LEVERAGING RANK SET SAMPLING FOR ENHANCED STRESS-STRENGTH ESTIMATION IN THE CONTEXT OF NAKAGAMI DISTRIBUTION Текст научной статьи по специальности «Математика»

LEVERAGING RANK SET SAMPLING FOR ENHANCED STRESS-STRENGTH ESTIMATION IN THE CONTEXT OF NAKAGAMI DISTRIBUTION

Surinder Kumar1, Rahul Shukla1, Bhupendra Meena1, Shivendra Pratap Singh1

1 Department of Statistics, Babasaheb Bhimrao Ambedkar University, 226025, U.P, India [email protected] Corresponding Author: [email protected] [email protected] [email protected]

Abstract

This study addresses the estimation of the stress-strength reliability model, where stress and strength both following the Nakagami distribution. While conventional approaches have relied on simple random sampling (SRS) for estimating reliability models, recent research suggests that ranked set sampling (RSS) offers a more efficient alternative. RSS yields more informative samples compared to SRS, potentially enhancing the accuracy of reliability estimations. Our investigation focuses on deriving maximum likelihood estimators (MLEs) for stress-strength under both SRS and RSS methodologies. To evaluate the comparative efficacy of these sampling techniques, we conduct a comprehensive Monte Carlo simulation study. The results of this analysis provide compelling evidence that RSS-based estimators outperform their SRS counterparts in terms of efficiency and precision. This research contributes to the growing body of literature supporting the adoption of RSS in reliability engineering. By demonstrating the superior performance of RSS in the context of Nakagami-distributed stress-strength models, we offer valuable insights for researchers and practitioners seeking to optimize their estimation procedures in reliability analysis.

Keywords: Stress-strength reliability, simple random sampling, ranked set sampling, Nakagami distribution, maximum likelihood estimation.

1. Introduction

The concept of stress-strength reliability plays a pivotal role in engineering decision-making, design optimization, and risk evaluation, particularly where safety, performance, and longevity are paramount. This analytical approach is indispensable for ensuring that engineered systems, structures, and components not only meet functional requirements but also withstand the challenges posed by fluctuating loads, environmental influences, and operational dynamics. At the core of reliability engineering and statistics lies the stress-strength model, which primarily aims to quantify the probability of system success or failure when both stress and strength are subject to

random variations. This methodology finds applications across diverse sectors, including engineering, materials science, quality control, and even finance. Within the framework of the stress-strength paradigm, the expression "Pr(Y < X)" represents the likelihood that a system's stress remains below its inherent strength. Essentially, this metric gauges the probability of system survival in the stress-strength model. Conversely, system failure occurs when the applied stress exceeds the material or component strength.

In the literature, the work on stress-strength model was first done by Birnbaum [2] and Birnbaum and McCarty [3]. The word stress-strength was first used by Church and Harris [4] in their research article, and they done a remarkable work under parametric and non-parametric inference. After that various authors choose different probabilistic models for estimating the stress-strength models. Some of these choices were summarised by Johnson [5]. A summary of all approaches and findings on the stress-strength model during the previous four decades was published by Kotz et al. [6]. The situation where X and Y are independent Type XII Burr random variables was examined by Awad and Gharraf [7]. For the recent development on this topic, one may refer to Chaturvedi and Kumar [8], Kundu and Gupta [9], Kundu and Raqab [10], Krishnamoorthy and Lin [11], Lio and Tsai [12], Barbiero [13], Chaturvedi and Kumari [14]. In all the above studies the authors have used the simple random sampling technique.

The ranked set sampling introduced by Mclntyre [1][1], gained importance when Halls and Dell [15] applied ranked set sampling to estimate forage yields under pine-hardwood forest. Takahashi and Wakimoto [16], Dell and Clutter [17], David and Levine [18] focused on the efficiency of the estimators based on RSS and they established that RSS outperforms its counterpart simple random sampling with an identical sample size. Expanding the horizons of RSS, Yu and Lam [19] and Chen [20] explored regression estimation based on this methodology, providing notable examples and results. Additionally, studies on the estimation of distribution functions under various RSS techniques were conducted by Stokes and Sager [21], Kvam and Samaniego [22], and Chen [23]. Zamanzade and Vock [24], Zhang et al. [25] and Ozturk [26] have yields insights into inferential procedures reliant on ranked set sampling.

To delve deeper into this specialized data collection technique, one may refer to the review papers of Kaur et al. [27], Bai and Chen [28], and Wolfe [29]. These review papers include all pertinent references on RSS, including historical development, current status and future research direction. Hassan et al. [30] obtained the point and interval estimators of P = Pr(Y < X) for the case of independent Gompertz random variables with common scale and different shape parameters based on RSS.

Here, we have consider the estimation of P = Pr(Y < X) with a focus on situation where the random stress Y and random strength X are two independent Nakagami random variables with shape parameters (a1,a2) and scale parameters (A, A), respectively. The point estimator of P = Pr(Y < X), is obtained using the maximum likelihood method based on both SRS and RSS, and the efficiency of this method based on SRS and RSS is compared. In Section 2, we present a brief overview about the Nakagami distribution and its relationship with other probability distributions. Point estimation of the parameters is given in Section 3. Section 4 and Section 5 comprises the point estimation of stress-strength model under SRS and under RSS, respectively. A simulation study employing the Monte Carlo method is discussed in Section 6. Section 7 details an empirical data analysis, and lastly Section 8 provides concluding remarks for the paper.

2. Preliminary

Consider a random variable X that adheres to the Nakagami distribution, denoted as NAD (a, A). In this distribution, a represents the shape parameter, which is bounded by the condition a > 0.5, while A symbolizes the scale parameter, constrained to be strictly positive (A > 0). For this distribution, the probability density function (PDF) and cumulative distribution function (CDF) are characterized as follows:

f(x-,a,X) x(2a~1)exp^-^^x2y,x > 0,a > 0.5,A > 0

(1)

and

F{x) = ix2)\x > 0, a > 0.5, A> 0

(2)

Where, y(a,x) = r ta~1e~tdt is the lower incomplete gamma function.

The reliability function of NAD (a, A) is

R(t) = 1 Xt2);t > °-a > > 0

(3)

The hazard rate of NAD (a, A) is.

(4)

Other distribution relationships

1) If a = 0.5, then Nakagami distribution (a, A) becomes Half Normal Distribution.

2) For a = l, then Nakagami distribution (a, A) reduces to Rayleigh Distribution.

3) If random variable Y ~ Gamma (a, A) where a is shape parameter and A is scale parameter, then VF - NAD (a,aX).

Let us draw a random sample X1,X2, ...,Xn from the NAD (a,A) of size n. The likelihood function of the Nakagami distribution NAD (a, A) is given by

where la is integer-valued.

3. Point estimation of the parameters

Theorem 1. The Maximum Likelihood Estimator of scale parameter A is

n

/1=-M

n

Theorem 2. The Maximum Likelihood Estimator of shape parameter a is

a =

0.5

f n \

- 2

x.

log

£

¿=1

n

V

- 2

V n ,-=1

Z logj

Proof. If we suppose that A is known, then the likelihood function for the parameter a is given as

n ✓ -n V

(2aa)r

L(a|x) =

(ra)n(X)r

The log likelihood function is

n

logL = nlog2 — nlog(ra) + nalog(a) — nalog(A) + — log(Xj) — y ^

¿=i

Partially differentiating with respect to a, and equating it equal to zero, we get

¿=1

From theorem 1

OC =

0.5

log A - 21 2 logj

n

. ¿=1

f n \

UK

i=1

n

V J

-1

OC -

0.5

n

2

x

log

i=1

n

V J

- 2

Z logx,

Vn ,'=1 y

4. Point estimation of P = Pr(Y < X) in case of simple random sampling

To derive the stress-strength reliability model P = Pr{Y < X), here we assumed that X is the strength variable and Y is the stress variable, both are following the Nakagami distribution with common scale parameter A >0 and different shape parameters a± > 0.5 and a2 > 0.5, respectively. By notation X ~ NAD (a1,A) and Y ~ NAD (a2,A), then

P = I P (Y < X)f{x)dx o

f 1 / a2 n 2 (tti\ai ,, ( a1

= -via?, — xzj-[ —) xlza± vex-p\— — x'L)dx

J ra2 V 2' X ) ra^X > >

o

I _ V*' x'm \

where y(n, x) = Tn 1 — e x J, - is the lower incomplete gamma function.

\ m=0 "i! /

1

1

S Kumar, R Shukla, B Meena, SP Singh RT&A, No 1 (82)

RANK SET SAMPLING FOR STRESS-STRENGTH Volume 20, March 2025

fa2x2/ \

- f x(^-i)exp(-^x2)exp(-^x2) £ V ^

dx

m\

m=0

- 2 1 — i - (-)ai r - "Y lai ( f(gl + m) )

yi) ra1\2\a1) a± Zj 2m! \{a1 + a2)a i+mJ

m=0

(«i)"1 r(a1+m)

ra1 Z.j ml (a1 + a2)a i+m

m=0

Let two independent random samples X and Y of size n and m are drawn from Nakagami distribution with parameters (a1,A)and (a2,A), respectively. For known À, the invariance characteristic of the maximum likelihood estimator provides the maximum likelihood estimator for P. The maximum likelihood estimators of a1 and a2 are

0.5 0.5

a

1SRS

f

log

A

and a

2SRS

i=1

f

- 2

V n i=i

I log-

A

log

f m \

Z yk2

k=1

m

(

- 2

1 m

1 Yj l0g yk

A

V mk=i

Maximum likelihood estimator of P in case of simple random sampling is given by

«1SRS~ 1

OW)™ v («2sfls)m r(a2SRS + m)

PSRS — 1 _

1 SRS

I

m\ (a1SRS + (22SflS)ais«s+m

2

X

n

5. Point estimation of P = Pr(Y < X) in case of ranked set sampling

1. Standard ranked set sampling

Ranked set sampling (RSS) represents a cutting-edge approach in statistical sampling, designed to boost the accuracy of parameter estimation, particularly in scenarios where resources are scarce or data collection costs are prohibitive. This method diverges from traditional random sampling by utilizing the ranked order or order statistics of sampled observations, thereby enhancing the quality and efficiency of estimations. The concept of RSS, initially proposed in the mid-20th century, has since gained traction across diverse fields such as environmental science, forestry, and ecology. Its popularity stems from its ability to yield robust statistical insights even when comprehensive population surveys are unfeasible. By offering a pragmatic and economical alternative to conventional sampling techniques, RSS has become an invaluable asset for researchers and statisticians aiming to refine their sampling strategies. The implementation of RSS to generate a sample of size n = r*m involves a series of structured steps, where m denotes the number of sample units selected in each cycle (of fixed size) and r represents the total number of cycles. These steps are executed sequentially as follows:

1. A random subset of the population consisting of m2 units is selected.

2. The m2 units are then divided arbitrarily into m sets, each containing m units.

3. The units within each set are ranked based on either professional judgment or correlation

with the variable of interest.

4. An individual quantile sample is constructed by taking the lowest ranked unit from the first set, the second lowest ranked unit from the second set, and continuing in this fashion.

5. To obtain a larger sample of size n = r*m, steps 1 through 4 can be repeated for r cycles.

The ranked set sampling (RSS) method takes only one observation from each set in each cycle. In the first cycle, it chooses the lowest observation X^11-)r. In later cycles, it independently selects the second lowest X(-12)r from a different set of m observations and the highest X(mm-jr from the final set of m. Let i = 1,2, ...,m; k = 1,2,... ,r, bea ranked sample set with set size m and r cycles. For

convenience, this paper will use the notation in place of the full description.

2. The maximum likelihood estimation of P = Pr(Y < X) in case of RSS

Let X(ij),i = 1,2,... ,r±; j = 1,2,... ,m1, denote the ranked set sample of size nt = r1m1 from Nakagami distribution with parameter {ai,X), where m1 is the set size and r-t is the number of cycles and Y(ki), k = 1,2,... ,r2) I = 1,2,... ,m2, denote the ranked set sample of size n2=r2m2 from Nakagami distribution with parameter (a2,A), where m2 is the set size and r2 is the number of cycles. Then the PDF of X^ and Y(k¿) are given by

fM = (¿-dX-0! - №)]mi_i/(*y)

3k(.yki) =

m2l

(fc-l)!(m2-fc):

[Fy(y)]k-1[l-FY(y)]m2-^g(ykl)

(6) (7)

Now the likelihood function is given as

7*1 m1 7*2 m2

= nn^nn^

1=1 j=1 r1 m1

= nn«

k = 1 1 = 1

¿=1 j=i

(t — 1)! (m1 — t)!

Tr.[Fx{x)]i-1[l-Fx{x)]m^-if{xij)

7*2 m2

nn

k=1 1=1

m2\

ri m!

(fc-l)!(m2-fc)!

7*2 m2

[FY(y)]k-1[l-FY(y)]m2-^g(ykl)

Letu

=nn (t — 1)! (m1 — t)! ' =nn

i=l ]=1 k=1 1=1 rt m1

1 = I! n«^)]'"1!1 - M*y)P_l/(*y)

r2 m2

\v[FY{ykl)]k-1[l-FY{ykl)}m--kg{ykl)

i=l j=1

(8)

k=1 1 = 1

r1 m1

i=1 j=1 2

Y

alxij'

ra1-Yya1,

r2 m2

alxij'

m1—i

v 2a1—l

Fa2 - Y I a2,

a2yu'

m2—k

k = 1 1 = 1

A

1 N™2"1

2'

V\a2,

a2yki'

k-1

exp |

i-Ty*')

sum of lower incomplete gamma function and upper incomplete gamma function is a gamma function, which implies

a 1x

1 xij

+ r[a1,

alxij'

= fa-.

gives

Thus,

YI

alxij'

~ \7a~J \7a.yj)

r± m1

nn[>

i = 1 ;=1 L r2 m2

nn[-

¡,— 1 1 — 1 L

„ , aixij2\ „( aixij2 ra1-y[ alt—-— I = r I alt——

IrJ (tJ (r) uv

r[a1,

alxij'

m1—i

2 a1-l

exp

k=1 ¿ = 1

71 a2'"

a2yfci/

rla2,

a2Vki'

m2—k

ykl2a2 1exp(--^yfc;2)

Taking log on both sides logL = logLi +logL2 where

"l(mi-l)

r! m1

(^"'ernnt

m1—i

r 2a±-l lij

exp

1=1 j = 1 ("ir1«2)

71 «i.

alxij'

and

(10)

(11)

(12)

7*2 m2

1 x "2^2-1^ 2 \"2/a^"2"2 rr FT i z z k=1 ¿=1 L

71 «2-

a2yki

k-1

(13)

This implies,

logLi = logu + n1{m1 - l)(-logfa1) + n1(log2 - logf^) + n1a1(\oga1 - log!)

ri rn1

+

^^(i-l)log

i = 1 7 = 1

71«!

a1xij'

ri rn1

f! nil

fi "ii

i=l 7 = 1

rK,

a1xij'

+ (2a1-l)YJYj\ogxij-^YjJjxfj

¿ = 1 7 = 1

i = l 7 = 1

Differentiating Eq.(5.2.9) with respect to ax and a2 respectively, we get

d\ogL1 da-,

r1 m1

■ = —m

logra± + n1 (log«! + 1) - n±\ogX + Z _ ^ l0g [

ri rn1

i = l j = 1

71 «i-

alxij'

¿=1 7=1

rl«!,

aixij'

ri rn1

ri m1

¿=1 7=1

¿ = 1 7 = 1

(15)

and

dlog^ 3a9

r2 m2

= -m2—logPa2 +n2(loga2 + 1) -n2logA+ _ ^ 3^Tlog K i"2'

2 fc=i¿=i 2 L V

7*2 m2

rla2,

a2yfci'

7*2 m2

7*2 m 2

J fc=l ¿=1 1—1 1-1

fc=li=l z Differentiating Eq.(5.2.6) with respect to A, we get

ri mi

+1 -° li ^

; = 1 L X 7 J ¿ = 1 7 = 1

7*1 mi 7*2 rn2

II^-^+XXc*-!)^:

1 L

yfci

dlogL n1a1

ai

r! 1»!

fc=li=l

(16)

¿=1 7 = 1

fc=l i=l

V I a2'~

a2y

r2 m2

k=l1=1

r[a2,

a2y

+

r2 m 2

III

k=l k = l

(17)

A numerical approach is utilized to obtain the maximum likelihood estimates for ai and a2, denoted as by aiRSS and a2RSS, from equations 5.2.10 and 5.2.11, respectively using the ranked set sampling method. Applying the invariance property of maximum likelihood estimators, the maximum likelihood estimate of the reliability parameter P based on RSS, denoted PRSS, can then be derived as

PRSS — 1

«1RSS "I

(a1RSS)aiRSS ^ (a2RSS)

r(cc2RSS+m)

fa

irss " (a1RSS + a2RSS)aiRss+™

m=0

6. Simulation study

In this section we carried out a simulation study. Bias and mean square error (MSE) for P are provided by Bias^P^ = E(P — P) and MSE^P^ = E(P — P)2; respectively to compare our suggested reliability estimator P based on ranked set sampling RSS with the conventional reliability estimator of P based on SRS. The formula for calculating the relative efficiency RE of the estimator of P is

MSE(Psrs)

MSE^p—j • Relative efficiency values greater than one suggest that the PRSS is more efficient than

the PSRS. All computations are performed using the R programming language. The simulation study is explained in the following steps.

Step 1: We generate 1000 simple random samples of X1,X2, ... ,Xn±, and Y1,Y2, ... ,Yn2 from Nakagami distribution with the sample sizes of (nx, n2) = (15, 15), (15, 20), (15, 25), (20, 20), (20, 25), (25, 25) in Case 1 and (20, 20), (20, 30), (20,40), (30, 30), (30, 40), (40,40) in Case 2. Step 2: We generate 1000 ranked set samples of X1±, ... ,Xm±ri and Y11, ... ,Ym2r2 from Nakagami distribution for the first case when the number of cycles is taken as r1 = r2 = 5 with set sizes m1 = m2 = 3,4,5 and for the second case when the number of cycles is taken as r-^ = r2 = 10 with set sizes mt = m2 = 2,3,4.

Step 3: To generate the simple random samples and ranked set samples for Nakagami distribution,

we consider the true value of the common scale parameter A = 3 and the true values of the shape parameter ax and ay are (0.5, 0.9), (0.7, 1.2) and (0.9, 1.5), respectively for the strength variable X and the stress variable Y, respectively. For these values, the true value of stress-strength model P is 0.40238, 0.50290 and 0.58635, respectively.

Step 4: The Biases, MSES and relative efficiency are presented in the Table 1.

Table 1: Biases, MSES and RE of P under SRS and RSS when the common scale parameter A = 3

SRS RSS

Case-1 N = 5

(a±, a2) (n1,n2) (m1,m2) PTrue PSRS Bias MSE PRSS Bias MSE RE

(0.5,0.9) (15,15) (3,3) 0.40238 0.38205 -0.02033 0.007075 0.37479 -0.02759 0.005877 1.2037

(15,20) (3,4) 0.37976 -0.02262 0.006309 0.36312 -0.03926 0.005434 1.1609

(15,25) (3,5) 0.36858 -0.0338 0.005457 0.35188 -0.05050 0.005283 1.0328

(20,20) (4,4) 0.36992 -0.03247 0.005543 0.36500 -0.03738 0.004656 1.1903

(20,25) (4,5) 0.36951 -0.03287 0.004886 0.36047 -0.04190 0.004239 1.1522

(25,25) (5,5) 0.36761 -0.03477 0.004822 0.35752 -0.04486 0.004346 1.1095

(0.7,1.2) (15,15) (3,3) 0.50290 0.48768 -0.01521 0.0081549 0.48285 -0.02004 0.007379 1.1051

(15,20) (3,4) 0.48781 -0.01509 0.0071741 0.47443 -0.02846 0.006434 1.1149

(15,25) (3,5) 0.48497 -0.01792 0.0064565 0.46760 -0.03529 0.005642 1.1442

(20,20) (4,4) 0.48402 -0.01887 0.0065401 0.47210 -0.03079 0.005534 1.1816

(20,25) (4,5) 0.48118 -0.02171 0.0058446 0.46763 -0.03527 0.004607 1.2684

(25,25) (5,5) 0.47852 -0.02437 0.0056632 0.46007 -0.04282 0.005261 1.0762

(0.9,1.5) (15,15) (3,3) 0.58635 0.58608 -0.00026 0.009415 0.56558 -0.02759 0.008915 1.0560

(15,20) (3,4) 0.57906 -0.00729 0.008066 0.56648 -0.03926 0.006912 1.1669

(15,25) (3,5) 0.57434 0.57434 0.006819 0.55942 -0.05050 0.005731 1.1897

(20,20) (4,4) 0.57625 -0.01010 0.007531 0.55986 -0.03738 0.006197 1.2151

(20,25) (4,5) 0.56799 -0.01836 0.006249 0.54742 -0.04190 0.005974 1.0460

(25,25) (5,5) 0.56607 -0.02027 0.005941 0.55713 -0.04486 0.004836 1.2282

Case-2 rt = r2 = 10

(a±, a2) (n1,n2) (m1,m2) PTrue ?SRS Bias MSE Prss Bias MSE RE

(0.5,0.9) (20,20) (2,2) 0.40238 0.37497 -0.02741 0.005234 0.37116 -0.03121 0.005073 1.0316

(20,30) (2,3) 0.36741 -0.03497 0.004678 0.36359 -0.03879 0.004379 1.0682

(20,40) (2,4) 0.36257 -0.03980 0.004350 0.36089 -0.04149 0.003852 1.1290

(30,30) (3,3) 0.36183 -0.04055 0.004494 0.36021 -0.04216 0.004249 1.0574

(30,40) (3,4) 0.36209 -0.04028 0.004011 0.35657 -0.04581 0.003841 1.0444

(40,40) (4,4) 0.35891 -0.04346 0.004020 0.35649 -0.04589 0.003606 1.1146

(0.7,1.2) (20,20) (2,2) 0.50290 0.48040 -0.02249 0.007313 0.48090 -0.02200 0.006477 1.1290

(20,30) (2,3) 0.47824 -0.02465 0.005291 0.47161 -0.03129 0.004850 1.0908

(20,40) (2,4) 0.47323 -0.02966 0.004726 0.46726 -0.03563 0.004282 1.1037

(30,30) (3,3) 0.47157 -0.03132 0.004867 0.46328 -0.03961 0.004715 1.0322

(30,40) (3,4) 0.46866 -0.03423 0.004614 0.46346 -0.03943 0.004210 1.0958

(40,40) (4,4) 0.46736 -0.03553 0.004181 0.46037 -0.04253 0.003984 1.0494

(0.9,1.5) (20,20) (2,2) 0.58635 0.57266 -0.01369 0.007225 0.56863 -0.01771 0.006831 1.0576

(20,30) (2,3) 0.56358 -0.02276 0.006070 0.55271 -0.03364 0.005913 1.0265

(20,40) (2,4) 0.56086 -0.02548 0.004649 0.55686 -0.02948 0.004561 1.0192

(30,30) (3,3) 0.55881 -0.02753 0.005109 0.55648 -0.02986 0.005039 1.0137

(30,40) (3,4) 0.55972 -0.02662 0.004832 0.55016 -0.03619 0.004452 1.0853

(40,40) (4,4) 0.55734 -0.02901 0.004548 0.55338 -0.03296 0.003982 1.1421

It is evident from the Table 1 that the relative efficiency is greater than one in every case; so, we can say that the ranked set sampling is showing more efficient results in comparison to simple random sampling in estimating the stress-strength reliability.

7. Real data application

In order to comprehend and provide a broad illustration of the processes covered in the preceding sections, we now take two real data sets. The first data set is used for the strength variable X and second data set is used for the stress variable Y in the stress-strength model P = Pr(Y < X).

7.1 First Data Set

Lawless (2003, pp. 267) is the source of the data set. The first report on this was published in 1987 by Schat, Staton, Mandel, and Shott. The hours to failure of 59 conductors with a length of 400 micrometres are represented by this data. The specimens are tested at the same temperature and current density, and at a specific high temperature and current density, they all failed. The MLES of the parameters a and A for this dataset is ax = 4.6731 and Ax = 51.2823

7.2 Second Data Set

The second data set is taken from Murthy et al. (2004, pp.180). This data represents 50 items that are put on use at time t = 0 and failure times are recorded (in weeks). The MLES for the parameters a and A for this dataset is ay = 0.1924 and Xy = 144.2292. Both the datasets are shown in Table 2.

Table 2: Dataset - 1 supposed to be X - Population and Dataset-2 supposed to be Y - Population

Dataset-1 : X-Population___Dataset-2 : Y-Population

6.545 6.522 7.945 7.224 0.013 2.838 7.291

9.289 4.137 6.869 7.365 0.065 3.269 7.087

7.543 7.459 6.352 6.923 0.111 3.977 7.787

6.956 7.495 4.7 5.64 0.111 3.981 8.596

6.492 6.573 6.948 5.434 0.163 4.52 9.388

5.459 6.538 9.254 7.937 0.309 4.789 10.261

8.12 5.589 5.009 6.515 0.426 4.849 10.713

4.706 6.087 7.489 6.476 0.535 5.202 11.658

8.687 5.807 7.398 6.071 0.684 5.291 13.006

2.997 6.725 6.033 10.491 0.747 5.349 13.388

8.591 8.532 10.092 5.923 0.997 5.911 13.842

6.129 9.663 7.496 1.284 6.018 17.152

11.038 6.369 4.531 1.304 6.427 17.283

5.381 7.024 7.974 1.647 6.456 19.418

6.958 8.336 8.799 1.829 6.572 23.471

4.288 9.218 7.683 2.336 7.023 27.777

(a) PDF Plot (b) CDF Plot (c) P-P Plot

Figure 1: The PDF, CDF and P-P Plots of the Nakagami distribution for First dataset

Figure 2: The PDF, CDF and P-P Plots of the Nakagami distribution for Second dataset

Before we dive into the core of our investigation, it's crucial to thoroughly examine the key characteristics of our data. To validate the strength of our results, we employ a powerful statistical instrument: the Kolmogorov-Smirnov (K-S) test, along with its corresponding P-value (P-V). This approach enables us to measure how well our empirical observations align with theoretical expectations.

Our analysis yields promising outcomes. For the first dataset, we calculate a K-S distance of 0.06779 and a P-V of 0.99940. The second dataset produces similar results, with a K-S distance of 0.12 and a P-V of 0.86428. These metrics provide compelling evidence that our model closely matches the observed data.

To enhance our understanding and provide visual context, we have created a series of graphical representations. These illustrations, found in the accompanying Figure 1 and 2, offer a comprehensive view of our statistical findings. They include probability-probability (PP) plots, as well as visualizations of the estimated probability density function (PDF) and cumulative distribution function (CDF) for both datasets. These visual aids serve to reinforce and clarify the numerical results of our analysis

We consider these two datasets as our random strength X and random stress Y, respectively. The MLES for a and A i.e. ax= 4.6731, Ax = 51.2823 and ay = 0.1924 and Xy= 144.2292 is taken as the true value of the parameters for this study. Now if ax = a± = 4.6731 and ay = a2 = 0.1924 then the true value of the stress-strength model from Eq.(4.1) is P = 0.1718.

In this analysis, we draw simple random samples of size 10 from each dataset and estimate the MLES for a and X, respectively. The simple random samples and MLES are presented in Table 3 and Table 4, respectively.

Table 3: MLES of a and A for each random sample of X-Population _Simple Random Samples from X-Population_

Sample 1 7.489 5.589 7.495 4.137 6.492 8.687 6.538 8.532 7.683 7.459 6.7031 50.8427

Sample 2 4.531 6.956 6.923 10.491 7.937 8.12 4.137 4.7 5.589 6.129 3.1983 46.3841

Sample 3 8.532 6.087 7.489 6.958 6.522 6.545 7.398 8.591 6.923 7.496 7.9133 42.4977

Sample 4 6.033 6.352 8.799 2.997 5.64 6.956 11.038 6.522 7.024 5.009 2.6220 48.2145

Sample 5 5.807 7.398 5.923 7.945 10.092 4.706 11.038 6.369 7.459 5.589 3.7265 55.9926

Sample 6 10.491 6.071 8.591 7.024 7.495 4.706 4.137 4.531 6.369 6.352 3.2477 46.6938

Sample 7 2.997 4.137 4.7 9.254 7.683 6.923 7.974 8.687 6.573 6.545 2.6515 46.5869

Sample 8 5.923 9.663 6.515 5.589 10.491 6.948 7.495 11.038 4.531 6.492 3.3188 60.1161

Sample 9 5.807 6.956 7.459 7.496 8.687 5.64 5.434 4.288 11.038 5.381 3.5093 49.9915

SamplelO 6.948 6.538 5.434 7.365 5.589 7.543 4.706 4.7 7.489 9.218 5.7396 44.8218

Table 4: MLES of a and A for each random sample of Y-Population

Simple Random Samples from Y-Population

Sample 1 7.787 6.456 7.087 5.202 5.911 7.023 0.684 23.471 19.418 7.291 0.4860 124.543

Sample 2 5.202 7.023 1.284 6.018 9.388 1.647 4.789 0.684 10.713 0.163 0.3309 34.329

Sample 3 13.842 0.997 5.911 13.006 23.471 0.535 1.284 0.684 5.349 7.787 0.2469 103.923

Sample 4 11.658 17.152 23.471 7.787 8.596 0.163 6.456 13.006 4.789 0.013 0.2078 134.931

Sample 5 5.291 1.284 13.006 8.596 0.426 11.658 9.388 4.789 17.283 7.787 0.4596 87.918

Sample 6 0.426 8.596 2.838 1.647 6.572 0.111 7.291 0.111 1.829 13.388 0.2006 36.379

Sample 7 0.997 3.981 7.087 0.426 6.456 8.596 13.388 6.427 13.006 4.789 0.4675 59.545

Sample 8 23.471 2.336 3.269 6.427 0.163 4.849 4.52 7.787 5.291 5.349 0.3125 76.955

Sample 9 9.388 6.456 17.152 17.283 24.777 6.018 13.388 4.789 0.163 10.713 0.4078 168.979

SamplelO 0.747 2.838 0.309 8.596 5.202 10.713 13.006 19.418 23.471 1.647 0.2632 132.424

Now, we draw 10 samples using ranked set sampling technique. We run two cycle (r = 2) of set size m = 5to get a ranked set sample of size n = r * m =10.

Table 5: MLES of a and A for each ranked set sample of X-Population

Ranked Set Samples from X-Population

âx ¿x

Sample 1 4.137 6.476 6.369 6.515 8.799 5.381 6.948 5.807 8.532 8.687 5.3408 47.8695

Sample 2 4.288 6.492 6.725 6.071 10.491 4.288 6.369 8.591 7.024 9.254 3.3715 52.0406

Sample 3 4.288 6.869 6.071 7.496 11.038 5.009 6.071 7.495 7.495 11.038 3.0263 57.6589

Sample 4 7.683 7.489 6.515 9.218 6.476 6.958 4.288 6.492 7.543 7.543 8.1074 50.7209

Sample 5 4.137 6.958 6.545 6.956 11.038 4.706 6.476 8.12 7.543 9.289 3.5078 55.1790

Sample 6 4.531 6.522 7.683 7.024 10.092 4.288 6.033 7.489 7.937 7.398 4.5176 50.1875

Sample 7 4.137 6.515 6.129 6.129 8.591 2.997 6.869 5.807 7.489 9.218 3.0288 43.9437

Sample 8 4.137 5.807 6.948 7.224 10.092 4.531 4.706 9.289 7.945 10.491 2.5873 55.5291

Sample 9 6.071 6.948 7.224 6.948 9.289 4.531 6.492 5.009 7.398 11.038 3.9824 53.6212

Sample 10 5.589 5.589 6.087 7.945 9.218 5.009 4.531 6.545 7.459 8.799 4.8530 46.9136

Table 6: MLES of a and A for each ranked set sample of Y-Population

Ranked Set Samples from Y-Population

ây K

Sample 1 5.291 0.997 4.52 17.152 9.388 0.065 1.284 1.647 13.842 11.658 0.24212 76.3620

Sample 2 0.013 7.787 2.336 6.456 17.152 3.977 0.747 11.658 19.418 17.283 0.21862 123.001

Sample 3 0.535 4.52 2.336 9.388 13.388 0.747 0.065 4.849 13.006 23.471 0.20209 103.7666

Sample 4 3.981 0.426 4.52 4.789 23.471 0.684 4.789 7.023 7.023 48.105 0.19412 304.6421

Sample 5 1.284 3.269 11.658 8.596 10.713 0.065 4.789 6.572 6.456 13.842 0.34995 63.6314

Sample 6 1.647 1.647 5.911 13.842 7.023 0.111 4.789 5.291 7.291 11.658 0.36021 52.1297

Sample 7 0.111 1.829 3.977 23.471 7.787 0.065 6.427 4.789 4.849 48.105 0.14340 303.2548

Sample 8 0.111 3.977 6.018 4.52 17.283 0.163 0.065 3.977 7.023 32.795 0.14298 151.186

Sample 9 0.111 5.202 0.535 4.52 7.023 2.336 0.426 2.336 13.006 5.911 0.25672 31.2303

Sample10 1.304 4.789 8.596 7.023 32.795 0.684 0.997 6.456 32.795 19.418 0.23438 271.9073

To obtain 10 samples of size 10, we conducted 20 cycles. Every pair of consecutive cycles makes up one sample of size 10. The 20 cycles we performed to get the 10 ranked samples from Population X. The ranked set samples from Population X, along with the corresponding maximum likelihood estimates of ax and Ax, are shown in Table 5. Similarly, we run the 20 cycles to draw ranked set samples from Population Y and the ranked set samples with maximum likelihood estimates of ay and Ay for Y Population are shown in Table 6.

Table 7: Bias, MSE and Relative efficiency of MLE of stress-strength model P in case of SRS and RSS

SRS RSS

a* ay P.'TRX Bias MSE a* ay Prss Bias MSE RE (%)

Sample 1 6.7031 0.4860 0.2725 0.1007 0.01494 5.3408 0.2421 0.2016 0.0298 0.00359 415.43%

Sample 2 3.1983 0.3309 3.3715 0.2186

Sample 3 7.9133 0.2469 3.0263 0.2020

Sample 4 2.6220 0.2078 8.1074 0.1941

Sample 5 3.7265 0.4596 3.5078 0.3499

Sample 6 3.2477 0.2006 4.5176 0.3602

Sample 7 2.6515 0.4675 3.0288 0.1434

Sample 8 3.3188 0.3125 2.5873 0.1429

Sample 9 3.5093 0.4078 3.9824 0.2567

SamplelO 5.7396 0.2632 4.8530 0.2343

An analysis of the statistical outcomes presented in the Table 7. This summary reveals a notable difference in the Mean Square Error (MSE) of the stress-strength model P between two sampling techniques. The Ranked Set Sampling (RSS) method demonstrates a significantly lower MSE compared to that obtained through Simple Random Sampling (SRS). Quantitatively, the relative efficiency (RE) of RSS surpasses SRS by a remarkable 415.43%. This substantial improvement in efficiency underscores the superior performance of RSS in practical applications. The findings strongly suggest that RSS offers more reliable and accurate results in real-world scenarios, outperforming the conventional SRS approach in the context of stress-strength modeling.

8. Conclusion

Delving into the realm of reliability engineering, this study sheds new light on the estimation of stress-strength models, with a particular focus on the intriguing Pr(Y < X) paradigm. Here, we explore the behavior of independent random variables Y and X, both dancing to the tune of the Nakagami distribution. While conventional wisdom has long favored simple random sampling, our research unveils a game-changing approach: ranked set sampling. By deriving maximum likelihood estimators for P under both sampling regimes, we set the stage for a riveting comparison.

Our simulation studies paint a vivid picture of ranked set sampling's superiority, showcasing its ability to outperform its traditional counterpart in efficiency. But we don't stop at theoretical musings - we put our findings to the test in the crucible of real-world data, where ranked set sampling continues to shine brightly.

As we draw the curtain on this investigation, one conclusion stands tall: in the arena of Nakagami stress-strength model estimation, ranked set sampling emerges as the undisputed champion over simple random sampling. Yet, this is not the end of our journey. The horizon beckons with tantalizing possibilities, as we set our sights on exploring the potential of other ranked set sampling methods in this critical field of study. The quest for ever-more efficient estimation techniques in stress-strength modeling continues, promising exciting developments in the future of reliability engineering.

Author Contributions: All authors have equal contribution. All authors reviewed the results and approved the final version of the manuscript.

Funding: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors

Conflict of interest: The Authors declares that there is no conflict of interest.

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