mean vector and dispersion matrix. Reiser and Farragi [14] derived the lower confidence bounds for R=P(a'x* > b'y*) and solved it iteratively and also derived an approximate lower confidence bounds for R. Enis and Geisser [15] have demonstrated that, how to obtain the exact confidence bounds for R.

In many instances, it is seen that the components of a system are equi-correlated. Generally, the form of the equi-correlation structure is known for a given situation within the components of a system, however parameters that are involved in the equi-correlation structure are always unknown. Thus, we compute the stress strength reliability analytically for the special case of equi-correlated multivariate normal setup. We consider the principal component analysis to estimate the a' and b' where as Gupta and Gupta [13] considered only spatial cases of a' and b' and we present estimation of R using method of moment (MOM) estimates of the parameters for equi-correlated multivariate normal setup in Section 2.1. Determine the asymptotic distribution of S in Section 2.2. Finally, Simulation studies and data analysis are carried out in Section 3 for performance of MOM of R in teams of mean squared errors (MSE), relative bias (RB) and mean absolute error (MAE).

2. Estimation of stress-strength reliability (R)

Let, xpiXl and yP2Xl be two random vector such that the distribution of ~ Npl+p2 X)

//1 p$ ....pA IP- P-.....P-\\

p$1 ....pA P- P-.....P-

where, ^ :

', ^ = (M1IP1X1) and £ = °

(p1+p2)xl

„P$P$ ....1) \P- P-.....P-,

(P- P-.....P-\ /1 P% ..p%>

P- P-.....P-\ P%1 ..Pl

V

,p- p-.....P-) \P2P2 ..■.1,

(Pl%P2)#(Pl%P2)

Sil = ^

r 1r -A ö Pii-Ai

AA -1

' £22 = s

(lPi ■■■ Piö p2l....p2

0 ( Pi XPi )

P2P2 ■•..1

& Z12 = £2, = s

21

J( P2 xp2)

f Ps Ps.....Ps ö

Ps Ps.....Ps

Vp3 A.....A 0 (p,xp2)

1 <P1 <1 <p2 <1 &-1<p- < ^=[1+(Pl-l)1lf2-l)12

Pl-l

P 2-1

vPlP2

Now, we are interested to find out the overall strain vector is more than overall stress vector and a gross idea of doing this is to find that in terms of their principal components a'x and b'y.

2

2

A. Goswami, B. Seal RT&A, No 4 (71) STRESS-STRENGTH RELIABILITY FOR EQUI-CORRELATED_Volume 17, December 2022

Then, we want to find the approximate reliability in terms of a'x and b'y. Then, R= Pr(a'x > b'y) = Pr(a'x - by > 0).

Now, the distribution of u = a'x — b'y follows NQ^ua22), where, pu=E (a'x — b'y) = ^a'lp^ — M'lp^i and = Var(a'x — b'y) = a'£1±a — 2a'£12 b + b'£22 b So, R= Pr(a'x — b y >0) = Pr(u > 0)

= /."vfk; «if % exp{— % *2}«*= (1)

The overall representation of the two sets or vectors are related to vectors a and b, such that they are approximated by a x and b y as in principal component analysis. Principal component analysis explaining the variance-Covariance structure £n & £22 of a set of variables x and y through a linear combination (a' & b') of these variables, i.e, explain maximum variability. It is noted that, the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it is orthogonal to the preceding components.

Let, the estimate of a' by ei normalized eigenvector of £n corresponding to eigen value and b' by l\ normalized eigenvector of £22 corresponding to eigen value TMs we have ei = i;ixl and ii = i;2Xl [16]

Then, ^u = lplXllpixl — M-2 lp2xllp2xl = V&1 — 7^2

and,

2 _ 1 ' 1 ' 1 ' °u=—ipjxi £n ipixi—2 e~p~p~ lpixi £i2 lp2xi+~p-2. lp2xi £z2 lp2xi

= a2 rPl(1 + (Pl-l)Pl) — 2 P1P2 0 + P2(1 + (P2~1)P2)1 g Pl VPlP2 - P2 1

= a2(2 + (pi — 1)pi + (p% — 1)p% — 2VP1P2PS)

Then, from equation (1)

R= Pr(a'x > b'y) = O

VPT:I-VP2:2

= O(S) (2)

7V(2 + (P1-1)P1 + (P2-1)P2-2VP^P3).

where O = Distribution function of univariate standard normal distribution.

Now to estimate the R, we need to estimate the parameters of p.1, ^2,p1:p2, p3 and a2 in equation (2). Thus, we obtain the method of moments estimator (MOM) of these unknow parameters to estimate S, denoted by S and obtain its asymptotic distribution.

2.1.Method of Moments Estimation

Suppose, m1, m2,....,m6 are the sample moments of the random sample of (y") ,a = 1,2, ...,n, where (y") ~ ^P1+P2 (^,£). The sample moments are defined as

1 "I (1V n m1 = — / I-/ xir P1±—ii=1 \nZ—i r=1 1 "2 (1V1 n

m2 " nD?1 Vnnr?1 ^

m =1\Pl (1\n X2

m- p^i = 1 (nZJ r?1 X'r

^P2 ^Vn 2

m4 = — ) (-> y:2

V2^-'i=1 \nt-i r?1 ,r

mG = ■

ZPi Pi /lv-"@ i=l/-ij=l \nL-ir=l'

PliPl - D— i=l^j=: i<j

2 V^ VP" AVn

m< lKl VnZ,

i<j

l VPi VP2 /lv@

■>r=l

m7 = ■

PlP%

ZPi ^P2 i'l @ i=l£->j=l \nL-ir=l'

Then the moment estimators (MOM) of ^l,^2,pi,p2,p- and a2 are define as

l vPi n

P-l = ml = — / [-/ Xi<

Plt-ii=l\ nL-i r=l

l P2 n ^2 = 1^2 = — ; [-/ yj,

P2t-lj = l\ nZ—l r=l j

lT l vPi /lv@ N l VP2 /lv@

yj2r ) - ^l2 - #22

^ ^ V"1 Pi @ 2\ l V"1 P2 /№

2[plZ_ii=l\ia=:l 'j=A^ «j

Pi n ^ l V"1 P2 /ly

2 [p^ Z_ii=^^ a=$ ir) p2£-tj=An/-t {n ni=l {nn@=lXir)) - l inn@=lyjr))

= n (m3 + m4 —

l2

YL n= l (SL xBrxjr)—pl2)

i<j J

Pl =S7.

82 \ pl{pl — l)

(m3 +m4 — ml — ml)

2 P2

2 ^ (m5 —

l

P2=l^

nP2 nP2 (lyn y. y.

a2 \ P2 (P2 — l)Li=lL,j=l KnLr=lyirJ]r i<j

(m< — m2)

2)

(m3 + m4 — ml — m2)

P3

l l Pi P2 l n

= F fe^l Kl inLr==l X.ryjr)—^)

r?l

(m7 —mlm2)

(m3 +m4 — ml — m2) Thus, the method of moment estimator of R by R = 0(5)

VPTMi-VP2M2

where, S =

9V{2 + (Pi-l)pi + (P2-l)p2-2VP^P3)

mi^Pi-m2^iP2

—m)—m*+m-+m.+m22Pi—m-pi+m2P2—m.P2—2mim20PiP2%2mjJpiP2

2.2. Asymptotic distribution of 8

In this Section, we obtain the asymptotic distribution of S using delta method [17]. Using this we may determine the confidence intervals.

Let us define as

S = g (m, , m =

(m$\ m2

m-

m4

m5

m6

Vm7

and S = h[ß],ß =

ß12 + a2 ß22 + a2 ßi2 + o2p$ ß%2 + o2p% ßiß% + o2p-

By using the central limit theorem,

I

Jn

\

r\ m2 / ß2

m- ß12 + a2

m4 ß22 +ff2

m5 ß12 + a 2P1

m6 ß22 + o2p2

Km7) ß1ß2 + 02p-

Ni

H

\ \

,2*

/ )

/m,

where, 2* = D

m2

m-

m4

m5

m6

m7

/Var(m1) Cov(m1,m2) Cov(m1,m-) ..... Cov(m1,m7)\

Cov(m1,m2) Var(m2) Cov(m2,m-) ..... Cov(m2,m7)

Cov(m1,m-) Cov(m2,m-) Var(m-) ..... Cov(m-,m7)

Cov(m1,m7) Cov(m2,m7) Cov(m-,m7) ..... Var(m7)

f 1 v Pi ¡1 v @

\

Var(m1 ) = Var l —) ) \P1Î->Î=1 \n/Li

ZPi fST'n \ v-1"1 V"1 Pl /V"1™ V"1™ \ Var () xit.\ + 2} ) Cov ( ) xir, ) xjr )

i=1 \i—tr=1' i—ti=1^-ti=1 \i—tr=1 'r=1 /

n2p12

= TT [1 + (Vi-1)Pi]

Pi

Var(m2) = Var

' 1 P" tl^çn \\

^2,]=1 (nLr?1 yir)j

= —[1 + (P2 -1)P2] P2

/ ^ Pi /1V1 ™ VarCm-,) = Varl — > | — / ^

Kn/-, r?1

Z"i /Vn N V"! V"1 /Vn Vn N Varl) x? ) + 2y y Cowl) x2,) x2)

i=1 \i—tr=1' i—ti=1^-ti = 1 \i—Ir=1 'r = 1 /

n2p12

K]

= —[4(j2ß12 + 2aF + (p1 - 1)(4ß12p1a2 + 2p12aF)] P1

Var(m4) = Var

KP21 p? 1 (1y-?1^

— [4a2^22 + 2a4 + (p2 — l)(4p22p2°2 + 2P22o4)] P2

Var(m5) = Var I

PliPl — ^

ZP^ Pi /lV"1 n

i = l£->j = li<j\ n/-! r=

X.rXjr 6

Pi Pi i=l j=l

n2pl2(pl — l)2

Pi Pi Pi Pi

Var I y xirxjr) | +

j = li<j \ \i—!r = l

ZPi^Pi st-IPi ST-IPi / fST'n V"1

y y y I Coviy X.rXjr, y XkrXh

i=l*—ij=l*—ik=l^-il=l\ \^—ir=l i—tr=l

i<j,k<l(i,j)T(k,l)

4 ¡PliPl — l)

[2pl2a2 + 2pl2pla2 + a4 + p$2o 4]

(Pl2iPl — l)2)[ 2

+ 4{(iPl — l)iPl — 2))(iPl — l)(Pl — 2) — 2)i4iil2PlG2 + 2pl2a4)}

+1 {{(PliPl — l))iPliPl — l) — 2) — (iPl — l)(Pl — 2))((Pl — l)(Pl — 2) — 2)" (3fil2Pla2 + p2o2 + Pl2a4 + p^4)]]

Var(m6) = Var

P2&2 — ^

ZP^ P2 /lV"1 n

y by y^r

i=l£-tj = li<j\n£-t r=l j

n2p22(p2 — l)2

ZP2 ^ P2 P2 P2 i CST^n V->n

y y y I Coviy yirVjr.y JkrVv

i<j,k<l(i,j)T(k,l)

4

(P22iP2 — l)2)

W2202 + 2^22p2°2 +°4 + P22*4] +

4'(iP2 — l)iP2 — 2))(iP2 — l)iP2 — 2) — 2)(4^22P2^2 + 2P22a4)} +

l

4{{(P2iP2 — l))iP2iP2 — l) — 2) — (ip2 — l)iP2 — 2))(iP2 — l)iP2 — 2) — 2)} i3V22p2°2 + V22a2 + P22a4 + P2°4)]]

Var(m7) = Var (■ l

l Pi P2 l n

\PlP2L->i=lL->j = l \n£jr=

xiryjr 6

n2pl2p22

Pi P2 n

[yi=l lj = l Hlr==l +

2y y y C0V(y Xiryjr,y ^ XirVkr) +

iTjTk

2 T yj ^ C0V iyr=l Xiryjr' Zr=l Xkryjr) +

iTjTk

$2y y y y Cov iY

jkl

r

rV],

Y

xkrJ\i

iTjTkTl

PlP2

[ip-l2a2 + H22°2 + 2p.l№2p- + a4 + p-2a4) +

(Pl — l)(V-l2o2 + 2^l№2p3 + V22PI° 2 + a 4Pl + a 4p32) + (P2 — l)(^22^2 + 2y.l№2p- + VI2P2°2 + °4P2 + o4P-2) + (Pl — l)(P2 — l)№l2v2P2 + 2iil№2p- + p22o2PI + O4PIP2 + o 4P32)]

1 P! 1 n 1 P2 1 Cov(m.1,m.2) = Covl — > I- > ^ir), — > (- > \p^Z_ii=1 \nL-ir?1 ) \n/Li

1 P2 1 n

y\r

n2p1P2

[n:?1 n

P2 n

Cov( ) Xjr,

i?1 'r=1

nn

= ^2Ps

i 1 sr1 pi AV1 n \ 1 V"I Avn

Cov(m1,m3) = Cov^- ^ ^^ ^ ^

EPi /'\_,n \_,n \ v-1"! V-1 Pi /V",n V",n \

Covl) xir,y Xi2r)+ 2> ) Covl) x^) Xj2r)

i = 1 'r=1 i—tr=1' i—ti=1^-ti = 1 \i—Ir=1 'r=1 /

n2p12

= —[№2 + (P1 — 1)a2^1P1] P1

i<i

1 P! 1 n 1 P2 1 Cov(mi,m4) = Covl — > (- > xir), — > (-/

\p1/Lii?1 \nL-ir?1 ) p^Aji=1 V^Ajr?1

P! P2 n n

■I) ) Covl) Xj r,> 3

n = 1^-ii = 1 Ir=1 'r=1

n2p1P2

= 2a2p-^2

I 1 Pi /1^ n \ 2 y^pi V"I AVn \ C0V(m1,ms) = ^ ni?1 (nZr?1 X' r/,pKp1—1) ni?1 n;?1 (nZr?1

Pi V"1 Pi /1vn

2 I vPi vn

Cov

ZPi Vn V"^ VPi n

n2p1(p1 — 1) 2

i = 1^-ir=1

i<i

Xir^ r

i<i

= — ^1^2[P1(P1 — 1) + 1] P1

I 1 Pi /1Vn Cov(m1,m6) = Covl — > ( — / I PiZ-i i?1 \nZ—i

n:2 n:2 (in"

= 1 i/' P2(P2 — 1)^-1 i=1^i=1

i<i

n n n CovCy Xir,n

¿—¡i^-lj^-lR \i—lr=1 ¿-Jr=

yjrykr

n2p1P2(P2 — 1) = 2№2P-

1 P! 1 n 1 P! P2 1 n

Cov(m1,m7) = Covl —) (- > ^ir),-/ / (-/

i<R

1 V Pi /1 V n

1 /vPi Vn

Cov

P! n P! P2 n

ryjr

n2p^p2

Xx ryjr

= — K^Ps +^2^2 + (P1 — 1)(№ 2p3 +№2P1)] P1

1 P2 1 n 1 P! 1 n

Cov(m2,m3) = Covl—) (-> yjr),—> (-/

\p^Aji=1 \nL-i r=1 j7 PrL-i i=1 \nZ_ir=

■Cov

n2pip2 2a2p3^1

P! n 2 P2 n \i—li=1L-lr=1 lr'¿-1 i=^Z_(r=

y

X

Cov(m2, m4) = Cov(ln ^Y y^Ay ^Y y^))

\P2^-ii=l \nZ-i r=l ) Tp2^i = l\ nt-i r=l JJ

ZP2 /Vn Vn N •sr-<P2 v-iP2 /Vn Vn \

Coviy yir,y y2r) + 2y y Cov() y.r, y y2)

i=l r=l r=l i=l j=l r=l r=l

i<j

2 2 2 = —[№2 + (P2 — lWV2P2] P2

n2p2

l P" l n 2 Pi Pi l n

Cov{m2,ms) = Cov^-^l ^^ yir)'^^—T) L = , L, = i XirXjr)

i<j

P" n Pi Pi n

n2PlP2 iPl — l)^\ =l Lr=lyir> Li=l Vj=l vr=lXirhr

i<j

= 2y.lO 2p3

/ l Y"1 P2 n \ 2 V^ VP2 Avn \

Cov(m2,m6) = Cov | — ^ ^^ y.^^—T) Ll=i Li=i ^Kl ^

i<j

P" n P" P" n

2^ — l)Covn l yr==l y^l Vj==I Vr==l y-yjr

n2p!ip2 — l) \*—'i=l^->r=l i-ti=l^-tj = l^-tr=l

i<j

2№2

2 [P2iP2 — l)+l]

P2

l P" l n l Pi P" l n

Covim2,m.7) = Cov(—y (-> ^rh^—/ / (-> xirVjr)

l P" n Pi P" n

Coviy y yir,y y y xiryjr

i l r l i l j l r l

n2plpl

l 2 2 2 2 = —[№2P- +№2 + (P2 — l)(№2P- +№ 2P2)] P2

Cov(jn3, m^) = Cov{l x2),' Y= i y2]

l Pi n 2 P" n 2 --Coviy y x?, y y y{T

i l r l j l r

2 t-j^/v i y y ir, / / yjr

n PlP2 =l^-'r=l ¿—Ij=l^-!r=l

= 4pl^2^2P3 + 2o4p-2

Cov(jn3,m5) = Cov^ x2),-^-^^ y= i ^

i<j

Pi n 2 Pi Pi n

n2pl ^ — l) C0V X Kl Lr ==l ^=l Li = i Lr = l XirXjr

i<j

l

= —[4(pl2apl + pl2a2 + a4pl) + 4pl2a2pl + 2o4pl2] Pl

/ 1 y"1 Pi AVn 2\ 2 V"^ V"2 AVn ^^ = C0V\f1 ni=1 (nnr=1 I) ni=^f i = 1 ^=1 ^ r

2 P! n 2 P2 P2 n = 1 C0V \ni=1 nr=1 Xl2-,ni=1 ni = 1 nr=1 yiryjr

V i<i

= 4^1^2^2p3 + 2<J4p32

1 P! 1 n 2 1 P! P2 1 n

Cov(m-,m7) = Covl— > (-> x2.),- > > (-> Xiryj r

\nL-ir=1 > V1V2^i=1^i=1 \nZ-ir=1 j

1 P! n 2 P! P2 n = 2 2 Covl) ) Xi2r,> ) ) Xiryjr n2pjp2 \Z_(i=1Z_ir=1 Z_(i=^/_(i=^/_(r=1 j

2

= — [G"120"2P3 + + a4P-) + (P1 — 1)(^12o 2p3 + V1№2P1 + °4P1P3)]

P1

1 P2 1 n 2 2 P! P! 1 n Cov(m.4,m5) = Cov{- ^ ^^ I) L,=i (nZr=1 ^r)

i<i

2 P2 n 2 P! P! n

= 1) C0V\ni=1 nr=1 yj2r,ni=1 ni = 1 nr=1 XirXjr

\ i<i t

= 4^1^2^2P- + 2a4p-2

^^ = ^ nP^1 (1nn=1 1 (1nn=1 ^

2 P2 n 2 P2 P2 n

= DCov \ ni=1 nr=1y2 ni=1 nr=1^

i<i

1

= —[4(^22op2 +\i22o2 + o4p2) + 4^22o2p2 + 2o4P22] P2

1 P2 1 n % 1 P! P2 1 n

1 P2 n 2 P! P2 n

= ^ |ni = 1 Zr =1 ^¿i=1 A = 1 nr=1 Xiryjr] 2

= — [G"220"2P3 + + a4P-) + (P2 — 1)(^22° 2p3 + ^1^2^2P2 + p4p2p-)]

P2

Cov(m5,m6) =

(2 VPi VPi A Vn ^ 2 V"2 Y"2 Ayn \

P1(P1 — DL i=^i=1 (nL r=1XirXjr), 1) L i=^i=1r=1yiryjr) i<i i<i

4 I y^pi V"1 Pi st>n Y"'P2 Y"1 "2 X"1 n

= i)(p7—i)Cov \ni=1 ni=1 nr=1XlrXjr, ni=1 ni=1 nr=1yiryj

V i<i i<i

= 4^1^2^2P- + 2a4p32

2 Pi Pi l n l Pi P" l n

= ^[¿jk—T) y=I VJ==I ^KI ^^ y=I VJ==I ^KI ^

i<j

2 Pi Pi n Pi P" n

--—--- Covl y y y XirXjr,y y y X.ryjr

n2p2ipl — 1)P2 \t-ii=lt-ij = lt-ir=l ¿—Ii = lt—Ij = lt—Ir=l j

i<j 2

= — [(2Pl2o2p- + Vi№2PI + Vl№2 + a4P- + °4PlP-) + Pl

(Pl — 2)([il2a2p3 + VI№2PI + °4PlP3)]

ST'P2 VP2 AVn \ 1 VPi VP2 AVn \

Cov{m6,m7') = ^ y^,— ^ Li=_1 (nLr=_1 ^

i<j

1P2 -ST-IP2 X-1 n VP^ V P2 V n

P" P" n Pi P" n

Coviy y y yirVjr,y y y X.ryjr

i l j l r l i l j l r l

n2PlP^(P2 — 1)

i<j

= ~ [(2[j-22v2p3 + vI№2p2 + Vl№2 + o4p- + o4p2p-) + P"

(P2 — 2)([i22a2P3 + Vl№2P2 + °4P2P3)]

Thus we find,

dg dg dg dg dg dg dg

/(S) = ( d9 dg dg dg dg dg dg ( ) (dml, dm2, dm3, dm4, dmG, dm<, dm

4 d"tG U"L6 u"L7/m=:

where,

dg \ _ pihI(sp$hI — /T2V2)

(—)

KdmJ

lmi=:i +a2(2 + (—1 + pl)pl + (—1 + p2)p2 — 2e^P3)f2

i dg) =__

(o2(2 + i—1 + Pl)Pl + i—1 + P2)P2 — 2^2P3)f2 dg \ —e/pl^l +

(—)

\dm3J

i \3/2

m3=(:i2'72) 2 (a2(2 + (—1 + pl)pl + (—1 + p2)p2 — 2j^P3))

dg \ _ —jpivI + /&2

(—)

\dm4J

, \3/2

m*=X:22'72) 2 (a2(2 + (—1 + pl)pl + (—1 + p2)p2 — 2j^P3))

( dg ) =_—(—i + pl)(/pl^l — _

^dmGm5=i:i2+72Pi) 2 (a 2(2 + (—1 + pjpl + (—1 + p^p^ — 2j^2p3)f2

( dg ) _ (—i + p2)(—/p$vI + JV2H2)

\dmj

< M-6 = (:22+72P2)

2 (°2(2 + (—1 + Pl)Pl + (—1 + P2)P2 — 2/PiP2P3))2

dg\ _ VP1P2(V&1 —

(—)

\dm7J

, >3/2

m7=(:i:2+72P3) [o2(2 + (—1 + P1V1 + (—1 + P2)P2 — 24P1^P3)J

Using the delta method, we have

^(8 — S)^ N(0,0$) or, N+S,

where, = g/(S) rg(8)

= ((a3ptP2P1 (pj + 2o2p!) — 2pl^lp1((2 + 4a)p1 + 3a3 p!) + O 3eP1P2^(—1 + P1)(—2 + P1 — (—1+P2)P2)) —

2o3pl/2pl/2p2(v1V2 + 2a2 P3) + P1P2(—4(1 + 26)^ +

2p1p1(2(1 + 3a)p2/4 — 3ff3P1) + a 3plp|(^| + 2a2p2) — 4^4PiP2PlP2(P^PfP2 — 02(—1 + P1)(—2 +P1 + P2) + P2(—p22(—1 + P2) + a2 (—1 + P1)P2)) — 2(2(1 + 2a)p2p2 + a 3pI(—44p1p2 — 24PiP^Pl (—1 + P2) + 64^2P2 + (3 — 24p1p2)p\ ) +

a 5(—2+p1 +p2)2(—2 + p1 +p2 + 44P1P2P3)) — 2ap\(a2 p\ p2(—2 + P1 — 2(—1 + 4P1P2)^) + P4(—1 + P1 — 2

(—1 + 4P1P2)P2) + °4PI(—6 + 3P1 + 3P2 + 4eP1P2P3)) + P2(4P4(OeP1P2 + (1 — a(—3 + eP1P2))P2) + a3p2(4 + 9p2 + 4(—1 + 34p1F2)p2 + (9 — B^p^) Pi + p1(—4 + (2 — 44^f2)p2)) + 2°SP2 (—2 + P1 + P2)(—6 + 3p1 + 3P2 + 8eP1P2P3))) — 2P-/2 4V2(—2W2(—2°3eP1P2 — 034P1P2P1(—1 + P2) + (3o34hf2 + (2 + 4a)p2 )p2 — a3(—3 + 4p1^)p|) + v3pipKW2 + 2a2 P-) + p2 (4(1 + 3o)plp2p1 + 4O4P1P-2.P1 PK—1 + pj — 2a34p1p2p2 p1(—3 + 2p1 + p2) + P1P2 (4P-2 (a4 P1P2 + (1 — °(—3 + eP1P2))P2) + a3 (4 + 9p1 + (—4 + 64^~2)p2 + (9 — 44P1P2)P2 — 2P1(2 + (—1 + 4^)p2))) + 2aS(—2 + P1 + p2)p-(—2 +P1 +p2+ 44P1P2P3)) — 20pl(24p1p2p1 pi p2 +P1P2(a 2 p2(—2 + P1 —

(—2 + eP1P2)P2) + PK—1 +P1 — 2(—1 + eP1P2)P2)) +

2° 4P2P3(—2 +P1 +P2 + 2eP1P2P3) — 0 2 pi (—2P3 + P1QP1P2P2 + 2P-)))) + aplp2(—2p4(—1 + 2P1 +p2— p2^) + 4p2plp2p- +

4°2 P1^P1(4 P1P2P1 — 2P2P+ p2 (—4a2 p2 + 2p1(—o2(—2+p2) + p2(p2 + a2 p2)) —

4a 2 p2p- ) + 0 2 p1(—6a2 pi + 8a 2 p2p- + P1(P2(Pl + 6°2 P2) — 2°2(—6 + 3P2 + 4eP1P2P3)))) — 4«P5/2 4^2(—a24P1P2P1^P1(—1 + P2) + p2(—pIp2(—1 + 2P1 + p2) +

a2 p1p2(—2p1 + P1(2 + (—1 + 4p1p2)p2) + 2(—1 + p2)p-) + a2 P1QP1P2&P1 — 2o2 P-(—2 +P1 +P2 + 2eP1P2P3))) + pl(plp2p2 + 2p\p2 p- + 2a2 P3(—P2 P1 + o2 (P1P2 + 2p-3)) + P1^(P2 P1 + a2(P1P2 — 2P-(P2 + P-))))) + p2 (—2p\p2((2 + 4a)p2 + 3a3p2) + p2(4(1 + 3a)p4lP1 + 4a4^f2p3p2(—1 + p2) — 4a34p1p2p1p2p1

(-3 + 2.p$ + p2) + pi (4P-l + (1 + 3^- oePiP2) p2) + a2(4 + 9p\ + 2p$(-2 + p2) - 4p2 + 9p%)) +

2a-Pl(-2ePlp2nK-l+P2) + a 2(-2+p$ + p2) (-6 + 3Pi + 3p2 + 8epip2p3))) + aplQptpi + 4pip-p3 + o 2 p2(pi P2 + 2a2(3pip2 + 4p%)) + P2 + °2(PiP2 - 2P-(2P2 + P3)))) -

^Vefc&fiM + Pi (°2 P2(-2 +Pi + + vK-2+3Pi+(3 - 2epiP2)P2))+

a2p2(2p2 + PiQ-2 + (1 - - 4(-1 + p2)p3) -

2°2№2 (-2P- + Pi(ePiP2P2 + 2P-)) +

2a4(3pi p2 + 2(-2 + p2)p- + pi (3p2 + 2p2 + p2(-6 + 4^piF2p-))))))/

(2na5PiP2(2 + (-1 + Pi)Pi + (-1 + p2)p2 - 2ePiP2P3) ))

2.3. Asymptotic Confidence Intervals for R

Based on the asymptotic distribution of 8, we obtain the asymptotic confidence interval of R. Here, the estimate of R by R = 0(5) , i.e. S = <i>-i(R) and we have S -q N +5, -") as n — <*>. In order to determine the two sided confidence Intervals, we find out the two numbers Liand L2 (Li < L2), such that, for a given a, we have

P(Li < 0(S) <L2) = 1-a or, P(0-i(Li)<S <«-i(L2)) = 1-a (3)

Then, an asymptotic (1-a) level confidence Intervals for S is given by

PI -za/2 < S>> < za/2{ = 1-a

- a

or P+- ^ <(S-6)< z40) = 1

or, p{§- <8 <8 + z-4p) = 1-a

We can replace asby ds to obtain asymptotic confidence Intervals for 8. Thus, we can write

P+S- ^ <s<S + z4J) = 1-a (4)

Comparing (3) and (4), we have Li and L2 respectively as

<S>-i{L1) = 8-Z-a2=Z

or h = *(*- ^

ani 12 = *(<* + -z^J)

Then, an asymptotic (1-a) level confidence Intervals for R is represented by

where, za/2 upper critical value for the standard normal distribution. Thus, an asymptotic (1-a) confidence lower bound for R as

3. Simulation study and Data analysis

3.1. Simulation study

Now, we compute convergence and performance of MOM estimator, we considered different scenarios, each corresponding to a different combination of distributional parameters with different reliabilities for p1=10 and p2=10, reported in Table 1. We set the six parameters in order to get a high value (>0.5) for the reliability, since one typically looks for high reliability for the study component or system in real practice. Through these scenarios we cover the large range of reliability, since the range of R from 0.5825 to 0.9736.

For this above purpose we compute the following measures:

(i) Sample mean of R using MOM

(ii) Mean square error (MSE) of R : E(R — R)2

(iii) Mean Relative Bias (RB) of R :

(iv) Mean absolute error (MAE) of R: E(\R — fi|)

It is difficult to obtain the analytical form of the equation (1) for various 'R'. So, we figure out these by using simulation study. Hence, we generate the random samples of size n from

(y) ~ ^P1+P2 for different scenarios. For each of sample drown of size n, we compute the R

using MOM by taking 500 replications each time and also compute the above measures. Here we consider the different sample sizes (n) as 10, 30, 50 and 100. For this purpose, here, R programming language is used. The simulation results are reported in Table 2. It is noted that, the MSE, RB and MAE of R are reduces as the sample size increases decrease as expected and when n=100, R achieved the true value of R under each scenario. Thus, the result seems to be supportive for R in larger sample. Also, the performance of the R using MOM is quite satisfactory in terms of MSE, RB and MAE for small sample sizes. Hence, it is satisfying the consistency property of the MOM of R.

We take the components as p1=14, p2=12 and set the parameters are ^1=4, p2=3.5, p1=0.8, p2 =0.6, p3=0.7, a2 =4 in order to verify the asymptotic distribution of S as follows normal distribution, described in section 2.2. Then, we have S~N(1.538,0.0196) and generate n=500 samples using this as theoretical quantiles. For each of sample drown of size n=500, we compute the S using MOM by taking 500 replications each time is treated as sample quantiles. Figure 1, Q-Q plot [18] and Shapiro-Wilk normality test [19] (result as W = 0.99686, p-value = 0.4469) provided satisfactory result that the S follows asymptotic normal distribution.

The results of the simulation study for the confidence intervals as lower (L1) limit, upper (L2) limit and lower bound (LB) are recorded in Table 3. Table 1 and 2, represent the asymptotic and bootstrap confidence belt at 90%, 95% and 99% levels. It has been observed that for a small sample size, the estimate of R is getting high and also confidence intervals in case of asymptotic. The results get better as the sample sizes increase and the reliability R gets closer to true value. The overall band of asymptotic confidence is going to sink as the sample sizes increase and it has consistent variation.

Table 1: Parameters for the equi-correlated set-up_

Scenario Parameters and values

Mi Pi Pi P3 a2 R

1 1.5 1 0.2 0.2 0.1 16 0.5825

2 2.5 1.5 0.3 0.2 0.1 16 0.6453

3 3 2 0.3 0.2 0.2 16 0.6915

4 4 2.5 0.5 0.4 0.3 16 0.7209

5 4 2.5 0.5 0.4 0.3 8 0.7962

6 4 2.5 0.6 0.4 0.4 8 0.8335

7 4 2.5 0.6 0.5 0.5 8 0.8881

8 4 2.5 0.7 0.6 0.5 4 0.8912

9 4 2.5 0.8 0.6 0.6 4 0.9293

10 4 2.5 0.8 0.7 0.7 4 0.9736

Table 2: Simulation results: Coverage Probability, MSE, RB and MAE

Scenario Sample Size Sample Mean (A) MSE RB MAE

1 10 0.5840 0.0177 0.0025 0.1064

30 0.5833 0.0054 0.0013 0.0576

50 0.5831 0.0034 0.0010 0.0467

100 0.5828 0.0017 0.0006 0.0328

2 10 0.6491 0.0154 0.0060 0.1002

30 0.6456 0.0048 0.0005 0.0537

50 0.6455 0.0029 0.0004 0.0431

100 0.6454 0.0015 0.0001 0.0315

3 10 0.6918 0.0171 0.0006 0.1034

30 0.6918 0.0052 0.0005 0.0582

50 0.6917 0.0027 0.0004 0.0423

100 0.6917 0.0015 0.0003 0.0309

4 10 0.7238 0.0161 0.0040 0.1024

30 0.7222 0.0046 0.0017 0.0538

50 0.7209 0.0026 -0.0001 0.0409

100 0.7206 0.0013 -0.0004 0.0295

5 10 0.7977 0.0110 0.0018 0.0856

30 0.7973 0.0033 0.0014 0.0457

50 0.7969 0.0022 0.0008 0.0385

100 0.7965 0.0012 0.0004 0.0271

6 10 0.8376 0.0091 0.0048 0.0776

30 0.8347 0.0033 0.0014 0.0457

50 0.8341 0.0020 0.0007 0.0356

100 0.8336 0.0009 0.0001 0.0245

7 10 0.8903 0.0057 0.0024 0.0603

30 0.8895 0.0019 0.0016 0.0354

50 0.8888 0.0014 0.0008 0.0298

100 0.8884 0.0006 0.0003 0.0192

8 10 0.8934 0.0057 0.0025 0.0622

30 0.8926 0.0020 0.0016 0.0355

50 0.8918 0.0013 0.0007 0.0284

100 0.8917 0.0006 0.0005 0.0205

A. Goswami, B. Seal STRESS-STRENGTH RELIABILITY FOR EQUI-CORRELATED RT&A, No 4 (71) Volume 17, December 2022

Scenario Sample Size Sample Mean Ш) MSE RB MAE

9 10 0.9310 0.0031 0.0018 0.0434

30 0.9304 0.0013 0.0012 0.0278

50 0.9298 0.0008 0.0005 0.0224

100 0.9296 0.0004 0.0003 0.0159

10 10 0.9740 0.0008 0.0004 0.0213

30 0.9737 0.0003 0.0001 0.0139

50 0.9736 0.0002 0.0000 0.0119

100 0.9735 0.0001 -0.0001 0.0085

1.8-

1.7-

1.6-

аз _>

а

а> о.

1.4-

1.3-

1.25 1.50 1.75 2.00

Theoretical Quantiles

Figure 1: Normal Q-Q Plot

Table 3: Asymptotic Confidence Intervals

Scen- Sample R 90% 95% 99%

ario Size L U LB L U LB L U LB

1 10 0.5884 0.5167 0.6572 0.5326 0.5028 0.6699 0.5167 0.4757 0.6942 0.4867

30 0.5842 0.5624 0.6058 0.5672 0.5582 0.6099 0.5624 0.5499 0.6179 0.5533

50 0.5838 0.5706 0.5969 0.5735 0.5681 0.5994 0.5706 0.5631 0.6042 0.5651

100 0.5830 0.5762 0.5897 0.5777 0.5749 0.5910 0.5762 0.5724 0.5935 0.5734

2 10 0.7109 0.6465 0.7692 0.6612 0.6336 0.7796 0.6465 0.6079 0.7991 0.6184

30 0.6978 0.6712 0.7235 0.6771 0.6659 0.7283 0.6712 0.6557 0.7375 0.6598

50 0.6876 0.6744 0.7005 0.6774 0.6719 0.7029 0.6744 0.6669 0.7077 0.6689

100 0.6444 0.6371 0.6516 0.6387 0.6357 0.6530 0.6371 0.6330 0.6557 0.6341

3 10 0.7583 0.6551 0.8420 0.6793 0.6335 0.8555 0.6551 0.5902 0.8797 0.6079

30 0.7235 0.6925 0.7530 0.6995 0.6863 0.7584 0.6925 0.6742 0.7688 0.6792

50 0.7135 0.6859 0.7400 0.6921 0.6805 0.7449 0.6859 0.6698 0.7543 0.6741

Scen- Sample R _90%_95%_99%_

ario Size_ L U LB L U LB L U LB

_100 0.6954 0.6853 0.7054 0.6875 0.6833 0.7073 0.6853 0.6795 0.7110 0.6810

4 10 0.8041 0.6534 0.9063 0.6903 0.6202 0.9202 0.6534 0.5530 0.9429 0.5805 30 0.7571 0.7004 0.8074 0.7134 0.6889 0.8162 0.7004 0.6659 0.8328 0.6753 50 0.7420 0.7226 0.7606 0.7269 0.7188 0.7641 0.7226 0.7113 0.7709 0.7143

_100 0.7243 0.7140 0.7344 0.7163 0.7121 0.7363 0.7140 0.7081 0.7400 0.7097

5 10 0.8613 0.7002 0.9502 0.7418 0.6619 0.9603 0.7002 0.5823 0.9753 0.6151 30 0.8570 0.7952 0.9048 0.8101 0.7818 0.9124 0.7952 0.7542 0.9260 0.7656 50 0.8352 0.8057 0.8616 0.8125 0.7997 0.8663 0.8057 0.7877 0.8751 0.7926

_100 0.7996 0.7854 0.8133 0.7886 0.7826 0.8158 0.7854 0.7771 0.8208 0.7793

6 10 0.8800 0.7306 0.9586 0.7699 0.6940 0.9673 0.7306 0.6170 0.9799 0.6490 30 0.8613 0.7941 0.9117 0.8104 0.7793 0.9195 0.7941 0.7488 0.9334 0.7614 50 0.8506 0.8098 0.8852 0.8193 0.8012 0.8911 0.8098 0.7839 0.9020 0.7910

_100 0.8340 0.8111 0.8551 0.8163 0.8064 0.8589 0.8111 0.7972 0.8662 0.8010

7 10 0.9310 0.5747 0.9973 0.6824 0.4762 0.9988 0.5747 0.2930 0.9998 0.3639 30 0.9253 0.7169 0.9896 0.7780 0.6581 0.9934 0.7169 0.5327 0.9975 0.5846 50 0.9079 0.8218 0.9586 0.8442 0.8008 0.9650 0.8218 0.7557 0.9752 0.7746

_100 0.8839 0.8527 0.9101 0.8601 0.8462 0.9145 0.8527 0.8328 0.9228 0.8383

8 10 0.9146 0.6030 0.9934 0.6935 0.5194 0.9964 0.6030 0.3570 0.9991 0.4214 30 0.8957 0.7818 0.9587 0.8117 0.7538 0.9662 0.7818 0.6940 0.9776 0.7190 50 0.8939 0.8218 0.9421 0.8399 0.8051 0.9490 0.8218 0.7699 0.9605 0.7846

_100 0.8937 0.8723 0.9124 0.8772 0.8678 0.9157 0.8723 0.8589 0.9218 0.8626

9 10 0.9602 0.7672 0.9973 0.8304 0.7032 0.9985 0.7672 0.5597 0.9996 0.6200 30 0.9486 0.7554 0.9949 0.8157 0.6954 0.9970 0.7554 0.5633 0.9990 0.6186 50 0.9451 0.8894 0.9759 0.9043 0.8751 0.9797 0.8894 0.8438 0.9857 0.8571

_100 0.9306 0.9004 0.9532 0.9078 0.8936 0.9568 0.9004 0.8795 0.9631 0.8853

10 10 0.9893 0.0092 1.0000 0.0922 0.0006 1.0000 0.0092 0.0000 1.0000 0.0000 30 0.9821 0.6637 0.9999 0.7862 0.5403 1.0000 0.6637 0.2991 1.0000 0.3926 50 0.9770 0.8131 0.9990 0.8715 0.7509 0.9995 0.8131 0.6038 0.9999 0.6668 100 0.9768 0.9252 0.9945 0.9409 0.9091 0.9960 0.9252 0.8705 0.9979 0.8873

3.2. Data Analysis

In this section, we apply the above methods to find out values of the estimators as p.$, fi2, Pi, P2, p3, a2 and R from a given data set. The secondary data set of "Wave Energy Converters Data Set" is taken from the UCI Machine Learning site. The data set can be downloaded at https://archive.ics.uci.edu/ml/datasets/Wave+Energy+Converters. This data set consists of positions and absorbed power outputs of wave energy converters (WECs) in four real wave scenarios from the southern coast of Australia (Sydney, Adelaide, Perth and Tasmania). From this date set we take only two place of data set as Adelaide and Perth. We consider the eleven variables names as WECs absorbed power from each data set, which are consistent with the positive correlation among the variables. Then, we find out the stress strength reliability of absorbed power between the Adelaide and Perth respectively. Here we select the number of variables as p$=11, p2=11 and the MOM estimates as =88175.2, fi2 =87244.27, £$=0.06567, p2 =0.05251, p3=-0.04049, cr2 =107224128 and R=0.55872. Jennrich test [20] used to examine the differences between correlation matrices of elevens variables of Adelaide and Perth data sets. The result shows that, the sample and estimated correlations by MOM are equal, reported in Table 4 and 5. This means that there is an equi-correlation between variables of the above data sets. The mean vectors of each data set reported in Table 6 and all are mostly equal. The performance of MOM quite good for sample size. The confidence intervals result on "Wave Energy Converters Data Set" shows in Table 7. The asymptotic confidence intervals in terms of lower limit, upper limits and lower bound are almost same and also

A. Goswami, B. Seal RT&A, No 4 (71)

STRESS-STRENGTH RELIABILITY FOR EQUI-CORRELATED_Volume 17, December 2022

band at different levels, but confidence interval and band of bootstrap is lesser then asymptotic confidence intervals values.

Table 4. Correlation Matrix and Estimated Correlation Matrix of Adelaide data set

Correlation Matrix

Variable V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11

V1 1 0.022 0.009 0.044 0.051 0.022 0.018 0.12 0.07 0.115 0.07

V2 0.022 1 0.017 0.032 0.019 0.069 0.013 0.049 0.0004 0.07 0.039

V3 0.009 0.017 1 0.048 0.047 0.047 0.082 0.07 0.062 0.091 0.056

V4 0.044 0.032 0.048 1 0.052 0.067 0.06 0.041 0.059 0.085 0.098

V5 0.051 0.019 0.047 0.052 1 0.05 0.101 0.101 0.118 0.092 0.045

V6 0.022 0.069 0.047 0.067 0.05 1 0.063 0.068 0.078 0.098 0.026

V7 0.018 0.013 0.082 0.06 0.101 0.063 1 0.093 0.14 0.125 0.065

V8 0.12 0.049 0.07 0.041 0.101 0.068 0.093 1 0.071 0.145 0.052

V9 0.07 0.0004 0.062 0.059 0.118 0.078 0.14 0.071 1 0.071 0.092

V10 0.115 0.07 0.091 0.085 0.092 0.098 0.125 0.145 0.071 1 0.12

V11 0.07 0.039 0.056 0.098 0.045 0.026 0.065 0.052 0.092 0.12 1

Estimated Correlation Matrix using MOM

V1 1 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066

V2 0.066 1 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066

V3 0.066 0.066 1 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066

V4 0.066 0.066 0.066 1 0.066 0.066 0.066 0.066 0.066 0.066 0.066

V5 0.066 0.066 0.066 0.066 1 0.066 0.066 0.066 0.066 0.066 0.066

V6 0.066 0.066 0.066 0.066 0.066 1 0.066 0.066 0.066 0.066 0.066

V7 0.066 0.066 0.066 0.066 0.066 0.066 1 0.066 0.066 0.066 0.066

V8 0.066 0.066 0.066 0.066 0.066 0.066 0.066 1 0.066 0.066 0.066

V9 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 1 0.066 0.066

V10 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 1 0.066

V11 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 0.066 1

Jennrich test: x2= 30.1314, p-value= 0.9974615 (Ho: all the correlations are equal)

Table 5. Correlation Matrix and Estimated Correlation Matrix of Perth data set

Correlation Matrix

Variable V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11

V1 1 0.052 0.063 0.121 0.113 0.03 0.023 0.117 0.003 0.075 0.109

V2 0.052 1 0.048 0.037 0.031 0.05 0.018 0.065 0 0.018 0.052

V3 0.063 0.048 1 0.11 0.05 0.032 0.04 0.034 0.048 0.029 0.044

V4 0.121 0.037 0.11 1 0.086 0.048 0.029 0.096 0.019 0.068 0.038

V5 0.113 0.031 0.05 0.086 1 0.022 0.002 0.118 0.003 0.128 0.049

V6 0.03 0.05 0.032 0.048 0.022 1 0.024 0.045 0.029 0.041 0.061

V7 0.023 0.018 0.04 0.029 0.002 0.024 1 0.026 0.088 0.068 0.09

V8 0.117 0.065 0.034 0.096 0.118 0.045 0.026 1 0.008 0.069 0.114

V9 0.003 0 0.048 0.019 0.003 0.029 0.088 0.008 1 0.011 0.029

V10 0.075 0.018 0.029 0.068 0.128 0.041 0.068 0.069 0.011 1 0.077

V11 0.109 0.052 0.044 0.038 0.049 0.061 0.09 0.114 0.029 0.077 1

Estimated Correlation Matrix using MOM

V1 1 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053

V2 0.053 1 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053

V3 0.053 0.053 1 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053

V4 0.053 0.053 0.053 1 0.053 0.053 0.053 0.053 0.053 0.053 0.053

V5 0.053 0.053 0.053 0.053 1 0.053 0.053 0.053 0.053 0.053 0.053

V6 0.053 0.053 0.053 0.053 0.053 1 0.053 0.053 0.053 0.053 0.053

V7 0.053 0.053 0.053 0.053 0.053 0.053 1 0.053 0.053 0.053 0.053

V8 0.053 0.053 0.053 0.053 0.053 0.053 0.053 1 0.053 0.053 0.053

V9 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 1 0.053 0.053

V10 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 1 0.053

V11 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 0.053 1

Jennrich test: x2= 30.1976, p-value= 0.9973857 (Ho: all the correlations are equal)

Table 6. Mean Vector of data sets

Data Mean Vector

Set V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11

Adela 87821 87785 88185 87680 88436 87564 88660 88424 87703 89191 88471

ide .85 .72 .84 .8 .53 .24 .64 .98 .94 .15 .47

88115 86299 87054 87490 87172 87227 87479 87259 86110 88026 87450

Perth .64 .28 .98 .71 .86 .25 .91 .51 .12 .43 .23

Table 7: Confidence Intervals for tests of the ' "Wave Energy Converters Data Set

Confidence 90% 95% 99%

Intervals L U LB L U LB L U LB

Asymptotic 0.5567 0.5587 0.5577 0.5566 0.5588 0.5587 0.5565 0.5589 0.5588

Bootstrap 0.5563 0.5612 0.5568 0.5558 0.5617 0.5563 0.5549 0.5626 0.5553

4. Conclusions

In this article, we proposed a method to estimate the stress-strength reliability and all unknown parameters under the equi-corelated multivariate normal setup. We provide MOM method to estimate these unknown parameters and use them to estimate of S and R. We also obtain the asymptotic distribution of estimated S. The simulation results indicate that performance than MOM in terms of MSE, RB and MAE for different choices of the parameters. Simulation studies illustrate that, the MSE, RB and MAE of this estimator reduce as the sample size increases and they almost achieved the true value of R. Also, the simulation studies illustrate that the proposed method has the best coverage probability and also produces the shortest band of confidence intervals. The stress-strength reliability of the given data set is R=0.55872. The performance of method of moments estimator (MOM) of R is consistent for different sample size and quite good for small sample size.

References

[1] Birnbaum, Z. W. (1956). On a use of the Mann-Whitney statistic. Proc. of the third Berkeley Symposium on Mathematical Statistics and Probability, 1:13-17.

[2] Z. W. Birnbaum and R. C. McCarty. (1958). A distribution-free upper confidence bound for P(Y<X) based on independent samples of X and Y. Annals of Mathematical Statistics, 29:558-562.

[3] Nadarajah, S. (2002). Reliability for beta models. Serdica Mathematical Journal, 28: 267-282.

A. Goswami, B. Seal RT&A, No 4 (71) STRESS-STRENGTH RELIABILITY FOR EQUI-CORRELATED_Volume 17, December 2022

[4] Nadarajah, S. (2003). Reliability for extreme value distributions. Mathematical and Computer Modelling, 37:915-922.

[5] Nadarajah, S. (2003). Reliability for lifetime distributions. Mathematical and Computer Modelling, 37:683-688.

[6] Nadarajah, S. (2004). Reliability for Laplace distributions. Mathematical and Computer Modelling, 2:169-183

[7] Nadarajah, S. (2004). Reliability for Logistic distributions. Engineering Simulation, to appear, 26:8198.

[8] Weerahandi, S. and Johnson, R. A. (1992). Testing Reliability in a Stress-Strength Model When X and Y Are Normally Distributed. Technometrics, 34(1):83-91.

[9] Simonoff, J. S., Hochberg, Y., and Reiser, B. (1986). Alternative estimation procedures for Pr(X<Y) in categorized data. Biometrics, 42:895-907.

[10] Sen, P. K. (1967). A note on asymptotically distribution-free confidence bounds for P(X<Y) based on two independent samples. Sankhya, 29:95-102.

[11]Govindarazulu, Zakkula. (1967). Two-Sided Confidence Limits for P(X <Y) Based on Normal

Samples of X and Y. Sankhya, 29(1/2):35-40.

[12] Church, J.D., Harris, B. (1970). The estimation of reliability from stress strength relationship.

[13] Gupta, R. D. and Gupta, R. C. (1990). Estimation on Pr(a'x > b'y) in the Multivariate Normal Case. Statistics: A Journal of Theoretical and Applied Statistics, 21(1):91-97.

[14] Reiser, B. and Farragi, D. (1994). Confidence bounds for P(a'x>b'y). Statistics, 25:107-111.

[15] Enis, P. and Geisser, S. (1971). Estimation of the Probability that Y < X. J. Amer. Statist. Assoc., 66: 162-168.

[16] Johnson, R. A. and Wichern, D. W. Applied Multivariate Statistical Analysis, 6th Edition, Pearson, 2008.

[17] Srivastava, M. K., Khan, A.H. and Srivastava, N. Statistical Inference: Theory of Estimation. PHI Learning Pvt. Ltd, 2014.

[18] Chambers, J.M., Cleveland, W.S., Kleiner, B. and Tukey, P. A. (1983). Graphical Methods for Data Analysis. Duxbury Press, Boston, MA, 11-16.

[19] Royston, P. (1982). An extension of Shapiro and Wilk's W test for normality to large samples. Applied Statistics, 31:115-124.

[20] Jennrich, R. I. (1970). An Asymptotic x2 Test for the Equality of Two Correlation Matrices. Journal of the American Statistical Association, 65:904-912.