Estimation of P(X < Y) using ranked set sampling for ...

Quality Technology & Quantitative Management

ISSN: (Print) 1684-3703 (Online) Journal homepage: http://www.tandfonline.com/loi/ttqm20

Estimation of P(X  0, 𝜎 > 0,

(1)

and p ( ) x F x, p, 𝜎 = 1 − e− 𝜎 ,

x > 0, p > 0, 𝜎 > 0,

(2)

respectively. Here, p is the shape parameter and σ is the scale parameter. If the random ( var) iable X has a Weibull distribution with parameters p and σ, it is denoted by Weibull (p, 𝜎 .) System reliability ( R is)obtained as shown below under the assumption of X ∼ Weibull p, 𝜎1 and Y ∼ Weibull p, 𝜎2 ∞

R = P(X < Y ) =

∞

𝜎2 p p−1 − 𝜎xp p p−1 − 𝜎yp x e 1 y e 2 dydx = . ∫ ∫ 𝜎1 𝜎2 𝜎 1 + 𝜎2

(3)

x=0 y=x

In this paper, we obtain the ML estimates of the system reliability R by using two different approaches based on RSS, see also Akgül and Şenoğlu (2014) and Akgül (2015). First, we solve the likelihood equations iteratively. Second, an approximate solution for the likelihood equations is obtained by using a methodology known as modified maximum likelihood (MML) which is non-iterative, see Tiku (1967, 1968). Resulting estimators are called MML estimators. They have closed form and are easy to compute. It should also be noted that they are asymptotically equivalent to ML estimators and have high efficiencies, even for small sample sizes, see Tiku and Akkaya (2004). To the best of our knowledge, there has been no previous work on estimating the system reliability R based on RSS data when the stress X and the strength Y are independent Weibull. We compare the performances of the proposed estimators with the corresponding ML estimators based on SRS. An estimation of R, based on SRS, is considered by Kundu and Gupta (2006). They obtain the ML and the approximate maximum likelihood (AML) estimators of R. Therefore, we did not reproduce the results regarding an estimation of R based on SRS for the sake of brevity. We only provide results based on RSS.

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 3

The remainder of this paper is organized as follows: In Section 2, after the description of RSS, we obtain the ML and the MML estimators of R based on RSS. The results of a Monte Carlo simulation study are given in Section 3. In Section 4, the effect of imperfect ranking in RSS is investigated. A real-life example is given in Section 5, followed by comments and conclusions in Section 6.

2.  Estimators for the system reliability R In this section, we obtain the estimators for the system reliability R by using the ML and the MML estimation under RSS. We first give a brief description of RSS in the following subsection. 2.1.  Description of the RSS method RSS was first proposed by McIntyre (1952) to estimate the population mean. In many studies, it is used for estimating the population parameters. The results of these studies showed that estimators based on RSS are more efficient than their counterparts based on SRS. It is also beneficial in terms of cost and time. The procedure of RSS is summarized as follows. Step 1.  Select m random sets via SRS, each of size m. Step 2.  Without any certain measurements, rank the units with respect to virtual comparisons, expert opinion or auxiliary variables. Step 3.  Select a sample of size m containing the smallest ranked unit in the first set, the second smallest ranked unit in the second set and continue this process until the largest ranked unit in the last set is selected. Step 4.  Repeat Steps 1–3 r times, until the sample size is obtained as n = mr. Here, r and n refer to the number of cycles and the sample size, respectively. 2.2.  ML estimator of R Let X(i)ic, i = 1, …, mx, c = 1, …, rx be a ranked set sample of size ( n = m ) xrx for X where X(i)ic is the ith order statistics in the ith set of cth cycle from Weibull p, 𝜎1 and Y(j)jl, j = 1, …, my, l = 1, …, ry be a ranked set sample of(size m = m r for Y where Y(j)jl is the jth order statistics y y ) in jth set of lth cycle from Weibull p, 𝜎2 . Here, mx and my are the set sizes for X and Y, respectively. rx and ry are the number of cycles for X and Y, respectively. In the rest of the paper, we use the notations Xic and Yjl instead of the notations X(i)ic and Y(j)jl, respectively, for easy understanding and the simplicity. The pdf of Xic and Yjl are given by

( ) fXi xic =

( )]m −i ( ) [ ( )]i−1 [ mx ! 1 − F xic x f xic ( ) F xic (i − 1)! mx − i !

and

[ ( )]j−1 [ ( )]my −j ( ) ( ) my ! F y 1 − F yjl f yjl , fYj yjl = ( ( ) jl ) j − 1 ! my − i !

(4)

4 

  F. G. Akgül and B. Şenoğlu

respectively. Then the likelihood function based on RSS is given by ( )j−1 ( p )my −j+1 p r y my ( p )i−1 ( p )mx −i+1 rx m x y y x x jl jl ∏ p−1 ∏ ∏ p−1 pn+m ∏ − 𝜎ic −𝜎 − 𝜎ic − xic 1 − e 1 yjl 1−e 2 e 1 ⋅ e 𝜎2 L=C n m 𝜎1 𝜎2 c=1 i=1 l=1 j=1 (5)

where C =

∏rx ∏mx

mx ! i=1 (i−1)!(m −i)! x

∏ r y ∏ my

m!

y . (j−1)!(my −j)! ML estimators of the parameters p, σ1 and σ2 are the solutions of the following likelihood equations

𝜕 ln L 𝜕p

c=1

=

n+m p

− +

+ 1 𝜎1 1 𝜎2

r x mx ∑ ∑ c=1 i=1 rx m x

l=1

ln xic +

1 𝜎1

j=1

r x mx ∑ ∑

p

(i − 1)

c=1 i=1

xic ln xic p exic ∕𝜎1 −1

r m � p ∑y ∑y ln yjl mx − i + 1 xic ln xic +

∑ ∑�

c=1 i=1 ry my

∑∑�

l=1 j=1

� yjlp ln yjl

j−1

l=1 j=1

e

p y ∕𝜎2 jl −1

−

1 𝜎2

r y my

∑∑� l=1 j=1

(6)

� p my − j + 1 yjl ln yjl = 0,

p r x mx rx mx xic ) p 𝜕 ln L 1 ∑∑( n 1 ∑∑ + 2 mx − i + 1 xic = 0, =− − 2 (i − 1) p x ∕𝜎 𝜕𝜎1 𝜎1 𝜎1 c=1 i=1 e ic 1 − 1 𝜎1 c=1 i=1

(7)

p r y my r y my ( ) ) yjl 𝜕 ln L m 1 ∑∑( 1 ∑∑ p j−1 p my − j + 1 yjl = 0. =− − 2 + 2 y ∕𝜎 𝜕𝜎2 𝜎2 𝜎2 l=1 j=1 e jl 2 − 1 𝜎2 l=1 j=1

(8)

It is clear that there is no explicit solution for the likelihood equations. Therefore, we resort to iterative methods. The ML estimators of p, σ1 and σ2, based on RSS, are denoted by p̂ ML,RSS, 𝜎̂ 1ML,RSS and 𝜎̂ 2ML,RSS ,

respectively. By using the invariance property of the ML estimators, the ML estimator of the system reliability R based on RSS is as shown below

R̂ ML,RSS =

𝜎̂ 2ML,RSS 𝜎̂ 1ML,RSS + 𝜎̂ 2ML,RSS

.

(9)

2.3.  MML estimator of R In the previous subsection, we obtained the ML estimator of the system reliability R by using iterative methods. However, iterative methods can be problematic because of the following reasons: (i) convergence to wrong roots, (ii) convergence to multiple roots, and (iii) non-convergence of iterations, see, for example, Smith (1985); Puthenpura and Sinha (1986) and Vaughan (2002). To overcome these difficulties, we also used MML (Tiku, 1967, 1968), which is non-iterative and gives us explicit estimators of the unknown parameters. MML estimators of the parameters of the Weibull distribution are obtained via the Extreme Value (EV) distribution, see Tiku and Akkaya (2004). This is because, MML is applied to distributions belonging to the location-scale family. It should be noted that the distribution of the natural logarithm of a random variable X, having Weibull distribution is EV and hence belongs to the location scale family. The EV distribution has the following pdf

Quality Technology & Quantitative Management 

1 fU (u, 𝜇, 𝜂) = e 𝜂

(

u−𝜇 u−𝜇 −e 𝜂 𝜂

 5

)

− ∞ < u < ∞,

(10)

where μ ∊ R is the location parameter and η ∊ R+ is the scale parameter. If the random variable U has the EV distribution with the location parameter μ and the scale parameter η, then it is denoted by U ∼ EV(μ, η). Parameters of the EV distribution can be expressed in terms of the parameters of the Weibull distribution, i.e. μ = 1/p ln σ and η = 1/p. It is clear from these equalities that after obtaining the estimators of the parameters of the EV distribution, the scale and the shape parameters of the Weibull distribution are obtained by using the inverse transformations

𝜎 = e𝜇p and p =

1 , 𝜂

(11)

respectively. To obtain the MML estimators of the parameters μ1, μ2 and η, the likelihood function can be written as follows

L∝

��i−1 � � u −𝜇 ��mx −i � u −𝜇 � r x mx � � ∏ ∏ u −𝜇 F (i)c𝜂 1 1 − F (i)c𝜂 1 f (i)c𝜂 1 c=1 i=1 r y my � � ��j−1 � � v −𝜇 ��my −j � v −𝜇 � ∏ ∏ v −𝜇 . 1 − F (j)l𝜂 2 f (j)l𝜂 2 . F (j)l𝜂 2

1 𝜂

n+m

(12)

l=1 j=1

Here, z(i)c =

(( ) ) ) ) (( u(i)c − 𝜇1 ∕𝜂 and w(j)l = v(j)l − 𝜇2 ∕𝜂 are the standardized order sta-

tistics, respectively. U(i)c, i = 1, …, mx, c = 1, …, rx and V(j)l, j = 1, …, my, l = 1, …, ry are the ith and jth order statistics in the cth and lth cycles under the transformation U = ln X and V = ln Y. They are obtained by ordering the rank set samples Uic and Vjl in ascending order of magnitude. Then the likelihood equations are obtained as given below 𝜕 ln L 𝜕𝜇1

= − 𝜂1

𝜕 ln L 𝜕𝜂

r x mx � � � � � � ∑ ∑ mx − i g2 z(i)c (i − 1)g1 z(i)c + 𝜂1

c=1 i=1 r x mx ∑ ∑ � � g3 z(i)c − 𝜂1 c=1 i=1

r

𝜕 ln L 𝜕𝜇2

r x mx ∑ ∑

= − 𝜂1

m

c=1 i=1

r m � � � � � � ∑y ∑y � my − j g2 w(j)l j − 1 g1 w(j)l + 𝜂1

∑y ∑y �

l=1 j=1 r y my � � 1 ∑∑ g3 w(j)l −𝜂 l=1 j=1

l=1 j=1

= 0,

r x mx � � � � � � ∑ ∑ mx − i g2 z(i)c z(i)c (i − 1)g1 z(i)c z(i)c + 𝜂1 c=1 i=1 c=1 i=1 r y my � rx m x � � � � � ∑ ∑ ∑ ∑ g3 z(i)c z(i)c − 𝜂1 − 𝜂1 j − 1 g1 w(j)l w(j)l c=1 i=1 l=1 j=1 ry my � r m � � � � ∑ ∑y ∑y � ∑ my − j g2 w(j)l w(j)l − 𝜂1 g3 w(j)l w(j)l = 0. + 𝜂1 l=1 j=1 l=1 j=1

= − n+m − 𝜂

(13)

= 0,

1 𝜂

(14)

r x mx ∑ ∑

(15)

6 

  F. G. Akgül and B. Şenoğlu

The Equations (13)–(15) have no explicit analytical solutions because of the non-linear functions ( ) f � (z ) ( ) f (z ) ( ) f (z ) g1 z(i)c = F z(i)c , g2 z(i)c = 1−F (i)cz , g3 z(i)c = f z (i)c , i = 1, …, mx, c = 1, …, rx; ( ) ( ) ( (i)c () ) ( ) f (i)cw ( ) (i)c f � w ( ) f w(j)l ( (j)l ) ( ) g1 w(j)l = F w , g2 w(j)l = 1−F w , g3 w(j)l = ( (j)l), j = 1, …, my,l = 1, …, ry. ( (j)l ) ( (j)l ) f w(j)l Appropriate linear approximations for these non-linear functions ( are)given by the first ( ) u and E w(j)l = t vj , namely, two terms of a Taylor series expansion around E z(i)c = t(i) ( ) ( ) ( ) () g1 z(i)c ≅ 𝛼1iu − 𝛽1iu z(i)c, g2 z(i)c ≅ 𝛼2iu + 𝛽2iu z(i)c, g3 z(i)c ≅ 𝛼3iu − 𝛽3iu z(i)c,

i = 1,(… , m)x , c = 1, … , rx ; g1 w(j)l ≅ 𝛼1jv − 𝛽1jv w(j)l,

( ) g2 w(j)l ≅ 𝛼2jv + 𝛽2jv w(j)l,

( ) g3 w(j)l ≅ 𝛼3jv − 𝛽3jv w(j)l,

j = 1, … , my , l = 1, … , ry , where (

𝛼1iu

=

𝛼2iu u t(i)

u f (t(i) ) u F (t(i) ) u t(i)

+

u u 𝛽1i , t(i)

𝛽1iu

u u t(i) t(i) e( , 𝛽2iu

=

) u 2 t u e (i) −1 f (t(i) )F (t(i)u )+f (t(i)u ) 2

u t(i)

e)), 𝛼3iu

= = e (− = ln − ln 1 − m i+1

,

u F (t(i) ) u ( t(i)

=1−e

1−

u t(i)

, ) u u , 𝛽3i = et(i) ,

(16)

i = 1, 2, … , mx.

x

( )( ) ( ) Here, it should be noted that the coefficients 𝛼1jv , 𝛽1jv , 𝛼2jv , 𝛽2jv and 𝛼3jv , 𝛽3jv and as well as the expected values of the standardized ordered statistic t vj for the variable V are similar () with those given for U variable. Therefore, we did not reproduce them for the sake of brevity. Incorporating (16) in (13)–(15), we obtain the modified likelihood equations ∂ ln L*/ ∂ μ1 = 0, ∂ ln L*/ ∂ μ2 = 0 and ∂ ln L*/ ∂ η = 0. Solutions of these modified equations are the following MML estimators Δ1 𝜂̂MML,RSS , √m1 2 −B+ B +4AC , 2(n+m)

𝜇̂ 1MML,RSS = K1 − 𝜂̂MML,RSS =

𝜇̂ 2MML,RSS = K2 −

Δ2 𝜂̂ , m2 MML,RSS

(17)

where

� � 𝛿iu = (i − 1)𝛽1iu + mx − i 𝛽2iu + 𝛽3iu , Δui

1)𝛼1iu

�

𝛼2iu

�

𝛼3iu ,

m1 = rx

mx ∑ i=1 mx

𝛿iu , K1 =

∑

r x mx ∑ ∑ c=1 i=1

𝛿iu u(i)c ∕m1 ,

Δui ,

− mx − i + Δ1 = r x i=1 m r m � � � � ∑y v ∑y ∑y v v v v v 𝛿j = j − 1 𝛽1j + my − j 𝛽2j + 𝛽3j , m2 = ry 𝛿j , K2 = 𝛿j v(j)l ∕m2 , = (i −

� � � � Δvj = j − 1 𝛼1jv − my − j 𝛼2jv + 𝛼3jv ,

j=1 my

Δ2 = ry

∑ j=1

A = n + m, r y my � r x mx � ∑ ∑ v� ∑ ∑ u� B= 𝛥j v(j)l − K2 𝛥i u(i)c − K1 + c=1 i=1 r x mx

C=

∑∑ c=1 i=1

l=1 j=1

r y my �2 �2 ∑ � ∑ v� 𝛿iu u(i)c − K1 + 𝛿j v(j)l − K2 . l=1 j=1

l=1 j=1

Δvj ,

(18)

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 7

Using the inverse transformations defined in (11), the MML estimators of the parameters σ1, σ2 and p based on RSS are defined as

𝜎̂ 1MML,RSS = e p̂ MML,RSS =

p̂ MML,RSS 𝜇̂ 1 1 𝜂̂ MML,RSS

MML,RSS

,

𝜎̂ 2MML,RSS = e

p̂ MML,RSS 𝜇̂ 2

MML,RSS

(19)

,

respectively. By incorporating estimators in (19) into (3), we obtain the MML estimator of the system reliability R based on RSS as shown below

R̂ MML,RSS =

𝜎̂ 2MML,RSS 𝜎̂ 1MML,RSS + 𝜎̂ 2MML,RSS

(20)

.

The MML estimators are fully efficient when the regularity conditions hold, in other words their variances are equivalent to the Rao–Cramer lower bound, see (Vaughan, 2002).

3.  A simulation study We compare the performances of the proposed estimators of R, based on RSS, with the traditional estimators based on SRS using the bias and the MSE (mean ( error) ) cri( ) square

̂ ̂ teria.(The and the ) bias ( )2MSE of an estimator R̂ are given by Bias R = E R −R and MSE R̂ = E R̂ −R , respectively. We also report the relative efficiencies (REs) of the estimators of R based on MSE defined by the following

RE1 =

MSE(R̂ ML,SRS )

, RE2 =

MSE(R̂ ML,SRS )

MSE(R̂ ML,RSS ) MSE(R̂ MML,SRS )

, RE3 =

MSE(R̂ MML,SRS ) MSE(R̂ ML,RSS ) RE4 = and RE5 = MSE(R̂ MML,RSS ) MSE(R̂ MML,RSS )

MSE(R̂ ML,SRS ) MSE(R̂ MML,RSS )

, (21)

See Kundu and Gupta (2006) for the estimators of the system reliability R based on SRS. Larger values (> 1) of RE indicate that the efficiency of the estimator given in the denominator outperforms the estimator given in the numerator. All the computations were in Matlab ( performed ) ( 2013.)In the context of RSS, we consider different number of cycles rx , ry and set sizes mx , my shown below rx

ry

mx

my

rx

ry

mx

my

rx

ry

mx

my

1

1

5

 5 10 15 20 20

5

5

2

2 3 4 4 4

10

10

2

2 3 3

15 20

3 4

3

It is clear that when rx = ry = 1, the sample sizes (n, m) corresponding to X and Y are ( ) equal to the set sizes mx , my , respectively, since the sample sizes are defined as n = mxrx and m = myry. Using these equalities, sample sizes (n, m) for rx = ry = 5 and rx = ry = 10 can be found in a similar manner. Note that we use (n, m) as sample sizes in the Monte Carlo simulations for estimating R based on SRS.

8 

  F. G. Akgül and B. Şenoğlu

In the simulation study, we consider σ1 = 1, σ2 = 1, 2, 3 and p = 1.5 as in Kundu and Gupta (2006). However, we extended their simulation set-up by including the shape parameters p = 3 and 6 into the simulation study. All the simulations are based on [|100, 000∕min(n, m)|] Monte Carlo runs. Here, [| |] represents the greatest integer value. In Table 1, we obtain the following biases and RE values for the system reliability R. In terms of the bias criterion, it is clear from Table 1 that all the estimators have a negligible bias. However, R̂ ML,RSS has the smallest bias in all cases overall. It view of the efficiencies, the following conclusions can be drawn from Table 1. The estimators of the system reliability R, based on RSS are more efficient than the corresponding estimators based on SRS for all sample sizes and the shape parameters, see the columns for RE1 and RE4. It should also be noted that ( ) the efficiencies of the estimators based on SRS decrease as the number of cycles rx , ry increase. However, their efficiencies increase as the set sizes increase. According to the column RE2, the efficiencies of the ML and the MML estimators based on SRS are more or less the same for all cases. Our findings are in agreement with Kundu and Gupta (2006). When we consider column RE5, in contrast to column RE2, we see that the ML estimator of R based on RSS is more efficient than the corresponding MML estimator. In column RE3, the classical ML estimator based on SRS is compared with the proposed MML estimator based on RSS. It can be seen that the proposed estimator is much more efficient than the well-known ML estimator. It should be noted that it is chosen to increase the number of cycles and to reduce the set sizes in order to avoid a ranking error in the context of RSS. In this way, it is believed that the probability of imperfect ranking encountered in RSS decreases.

4.  Imperfect ranking In previous sections, we have considered the estimation of R under perfect ranking. However, in practice, imperfect ranking or making error in ranking is ineluctable, particularly when the set sizes are large. Therefore, in this section, we investigate the effect of imperfect ranking on the efficiencies of the estimators of R. In order to compare the performances of the ML and the MML estimators of R, based on RSS, under the assumption of imperfect ranking, we use the simulation method originated by Dell and Clutter (1972). For the variables of interest X and Y, we let the ranking be done with respect to the variables X* = X + e and Y* = Y + e, respectively. Here, we ( concomitant ) 2 2 take e ∼ Normal 0, 𝜏 and τ  = 0, .25, .5 and 1 in the simulation set-up. It is clear that X* and Y* become equal to X and Y when τ2 = 0. This is the case for the perfect ranking. The error in ranking increases when τ2 increases, see Zheng and Al-Saleh (2002). In the simulation study, we consider the set sizes mx = my = 2, 4 and 6 and the number of cycles rx = ry = 10. In addition, the shape parameter p is taken to be 3 and the scale parameters σ1 = σ2 = 1. In Table 2, the biases and the MSEs of the ML and the MML estimators of R under imperfect ranking are given. Both the ML and the MML estimators of R have negligible biases. It can also be seen that the ML estimator of R, based on RSS, is less affected by the effect of imperfect ranking than the corresponding MML estimator. This is because of the fact that the ML estimator outperforms the MML estimator for estimating the scale parameters σ1 and σ2.

Quality Technology & Quantitative Management 

 9

Table 1. Biases and REs for the estimators of R under SRS and RSS. Bias rx = ry

(mx, my)

Relative Efficiencies

(n, m) R̂ ML,SRS R̂ MML,SRS R̂ ML,RSS R̂ MML,RSS

RE1

RE2

RE3

RE4

RE5

2.83 3.43 6.94 7.95 8.94 1.42 1.57 1.59 2.11 2.34 1.45 1.60 1.90

1.05 1.01 1.01 1.01 1.00 1.02 1.01 .99 1.01 1.00 1.00 1.00 .99

2.70 3.02 6.47 7.37 8.39 1.24 1.25 1.10 1.80 2.09 1.22 1.13 1.67

2.57 2.97 6.39 7.29 8.33 1.22 1.24 1.10 1.78 2.08 1.21 1.13 1.67

.95 .88 .93 .92 .93 .87 .79 .69 .85 .89 .84 .71 .88

2.66 3.22 6.54 6.98 8.53 1.37 1.59 1.65 1.93 2.26 1.41 1.49 1.80

1.03 1.03 1.00 1.01 1.00 1.01 1.01 1.01 1.00 .99 1.00 1.00 1.00

2.47 2.82 5.88 6.12 7.51 1.18 1.33 1.24 1.67 1.98 1.12 1.17 1.54

2.37 2.73 5.84 6.06 7.48 1.16 1.31 1.23 1.66 1.98 1.11 1.16 1.53

.92 .87 .89 .87 .88 .85 .83 .75 .86 .87 .79 .78 .85

2.60 3.13 6.11 6.63 7.73 1.37 1.58 1.64 1.90 2.01 1.41 1.55 1.78

1.02 1.03 1.00 1.00 1.00 1.01 1.01 1.01 1.00 1.00 1.00 1.00 .99

2.34 2.72 5.30 5.60 6.60 1.10 1.33 1.26 1.60 1.67 1.03 1.22 1.42

2.29 2.62 5.28 5.57 6.56 1.09 1.31 1.24 1.58 1.66 1.03 1.21 1.42

.90 .87 .86 .84 .85 .80 .83 .77 .84 .82 .73 .78 .79

2.82 3.32 7.45 8.34 9.27 1.43 1.57 1.61 2.02 2.27 1.45

1.04 1.01 1.01 1.00 1.01 1.02 1.01 .99 1.00 1.00 1.00

2.69 2.90 6.92 7.76 8.68 1.24 1.25 1.12 1.72 2.09 1.21

2.56 2.86 6.83 7.70 8.59 1.21 1.23 1.12 1.71 2.07 1.21

.95 .87 .92 .92 .93 .86 .79 .69 .85 .92 .83

p = 1.5 𝜎1 = 1, 𝜎2 = 1

1

5

10

1

5

10

1

5

10

1

5

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4) (2,2) (2,3) (3,3)

(5,5) −.0014 (5,10) .0131 (15,15) .0010 (15,20) .0022 (20,20) .0024 (10,10) .0011 (10,15) .0064 (10,20) .0067 (15,20) .0027 (20,20) .0003 (20,20) −.0005 (20,30) .0026 (30,30) .0010

−.0011 .0226 .0008 .0045 .0024 .0012 .0105 .0129 .0050 .0004 −.0004 .0050 .0011

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4) (2,2) (2,3) (3,3)

(5,5) (5,10) (15,15) (15,20) (20,20) (10,10) (10,15) (10,20) (15,20) (20,20) (20,20) (20,30) (30,30)

.0119 .0193 .0064 .0049 .0056 .0075 .0098 .0104 .0050 .0025 .0015 .0042 .0025

.0072 .0242 .0050 .0059 .0047 .0052 .0116 .0144 .0058 .0016 .0008 .0057 .0021

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4) (2,2) (2,3) (3,3)

(5,5) (5,10) (15,15) (15,20) (20,20) (10,10) (10,15) (10,20) (15,20) (20,20) (20,20) (20,30) (30,30)

.0144 .0195 .0050 .0066 .0038 .0092 .0089 .0092 .0058 .0042 .0051 .0054 .0023

.0080 .0223 .0031 .0067 .0026 .0061 .0097 .0119 .0057 .0030 .0039 .0062 .0017

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4) (2,2)

(5,5) (5,10) (15,15) (15,20) (20,20) (10,10) (10,15) (10,20) (15,20) (20,20) (20,20)

−.0019 .0141 .0017 .0030 −.0007 .0007 .0057 .0062 .0028 −.0013 −.0013

−.0020 .0233 .0019 .0053 −.0007 .0005 .0098 .0124 .0050 −.0013 −.0013

−.0009 −.0010 .0034 .0198 −.0004 −.0004 .0007 .0036 .0001 .0002 −.0008 −.0007 .0032 .0297 .0034 .0457 .0020 .0177 .0004 .0004 −.0011 −.0015 .0016 .0289 −.0005 −.0006 𝜎1 = 1, 𝜎2 = 2 .0123 .0119 .0101 .0262 .0027 .0059 .0019 .0077 .0018 .0053 .0075 .0008 .0066 .0235 .0065 .0377 .0042 .0136 .0025 −.0007 .0025 −.0053 .0051 .0230 .0027 −.0032 𝜎1 = 1, 𝜎2 = 3 .0151 .0142 .0102 .0238 .0031 .0073 .0022 .0089 .0016 .0062 .0092 −.0001 .0079 .0191 .0073 .0313 .0061 .0119 .0040 −.0011 .0023 −.0083 .0035 .0158 .0031 −.0043 p = 3 𝜎1 = 1, 𝜎2 = 1 −.0008 −.0011 .0051 .0217 .0005 .0005 .0001 .0028 .0007 .0007 −.0005 −.0012 .0036 .0294 .0042 .0454 .0019 .0175 −.0001 −.0001 −.0017 −.0019

(Continued)

10 

  F. G. Akgül and B. Şenoğlu

Table 1. (Continued) Bias rx = ry 10

1

5

10

1

5

10

1

5

10

1

5

Relative Efficiencies

(mx, my) (n, m) R̂ ML,SRS R̂ MML,SRS R̂ ML,RSS R̂ MML,RSS (2,3) (3,3)

(20,30) (30,30)

.0014 .0001

.0039 .0001

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4) (2,2) (2,3) (3,3)

(5,5) (5,10) (15,15) (15,20) (20,20) (10,10) (10,15) (10,20) (15,20) (20,20) (20,20) (20,30) (30,30)

.0112 .0201 .0048 .0054 .0036 .0085 .0100 .0088 .0062 .0038 .0024 .0043 .0027

.0067 .0252 .0035 .0064 .0026 .0063 .0118 .0129 .0070 .0029 .0016 .0056 .0022

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4) (2,2) (2,3) (3,3)

(5,5) (5,10) (15,15) (15,20) (20,20) (10,10) (10,15) (10,20) (15,20) (20,20) (20,20) (20,30) (30,30)

.0141 .0183 .0061 .0068 .0047 .0075 .0098 .0104 .0061 .0050 .0030 .0045 .0034

.0079 .0211 .0042 .0070 .0035 .0045 .0105 .0132 .0063 .0038 .0019 .0053 .0027

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4) (2,2) (2,3) (3,3)

(5,5) .0001 (5,10) .0140 (15,15) .0019 (15,20) .0011 (20,20) .0020 (10,10) .0022 (10,15) .0050 (10,20) .0040 (15,20) .0004 (20,20) −.0001 (20,20) .0005 (20,30) .0022 (30,30) −.0005

.0003 .0232 .0019 .0034 .0021 .0021 .0089 .0102 .0024 −.0001 .0005 .0046 −.0006

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4)

(5,5) (5,10) (15,15) (15,20) (20,20) (10,10) (10,15) (10,20) (15,20) (20,20)

.0105 .0183 .0047 .0060 .0029 .0065 .0099 .0092 .0060 .0020

.0060 .0235 .0034 .0068 .0022 .0041 .0117 .0133 .0068 .0012

.0002 .0277 .0012 .0011 𝜎1 = 1, 𝜎2 = 2 .0133 .0128 .0093 .0253 .0019 .0052 .0017 .0076 .0013 .0048 .0067 −.0009 .0070 .0245 .0084 .0398 .0045 .0143 .0036 .0008 .0029 −.0041 .0033 .0214 .0027 −.0023 𝜎1 = 1, 𝜎2 = 3 .0145 .0137 .0092 .0227 .0028 .0071 .0023 .0091 .0017 0.0064 .0082 −.0012 .0072 .0182 .0078 .0311 .0061 .0117 .0041 −.0006 .0053 −.0044 .0047 .0168 .0034 −.0047 p = 6 𝜎1 = 1, 𝜎2 = 1 .0001 .0001 .0054 .0221 .0004 .0004 .0003 .0031 .0003 .0003 −.0003 .0009 .0029 .0290 .0044 .0455 .0003 .0160 −.0001 −.0002 .0017 .0014 .0020 .0296 .0012 .0011 𝜎1 = 1, 𝜎2 = 2 .0114 .0110 .0083 .0240 .0022 .0055 .0019 .0080 .0011 .0047 .0071 .0000 .0079 .0243 .0092 .0399 .0038 .0129 .0027 −.0010

RE1 1.58 1.75

RE2

RE3

RE4

RE5

.99 1.00

1.14 1.52

1.14 1.52

.72 .87

2.72 3.27 6.78 7.21 8.48 1.42 1.55 1.65 1.99 2.15 1.41 1.51 1.72

1.03 1.03 1.01 1.00 1.00 1.01 1.01 1.01 1.01 1.00 1.00 1.00 1.00

2.54 2.89 6.03 6.31 7.59 1.18 1.30 1.22 1.70 1.91 1.16 1.19 1.44

2.45 2.79 5.96 6.27 7.57 1.16 1.27 1.20 1.68 1.90 1.15 1.19 1.44

.93 .88 .88 .87 .89 .83 .83 .73 .85 .88 .82 .78 .83

2.62 3.11 6.00 6.52 7.73 1.40 1.53 1.57 1.97 2.17 1.39 1.59 1.75

1.02 1.03 1.00 1.00 1.00 1.00 1.01 1.01 1.01 .99 1.00 1.00 .99

2.38 2.75 5.18 5.50 6.57 1.13 1.30 1.21 1.66 1.77 1.08 1.26 1.35

2.33 2.66 5.16 5.47 6.56 1.12 1.28 1.19 1.65 1.77 1.07 1.26 1.35

.90 .88 .86 .84 0,84 .80 .84 .77 .84 .81 .77 .79 .77

2.79 3.39 7.35 7.43 9.41 1.44 1.64 1.69 2.02 2.34 1.42 1.59 1.85

1.05 1.02 1.01 1.00 1.00 1.02 1.01 1.00 1.00 1.01 1.00 .99 .99

2.64 2.96 6.86 6.94 8.78 1.27 1.30 1.15 1.77 2.11 1.21 1.12 1.61

2.51 2.90 6.79 6.89 8.72 1.24 1.28 1.15 1.75 2.09 1.20 1.12 1.61

.94 .87 .93 .93 .93 .87 .79 .68 .87 .90 .85 .70 .87

2.74 3.28 6.76 7.18 8.55 1.39 1.57 1.62 1.98 2.17

1.03 1.03 1.00 1.00 1.00 1.01 1.01 1.01 1.00 1.00

2.55 2.91 6.05 6.29 7.60 1.18 1.32 1.20 1.72 1.90

2.46 2.81 5.99 6.23 7.57 1.15 1.30 1.18 1.71 1.89

.93 .88 .89 .87 .88 .84 .83 .73 .86 .87 (Continued)

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Table 1. (Continued) Bias rx = ry 10

1

5

10

(mx, my)

Relative Efficiencies

(n, m) R̂ ML,SRS R̂ MML,SRS R̂ ML,RSS R̂ MML,RSS

(2,2) (2,3) (3,3)

(20,20) (20,30) (30,30)

.0047 .0044 .0033

.0037 .0060 .0029

(5,5) (5,10) (15,15) (15,20) (20,20) (2,2) (2,3) (2,4) (3,4) (4,4) (2,2) (2,3) (3,3)

(5,5) (5,10) (15,15) (15,20) (20,20) (10,10) (10,15) (10,20) (15,20) (20,20) (20,20) (20,30) (30,30)

.0150 .0188 .0061 .0056 .0049 .0081 .0090 .0100 .0066 .0048 .0056 .0035 .0031

.0085 .0214 .0042 .0057 .0037 .0051 .0097 .0127 .0067 .0037 .0043 .0043 .0025

.0023 −.0056 .0051 .0237 .0019 −.0029 𝜎1 = 1, 𝜎2 = 3 .0150 .0142 .0109 .0249 .0023 .0065 .0027 .0093 .0013 .0059 .0084 −.0006 .0092 .0208 .0083 .0315 .0052 .0114 .0041 −.0003 .0039 −.0058 .0045 .0162 .0026 −.0050

RE1 1.34 1.55 1.58

RE2

RE3

RE4

RE5

1.00 1.00 1.00

1.03 1.17 1.38

1.03 1.16 1.38

.77 .75 .87

2.65 3.05 6.07 6.87 7.33 1.34 1.53 1.60 1.95 2.18 1.36 1.60 1.64

1.02 1.03 1.00 1.00 .99 1.00 1.01 1.01 1.00 .99 1.00 1.00 1.00

2.42 2.65 5.26 5.67 6.20 1.08 1.29 1.23 1.64 1.84 1.03 1.25 1.27

2.36 2.57 5.22 5.63 6.22 1.07 1.28 1.21 1.63 1.84 1.02 1.24 1.27

.91 .86 .86 .82 .84 .80 .84 .76 .84 .84 .75 .77 .77

Table 2. Biases and MSEs of the estimators of R based on RSS under imperfect ranking.

R̂ ML.RSS ) ( mx .my (2.2)

(4.4)

(6.6)

R̂ MML.RSS

τ2

Bias

MSE

Bias

MSE

.0 .25 .5 1.0 .0 .25 .5 1.0 .0 .25 .5 1.0

.0013 .0011 −.0002 .0012 .0001 .0002 −.0008 .0008 −.0011 .0011 .0004 −.0009

.0048 .0051 .0052 .0053 .0014 .0016 .0016 .0017 .0007 .0009 .0008 .0008

.0014 .0017 −.0005 .0014 .0003 .0011 −.0012 .0015 −.0009 .0018 .0014 −.0010

.0055 .0075 .0079 .0078 .0017 .0030 .0031 .0034 .0007 .0020 .0020 .0021

However, it should be noted that under the assumption of imperfect ranking the MML estimator is preferable to the ML estimator when our interest is in estimating the shape parameter p.

5.  A real data example Here, we use the well-known strength data taken from the literature as an application of the proposed estimators based on RSS. The data are originally reported by Badar and Priest (1982). We consider two data-sets on strength measured in GPA of single carbon fibres of lengths 20 mm (Data-Set I) and 50 mm (Data-Set II), respectively. Sample sizes for Data-Set I and Data-Set II are 69 and 65, respectively, see Ghitany et al. (2015). To identify the distributions of Data-Set I and Data-Set II, we use Q–Q plots, see Figure 1(a) and (b). Histograms for these data-sets are also given in the same figures. It

12 

  F. G. Akgül and B. Şenoğlu

Figure 1. (a) Q–Q plot and the histogram with fitted pdf for the Data-set I, (b) Q–Q plot and the histogram with fitted pdf for the Data-set II.

Table 3. The ML and the MML estimates of R for the strength data.

R̂

R̂

R̂

R̂

ML,SRS

MML,SRS

ML,RSS

MML,RSS

.3681 (.0717)*

.3657 (.0724)*

.3793 (.0523)*

.3756 (.0561)*

*Bootstrap MSE.

can be seen that the Weibull distribution provides good fit for the strength data (Data-Set I and Data-Set II). Under the assumption of equal shape parameter, p is estimated to be 5.759. Based on this value of p, scale parameters for the Data-Set I and the Data-Set II are estimated to be 𝜎̂ 1 = 280.363 and 𝜎̂ 2 = 157.695, respectively. Different from the earlier studies, we use the strength data-sets as populations of interest and sample from these populations using SRS and RSS to estimate R. In the context of SRS, we select 21 observations from each of the Data-Set I and Data-Set II and call these as X and Y, respectively. In the context of RSS, we take the set sizes mx = my = 3 and the number of cycles rx = ry = 7, so that the sample sizes n = m = 21 for both data-sets corresponding to the gauge lengths of 20 mm (X) and 50 mm (Y), respectively. We then compute the ML and the MML estimates of R, based on SRS and RSS, without taking into account whether the ranking is perfect or imperfect. By using bootstrap MSE values, we compare the efficiencies of them. The results are given in Table 3. It is clear from Table 3 that the bootstrapped MSE values, based on RSS, are smaller than the corresponding MSE values based on SRS. Therefore, we can say that the estimates based on RSS are more reliable than the estimates based on SRS. The estimate of P(X < Y ) is approximately .37, in other words the estimate of P(X > Y ) is .63. This implies that the single carbon fibres with length 20  mm are stronger than the single carbon fibres with length 50 mm.

Quality Technology & Quantitative Management 

 13

6. Conclusions In this paper, we have proposed the ML and the MML estimators of R = P(X < Y ), based on RSS, where X and Y are independent Weibull random variables. It is clear from the simulation results that both the ML and the MML estimators of R, based on RSS, are more efficient than their counterparts based on SRS. The ML estimator of R is more preferable than the MML estimator under RSS, however they are equally preferable under SRS. The effect of imperfect ranking is also investigated and the ML estimator is found to be more efficient than the MML estimator for estimating the system reliability R. In future studies, we hope to report our findings regarding the estimation of R using the modifications of RSS, which are useful in overcoming the ranking error encountered in RSS. We will include extreme ranked set sampling, median ranked set sampling and percentile ranked set sampling in our studies.

Acknowledgments We are grateful to the referees for their very helpful comments and suggestions.

Disclosure statement No potential conflict of interest was reported by the authors.

Notes on contributors Fatma Gül Akgül is an assistant professor at Artvin Çoruh University, Turkey. She received her BSc and MS degrees in Statistics and Computer Sciences from Karadeniz Technical University, Turkey. She also received her PhD degree in Statistics from Ankara University, Turkey. Her main areas of interest are reliability, sampling and robust statistics. Birdal Şenoğlu is a professor at Ankara University, Turkey. He received his BSc degree in Statistics from Middle East Technical University, Turkey and MS degree in Statistics from Iowa State University, USA. He also received his PhD degree in Statistics from Middle East Technical University, Turkey. His primary research areas are experimental design and robust statistics.

References Akgül, F. G. (2015). Robust estimation of system reliability in stress–strength model using ranked set sampling (Ph.D. diss.) Ankara University, Ankara. Akgül, F. G., & Şenoğlu, B. (2014). Estimation of system reliability using ranked set sampling for Weibull distribution. 15th ISEOS, 93, Isparta. Badar, M. G., & Priest, A. M. (1982). Statistical aspects of fiber and bundle strength in hybrid composites. In T. Hayashi, K. Kawata, & S. Umekawa (Eds.), Progress in Science and Engineering Composited (pp. 1129–1139). Tokyo: ICCM-IV. Beg, M. A. (1980). Estimation of P(Y