ESTIMATION OF A PARAMETER OF FARLIE-GUMBEL-MORGENSTERN BIVARIATE BILAL DISTRIBUTION BY RANKED SET SAMPLING Текст научной статьи по специальности «Математика»

RT&A, No 3 (74) Volume 18, September 2023

ESTIMATION OF A PARAMETER OF FARLIE-GUMBEL-MORGENSTERN BIVARIATE BILAL DISTRIBUTION BY RANKED SET SAMPLING

M. R. Irshad1, R. Maya2, A.I. Al-Omari3, Ahmad A. Hanandeh4, S. P. Arun5

1Cochin University of Science and Technology, Kochi-22, Kerala. [email protected] 2University College, Trivandrum-695 034. [email protected] 3Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq, Jordan.

[email protected] 4Yarmouk University, Irbid 21163, Jordan. [email protected] 5University Library, Research Centre, University of Kerala, Trivandrum-695 034.

[email protected]

A bivariate version of the Bilal distribution has been proposed in the literature, called the Farlie-Gumbel-Morgenstern bivariate Bilal (FGMBB) distribution. In this article, we have dealt with the problem of estimation of the scale parameter associated with the study variable Z of primary interest, based on the ranked set sample defined by ordering the marginal observations on an auxiliary variable W, when (W, Z) follows a FGMBB distribution. When the dependence parameter <p is known, we have proposed the following estimators, viz., an unbiased estimator based on the Stoke's ranked set sample and the best linear unbiased estimator based on the Stoke's ranked set sample for the scale parameter of the variable of primary interest. The efficiency comparison of the proposed estimators with respect to the maximum likelihood estimator have been carried out.

Keywords: Farlie-Gumbel-Morgenstern bivariate Bilal distribution, Concomitants of order statistics, Ranked set sampling, Best linear unbiased estimator

The Bilal distribution was introduced by [1], as a member of the families of distributions for the median of a random sample arising from an arbitrary lifetime distribution. Also, he shows that, this distribution belongs to the class of new better than average renewal failure rates and its probability density function (pdf) is always unimodal and has less of skewness and kurtosis than the pdf of the exponential distribution by about 25% and 28% respectively. The cumulative distribution function (cdf) of the Bilal distribution with the scale parameter a is given by

Abstract

1. Introduction

(1)

The corresponding pdf is given by

Furthermore, the author obtained the closed form expressions for the quantile function, the hazard rate function and simple expression for moments in terms of the exponential function. Even though the Bilal distribution has only one parameter, this distribution possess high fitting ability compared to other competing models for two different real datasets, namely, the dataset consisting of thirty successive values of precipitation (in inches) given by [14] and the data for waiting times before service of 100 bank customers reported by [13]. Based on type-2 censored sample, [2] provide certain estimators of the parameter of the Bilal distribution. According to [3], the one parameter Bilal model can be generalized into the two parameter Bilal model, whose applications are elaborately discussed. Now the Proficiency of univariate Bilal distribution compared to other competing models well established in the literature in the theoretical as well as applied perspective. But even a single work is not been seen so far in the available literature on bivariate Bilal model except the work of [17]. A bivariate extension of one parameter Bilal distribution using Morgenstern approach was proposed by [17], so-called the Farlie-Gumbel-Morgenstern Bivariate Bilal (FGMBB) distribution and elucidated its inferential aspects using concomitants of order statistics (COS).

A bivariate random variable (W, Z) is said to follow a FGMBB distribution, if its pdf is given by

f z)

36 -2W / -WL \ _2z --

-e "11- e a1 e a2 11- e "2

"1"2

- x

1 + 0 2e "13 - 2e

"} - o

_ 2z

2e "2

{3 - 2e "2 }

- 1

(3)

w > 0, z > 0; "1 > 0, "2 > 0; -1 < 0 < 1.

0, otherwise.

Clearly the marginal distributions of W and Z variables are univariate Bilal distributions with

pdf's are respectively given by

fW(w)

A e "1

1 - e ; otherwise

if "1 > 0, w > 0,

and

fz (z)

f.

e "2

"2

0,

1 - e " otherwise.

i f "2 > 0, z > 0,

(4)

Clearly,

E(W)

-"1, Var(W) 6

13 2 36 "1,

E(Z) = 6 "2, 13

Var(Z) = 13 "2.

(5)

(6)

The ranked set sampling (RSS) scheme was first developed by [19] as a process of increasing the precision of the sample mean as an estimator of the population mean. McIntyre's idea of ranking is possible whenever it can be done easily by a judgement method. For a detailed discussion on the theory and applications of RSS [11]. Basically the procedure involves choosing n sets of units, each of size n, and ordering the units of each of the set by judgement method or by applying some inexpensive method, without making actual measurement on the units. Then the unit ranked as one from the 1st set is actually measured, the unit ranked as two from the 2nd set is measured. The process continuous in this way until the unit ranked as n from the nth set is measured. Then the observations obtained under the afore mentioned criterion is known as ranked set sample (rss) and the procedure is known as RSS. For recent developments in RSS, one can refer [6], [4] and [5].

In some practical problems, the variable of primary concern say Z, is more intricate to measure, but an auxiliary variable W related with Z is easily measurable and can be ordered exactly. In this case, [22] developed another scheme of RSS, which is as follows: Choose n independent bivariate sets, each of size n. In the first set of size n, the Z variate associated with smallest ordered W is measured, in the second set of size n, the Z variate associated with the second smallest, W is measured. This process is continued until the Z associated with the largest W from the nth set is measured. The measurements on the Z variate of the resulting new set of n units chosen by the above method gives a rss as suggested by [22]. If W(r:n)r is the observation measured on the auxiliary variable W from the unit chosen from the rth set, then we write Z[r:n]r to denote the corresponding measurement made on the study variable Z on this unit so that Z[r:n]r, r = 1,2, ■ ■ ■ , n form the rss. Z[r:n]r was referred by [12] as the concomitant of the rth order statistic arising from the rth sample.

The rss mean as an estimator for the mean of the study variate Z, when an auxiliary variable W is used for ranking the sample units has suggested by [22], under the assumption that (W, Z) follows a bivariate normal distribution. Based on rss obtained on the study variate Z, [10] have improved the estimator of [22] by deriving the best linear unbiased estimator (BLUE) of the mean of the study variate Z. COS and its applications in RSS from Farlie-Gumbel-Morgenstern bivariate Lomax distribution is elaborately elucidated by [20]. The estimation of a parameter of Morgenstern type bivarite Lindley distribution by RSS has been discussed in [15]. Parameter estimation of Cambanis-type bivariate uniform distribution with RSS is studied by [16]. For review of various variants of RSS and their application in parameter estimation [11]. The remaining part of this paper is assembled as follows. In section 2, we have proposed an unbiased estimator a| of a2 using Stoke's rss. As mentioned earlier if (W, Z) has a FGMBB distribution as defined in (3), then the marginal distributions of both W and Z have Bilal distributions and the pdf of Z is given in (4). We have evaluated the Cramer-Rao Lower Bound

(CRLB) for the variance of an unbiased estimator of a2 involved in (4) based on a random sample

2

of size n and is given by 25 02. In this section, we have also shown that the variance of proposed

a2

unbiased estimator a| is strictly less than 25 at, the CRLB for the variance of an unbiased estimator of a~2 involved in (4), for all p E B, where B = [-1,1] — {0}. In this section, we have further discussed an efficiency comparison between a| and the maximum likelihood estimator (MLE) 02 of a2 based on a random sample of size n arising from (3). In section 3, we have derived the BLUE 02 of 02 involved in FGMBB distribution based on Stoke's rss and made an efficiency comparison of 02 relative to 02.

2. An unbiased estimator of a2 using Stoke's RSS.

Suppose the bivariate random vector (W, Z) follows a FGMBB distribution with pdf given in (3). Select a rss as per Stoke's RSS scheme. Let W(r:n)r be the observation obtained on the auxiliary variate W in the rth unit of the rss and let Z[r:n]r be the measurement made on the variate related with W(r:n)r, r = 1,2, ■ ■ ■ , n. Clearly Z[r:n]r is the rth COS of a random sample of size n arising from the FGMBB distribution. Using the results of [21], we obtain the pdf of Z[r:n]r, r = 1,2, ■ ■ ■ , n, and is given by

/ \ 6 — / — _z. f[r:n](z) = -e a2{1 — e 02

1 + PT+T1 (2e—1 {3 — 2e—02 } — 1

The mean and variance of Z[r:n]r for r = 1,2, ■ ■ ■ , n, is obtained as

E[Z[r:n]r] = 02

5 _ 19 (n — 2r + 1)

6 — 60 P (n + 1)

and

Var[Z[r:n]r] = 022

13 253 (n — 2r + 1) 361 2 (n — 2r + 1)21 p 7ITTT\ 3600p

36 1800 (n + 1)

(n +1)2

(7)

(8)

(9)

Since Z[r:n]r and Z[s:n]s for r = s are arising from two independent samples, we obtain

Cov[Z[r:n]r, Z[s:n]s] = 0, r = s.

Next, we derive an unbiased estimator of "2 and its variance using the rss observations Z[r:n]r for r = 1,2, ■ ■ ■ , n, on the variable Z of primary interest and are given by the following theorem.

Theorem 1. Let (W, Z) follows a FGMBB distribution with pdf given by (3). Let Z[r:n]r, r = 1,2, ■ ■ ■ , n be the rss observations on a study variate Z generated out of ranking made on an auxiliary variate W. Then

"2

6 n

^ Z[r:n]r

r=1

is an unbiased estimator of "2 and its variance is given by

Var ["2 ] = —

2n

361 02 y (n - 2r + 1x 2

13

25 2500 n

r=1

n+1

Proof By using the definition, we have

E" ]

6 n

y E[Z[r:n]r]

r=1

6 n — y

5n ^

r=1

5 _ 19 0 (n - 2r + 1)

6 - 60 0 (n + 1)

"2.

Using the result,

Applying (12) in (11) we get,

y (n - 2r + 1) = 0.

r=1

E["2 ] = "2.

Therefore, is an unbiased estimator of "2. The variance of "2 is given by,

36 n

Var["2] = y Var[Z[r:n]r].

Applying (9) and (12) in (13), we get

Var ["2 ] = —

2n

13 _ ^36102 y (n - 2r + 1x 2

25 2500 n

r=1

n+1

Hence the proof.

(10)

(11)

(12)

(13)

As mentioned above, if (W, Z) has the FGMBB distribution as defined in (3), then the marginal distribution of both W and Z are Bilal distributions and the pdf of Z is given in (4). The CRLB

for the variance of any unbiased estimator of "2 based on a random sample of size n drawn from

2

(4) is obtained as 13 "2. Now we compare the the variance of with the CRLB for the variance of

"2

an unbiased estimator of "2 involved in (4). If we write E1 ("2) to denote the ratio of 55 with Var ("2), then we have,

21

E1("2) = T-;-;-TtT . (14)

It is easily verified that

1 _ 361 02 n 1 1300 n yr=1

E1 ("2 ) > 1.

n-2r+1 \2 n+1 )

n

Thus we arrive at a conclusion that the estimator based on Stoke's rss is more efficient as it assert the statement that rss always provide more information than simple random sample even if ranking is imperfect [11]. It is very clear that Var(a2, ) is a decreasing function of <2 and hence the gain in efficiency of the estimator increases as |<| increases. Again on simplifying (14) we get,

Ei(o-| )

1

361<2 1300

2 (2+1/n \ _ i

3 I 1+1/n 1 1

Then,

lim Ei (o! ) = lim -2

n^œ n^œ 1 _ 361<2

1300

2 /2+1/n i _ .

3 I 1+1/n 1 1

1

361<2 3900

From the above expression it is clear that the maximum value for Ei (a2,) is attained when = 1 and in this case E1 (o|) tends to 3900/3539.

Next we discuss the efficiency comparison of with the asymptotic variance of MLE of a2 involved in the FGMBB distribution. If (W, Z) follows a FGMBB distribution with pdf given in (3), then

and

9 l°g f ^ y)

1

01

2w

-1 +--

a\ (1 — e V1 )

+ -

d log f (x, y) = 1 i 1 + 2z — 9^2 02 I 02

+-

2w 4<we V1 2z 3z w w 2z w 3z —3 + 18e — 12e V2 + 3e V1 — 18e o e V2 + 12e V1 e V2

+ < 2w w 1 — 2e V1 (3 — 2)e o 2z z 1 — 2e V2 (3 — 2)e V2 }

o2 (1 — e V2 )

2z 4<ze V2 2w 3w z z 2w z 3w —3 + 18e V1 — 12e o + 3e V2 — 18e V2 e o + 12e V2 e V1

+ < 2w w 1 — 2e V1 (3 — 2)e o 2z z 1 — 2e 02 (3 — 2)e V2 }

Then we have,

0 (<) = E

/S-bgf^y) V M

36

(1 — e—«) j —1 + 4u + ue—u(u — 2 + 2e—«)(1 — e—«)—2

0 0

1lau2e-2u [—2 + 12e—2v — 8e—3v + 3e—u — 18e—ue—2v + 12e—ue—3v]

{1 + a[1 — 2e—2u (3 — 2e—u )] [1 — 2e—2v (3 — 2e—v )]} l4aue-2u [—1 + 6e—2v — 4e—3v + e—u — 6e-ue-lv + 4e—ue—3v] {1 + a[1 — 2e—2u (3 — 2e—u )] [1 — 2e—2v (3 — 2e—v )]}2 x{1 + a [1 — 2e—2v (3 — 2e—v )] [1 — 6e—2u — 6ue-2u + 4e—3u + 6ue—3u ]} } e-lv (1 — e—v ){1 + a[1 — 2e-lu (3 — 2e—u )] [1 — 2e—2v (3 — 2e—v )]}dudv,

+

1

2

2

œ œ

0

CO CO

/e~2" - e-" H-1 + + (v - 2 + 2e-v )(1 - e-v )-2

2 0 0

12a v2e-2v [-2 + 12e-2" - 8e-3" + 3e-v - 18e-ve-2" + 12e-ve-3"]

{1 + a[1 - 2e-2" (3 - 2e-")][1 - 2e-2v (3 - 2e-v)]}

21a ve ,-1 + 6e - 4e + e - rf-2" 1 -3"

+

24ave-2v [-1 + 6e-2" - 4e-3" + e-v - 6e-ve-2" + 4e-ve-3"]

and

{1 + a[1 - 2e-2" (3 - 2e-")][1 - 2e-2v (3 - 2e-v)]}2 x{1 + a[1 - 2e-2" (3 - 2e-")] [1 - 6e-2v - 6ve-2v + 4e-3v + 6ve-3v]} } e-2v (1 - e-v){1 + a[1 - 2e-2" (3 - 2e-")][1 - 2e-2v (3 - 2e-v)]}d"dv

<« = E ( f )

0

"e-2"(1 - e-") { 144awve-2"e-2v [1 - e-" - e-v + e^e-0]

CC

36

<J\ 02

00

144a2"ve-2"e-2v [-1 + 6e-2v - 4e-3v + e-" - 6e-"e-2v + 4e-"e-3v] {1 + a[1 - 2e-2"(3 - 2e-")][1 - 2e-2v(3 - 2e-v)]}2 x{ [1 - 2e-2" (3 - 2e-")] [e-v - 1]}

} e-2v(1 - e-v){1 + a[1 - 2e-2"(3 - 2e-")] [1 - 2e-2v(3 - 2e-v)]}d"dv. Thus the Fisher information matrix associated with the random variable (W, Z) is given by,

I (4)

(4) -1^102 (4) - Î0102 (4) I02 (4)

(15)

We have computed the values of 0— 2I01 (0) and 0— V2 1 I0102 (0) numerically for 0 = ±0.25, ±0.50, ±0.75, ±1 (clearly of2I01 (0)=o—2I02 (0)) and are given below:

<p 0 21010) O- V-1 Î01O2 (4) <P 0 21010) O-10-1 Î01O2 (4)

0.25 1.9381 0.1373 -0.25 1.9381 -0.1373

0.50 1.9795 0.2772 -0.50 1.9795 -0.2773

0.75 2.0530 0.4230 -0.75 2.0530 -0.4236

1.00 2.1705 0.5815 -1.00 2.1705 -0.5841

Thus from (15), the asymptotic variance of the MLE 02 of 02 involved in the FGMBB distribution under a bivariate sample of size n is obtained as

1

Var(02) = n If1 (0), (16)

where If1 (0) is the (2,2)th element of the inverse of I(0) given by (15).

We have compute the efficiency E(a||o2) = ^(0*) o2 relative to 02 for n = 2(2)20; 0 = ±0.25, ±0.50, ±0.75, ±1 and are given in table 1. From the table, one can infer that the estimator o* is more efficient than 02 and efficiency increases with n and |0| for n > 4. Remark 2.1. For given value of 0 G (0,1], once the variance of o* is evaluated, then this variance is equal to the variance of a* for —0 because the variance given in (10) depends only on 0 by a term containing 02 only.

2

3. Best linear unbiased estimator of 02 using Store's RSS

In this section we derive the BLUE of 02 provided the dependence parameter 0 is known. Suppose Z[r:n]r for r = 1,2, ■ ■ ■ , n, are the rss observation generated from (3) as per Stoke's RSS scheme. Let

r -5_19,f, (n - 2r + 1)

6 60 r (n +1) ' = 13_^53 (n - 2r + 1) 361 2 (n - 2r + 1)2

Tr,r,n a£ 1 QH r\l i 1 \ 0£n nT

Using (17) and (18), we get and

Also we have

36 1800T (n +1) 3600 T (n +1)2

E[Z[r:n]r] = 02Zr,n, 1 < r < n

Var[Z[r:n]r] = o2Vr,r,n, 1 < r < n.

(17)

(18)

(19)

(20) (21)

Cov[Z[r:n]r, Z[s:n]s] = 0, r, s = 1,2, ■ ■ ■ , n and r = s.

Let Z[n] = (Z[1:n]1, Z[2:n]2, ■ ■ ■ , Z[n:n]n)' denote the column vector of n rss observations. Then from (19), (20) and (21), we can write,

E[Z[n]] = 02 Z (22)

and the dispersion matrix of Z[n] ,

D[Z[n]] = ofG, (23)

where Z = (Zi,n, Zi,n, ■ ■ ■ , Zn,n) and G = diag(t1,1,n, ^2,2,n, ■ ■ ■ , tn,n,n), where Zr,n and tr,r,n for r = 1,2, ■ ■ ■ , n are respectively given by equations (17) and (18). If 0 contained in Z and G are known then (22) and (23) together defines a generalized Gauss-Markov setup and then the BLUE of 02 is given by

02 = (Z'G-1Z )-1 Z' G-1 Z

[n]

and the variance of 02 is given by

On simplifying, we get

Var(02)

0

Z 'G-XZ

E

Zr,n

02

1 t

V-1 br,; rE1 ^

' ' Z

^-2 [r:n]r zr

n

r,r,n

and

Var(02)

0

n z2 v-1 br,;

r=1 ^

(24)

(25)

n

r,r,n

From (24), we have 02 is a linear functions of the rss observations Z[r:n]r, r = 1,2, ■ ■ ■ , n and hence

02 can be written as 02 = E arZ[r:n]r, where

r=1

Zr,n

_ ^Y,r,n

n z2 E ^r,n

-, r = 1,2, ■ ■ ■ , n.

r=1 ^r,r,n

We have evaluated the numerical values of means and variances using the expressions (17) and (18) respectively for 0 = 0.25(0.25)1 and for n = 2(2)20. Using these values we have evaluated

r

n

n

r

the variance of BLUE fy for $ = 0.25(0.25)1 and for n = 2(2)20. Also we have computed the ratio E(ct2 |(72) =

V«r(?2) to measure the efiiciency of Our estimatOr <72 relative to (72 for n — 2(2)20 and $ = 0.25(0.25)1 and are presented in table 1. From the table it is clear that, BLUE of 02 performs well compared to the MLE of a2, namely a2.

Remark 3.1. As in the case of variance of an unbiased estimator given in (10), for a given value of $ G (0,1], once the variance of is evaluated, then there is no need to again evaluate the variance of fy when $ = —$. To establish this argument we prove the following theorem.

Theorem 2. Let (W, Z) follows a FGMBB distribution with pdf given by (3). For a given $ G (0,1], Var[d~2($0)] is the variance of the BLUE cr2 of a2 involved in the FGMBB distribution, then

Var[cf2(—$0)] = Var[a2($0)]. (26)

Proof The terms Zr,n and tyr,r,n defined by (17) and (18) are functions of $, r and n and hence Zr,n and tyr,r,n can be denoted as Zr,n ($) and tyr,r,n ($) respectively. From (17) and (18), it is clear that

Zr,n ($) = Zn-r+1,n( —$), 1 < r < n (27)

and

1r,r,n ($) = in-r+1,n-r+1,n (-$), 1 < r < n. (28)

As a consequence of (27) and (28), we get

Varfy ($)]

<7% crZ

» Z2,n ($) f Z2n-r+1,n (-$)

^ h......($) ^

r= hr,r,n ($) =1 hn-r+1,n-r+1,n (-$)

Var [¿72 (-$)].

Hence the proof.

Remark 3.2. For FGMBB distribution defined in (3), we have evaluated the correlation coefficient between the two variables and is given by p = H00 $■ But in certain real life situations our assumption that $ is known may viewed as unrealistic■ Hence if we have a situation with $ unknown, we compute the sample correlation t from (Wr:n, Z[r:n]) for r = 1,2, ■ ■ ■ , n and introduce a moment type estimator $ for $ as,

f-1, if t < -61

$ = J 1300t if -361 < t< ^61 $ f 361 t, V 1300 < t < 1300

I1, if T > 1300. Table 1: Efficiencies of the estimators c^ and c72 relative to P2.

n $ e(721^2) $ ^21^2 ) e(721^2)

0.25 0.9992 0.9992 -0.25 0.9992 0.9992

2 0.50 0.9984 0.9984 -0.50 0.9984 0.9984

0.75 0.9957 0.9957 -0.75 0.9957 0.9957

1.00 0.9849 0.9849 -1.00 0.9850 0.9853

0.25 1.0008 1.0008 -0.25 1.0008 1.0008

4 0.50 1.0047 1.0047 -0.50 1.0029 1.0029

0.75 1.0103 1.0111 -0.75 1.0047 1.0047

1.00 1.0106 1.0139 -1.00 1.0110 1.0147

0.25 1.0012 1.0012 -0.25 1.0012 1.0012

6 0.50 1.0082 1.0082 -0.50 1.0082 1.0082

0.75 1.0168 1.0180 -0.75 1.0168 1.0180

1.00 1.0223 1.0273 -1.00 1.0230 1.0286

0.25 1.0015 1.0015 -0.25 1.0015 1.0015

n T e(°2 |02) e(0202) T e(°2 l°2) e(02102)

8 0.50 1.0094 1.0094 -0.50 1.0094 1.0094

0.75 1.0192 1.0209 -0.75 1.0192 1.0209

1.00 1.0282 1.0351 -1.00 1.0300 1.0367

0.25 1.0019 1.0019 -0.25 1.0019 1.0019

10 0.50 1.0098 1.0098 -0.50 1.0098 1.0098

0.75 1.0221 1.0241 -0.75 1.0221 1.0241

1.00 1.0312 1.0398 -1.00 1.0330 1.0419

0.25 1.0023 1.0023 -0.25 1.0023 1.0023

12 0.50 1.0094 1.0094 -0.50 1.0094 1.0094

0.75 1.0242 1.0266 -0.75 1.0242 1.0266

1.00 1.0376 1.0455 -1.00 1.0376 1.0455

0.25 1.0000 1.0000 -0.25 1.0000 1.0000

14 0.50 1.0110 1.0110 -0.50 1.0110 1.0110

0.75 1.0225 1.0254 -0.75 1.0225 1.0254

1.00 1.0380 1.0472 -1.00 1.0380 1.0472

0.25 1.0031 1.0031 -0.25 1.0031 1.0031

16 0.50 1.0126 1.0126 -0.50 1.0126 1.0126

0.75 1.0258 1.0291 -0.75 1.0258 1.0291

1.00 1.0403 1.0473 -1.00 1.0403 1.0473

0.25 1.0035 1.0035 -0.25 1.0035 1.0035

18 0.50 1.0106 1.0106 -0.50 1.0106 1.0106

0.75 1.0291 1.0291 -0.75 1.0291 1.0291

1.00 1.0415 1.0534 -1.00 1.0415 1.0534

0.25 1.0000 1.0000 -0.25 1.0000 1.0000

20 0.50 1.0118 1.0157 -0.50 1.0118 1.0157

0.75 1.0242 1.0283 -0.75 1.0242 1.0283

1.00 1.0420 1.0508 -1.00 1.0420 1.0508

4. Estimation of 02 based on censored ranked set sample

In this section, we obtain some estimators of a2 using censored RSS scheme. Suppose k units are censored in the Stoke's RSS scheme, then we may represent the rss observations on the study variate Z as S1Z[1:n]j, ¿2Zp:n]2, ■ ■ ■ , SnZ[n:n]n where,

Si =

if the ith unit is censored, otherwise.

6 En S-Z

and hence Ef=i Si = n - k. In this case the usual unbiased estimator of 02 is equal to —5=»-kj':n]<. It may be noted that one need not get Si = 0 for ' = 1,2, ■ ■ ■ ,k and Si = 1 for ' = k + 1, k + 2, ■ ■ ■ , n. Hence if we write mi, i = 1,2, ■ ■ ■ , n - k as the integers such that 1 < m1 < m2 < ■ ■ ■ mn-k and for which Sm.=1, then,

E

6 En

=1 SiZ[i:n]i

5(n - k)

02

1

190

n-k

50(n + 1)(n - k) 'S (n - 2mi + 1)

Thus it is clear that the in the censored case the usual unbiased estimator is not an unbiased

estimator of 02. However we can construct an unbiased estimator of 02 based on given in the following theorem.

6 En=1 SiZ[i:n]i

5(n-k)

is

Theorem 3. Suppose that the random variable (W, Z) has a FGMBB distribution as defined in

(3). Let Z

[mi ]m;,

1,2, ■ ■ ■ , n - k be the rss observations on the study variate Z resulting out

of censoring applied on the auxiliary variable W. Then an unbiased estimator of 02 based on

5(n-) En=iZ[m;]m; is given by

02 (k) =

60(n +1)

50(n + 1)(n - k) - 190En-(n - 2mi + 1)

n- k

Z[mi]mi

i=1

and its variance is given by Var[02* (k)] =

3600(n + 1)2 tff

50(n + 1)(n - k) - 190 En=-1k(n - 2mi + 1)

n- k

■ E tm

i=1

where 0mi is as defined in (18). Proof We have

E02 (k)]

60(n + 1)

50(n + 1)(n - k) - 190Ern=i(n - 2mi + 1) 60(n + 1)

n- k

EE[Z [mi ]m; J

i=1

50(n + 1)(n - k) - 190En-(n - 2mi + 1)

n-k r5 19 (n - 2mi + 1) 6 - 60 0"

x E

i=1

(n +1) 60(n + 1)

02

50(n + 1)(n - k) - 190Ern=i (n - 2mi + 1)

nk

5(n - k) 190 E( 2m. +1)

60(n + 1) i=1 (n 2mi + 1)

02

02.

Thus 0ii (k) is an unbiased estimator of 02. The variance of 0"| (k) is given by

3600 (n +1)2

Var[0-| (k)]

50(n + 1)(n - k) - 190En==1 (n - 2mi + 1)

n- k

E Var(Z[mi]mi)

3600(n + 1)2 cf

50(n + 1)(n - k) - 190 En=!(n - 2mi + 1)

i=1

n-k

2 E tmi

i=1

where 0mi is as defined in (18). Hence the theorem.

As a competitor of the estimator (k), next we propose the BLUE of 02 based on the censored rss, resulting out of ranking of observations on W.

If Z[n] (k) = (Z[m1]m1, Z[m2]m^ ' ' ' , Z[mn-k]mn-k X, then the mean vector and the variance-cOvariance

matrix of Z[n] (k) are given by

E[Z[n](k)] = 02 Z (k), (29)

D[Z[n](k)]= 02 G(k), (30)

where Z(k) = (Zm1, Zm1, ■ ■ ■ , Zmn-k)', G(k) = diag(tm1, tm2, ■ ■ ■ , tmn-k).

if the parameter 0 involved in Z(k) and G(k) are known then (29) and (30) together defines a generalized Gauss-Markov setup and hence the BLUE 02 (k) of a2 is obtained as,

02 (k) = [(Z (k))' (G(k))-1 Z (k)]-1 (Z (k))' (G(k))-1Z[n](k)

(31)

and the variance of 02 is given by

Var(02(k)) = [(Z (k))' (G(k))-1 Z (k)]-102.

(32)

2

x

2

On substituting the values of Z(k) and G(k) in (31) and (32) and simplifying we get,

72 (k) = Sik(Zm'/thr ) Z[mi]mi (33)

Si=1 (Zmi 1 Wmi)

and

1

Var( 72 (k)) =-;-o22. (34)

( 2( )) En=-1kZm/hm) 2 ( )

Remark 4.1. Since both the BLUE 72 (k) and the unbiased estimator (k) based on the censored ranked set sample utilize the distributional property of the parent distribution they lose the usual robustness property. Hence in this case the BLUE 72(k) shall be considered as a more preferable estimator than c| (k).

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