CC BY
3
ESTIMATING THE POPULATION MEAN USING
STRATIFIED DOUBLE UNIFIED RANKED SET SAMPLING FOR ASYMMETRIC DISTRIBUTIONS
Mohammed Ahmed Alomair1, Chainarong Peanpailoon2*, Roohul Andrabi3, Tundo4 and Khalid U1 Islam Rather5
department of Quantitative Methods, School of Business, King Faisal University 2*Department of Organization Curriculum and Teaching Program (Mathematics), Faculty of Education Sakon Nakhon Rajabhat University, Sakon Nakhon,Thailand 3Department of Management Studies, Dr. MGR Education and Reseach Institute Chennai Tamil
Nadu, India
4 Department Informatics, Sekolah Tinggi Ilmu Komputer Cipta Karya Informatika (STIKOM CKI) 5Division of Statistics and Computer Science, Main Campus SKUAST-J, Jammu ima.alomairtgkfu.edu.sa, 2*[email protected], [email protected], [email protected], [email protected]
Abstract
In this study, we propose a novel sampling technique known as Stratified Unified Ranked Set Sampling (SDURSS) and evaluate its efficiency for estimating population means. SDURSS is designed to enhance the estimation accuracy by integrating concepts from ranked set sampling with stratified sampling. Our results demonstrate that the SDURSS estimator generally exhibits superior efficiency compared to SRS, particularly in complex distribution scenarios. While SDURSS often performs more efficiently than SSRS and SRSS, its performance relative to these methods varies depending on the specific distribution and sample size. In several cases, SDURSS outperforms SSRS and SRSS, highlighting its potential benefits in practical applications. The findings suggest that SDURSS is a promising alternative to traditional sampling methods, offering improved efficiency and potentially more accurate estimates of population means. This research underscores the value of exploring advanced sampling techniques to enhance statistical estimation, particularly in scenarios involving asymmetric distributions where traditional methods may be less effective.
Keywords: Simple random sampling, Ranked set sampling, Median Ranked Set Sampling, Unified ranked set sampling, Double Unified ranked set sampling, Stratified Double Unified ranked set sampling
I. Introduction
The ranked set sampling (RSS) method to estimate the population mean of average yields prosed by [1]. Later, RSS was developed and modified by many authors to estimate the population parameters. The mathematical proof for RSS. They proved that the sample mean based on RSS is an unbiased estimator of the population mean, which gave smaller variance than the sample mean based on a simple random sample (SRS) with the same sample size provided [2]. The variance of the sample mean based on RSS is less than or equal to that of the SRS, whether or not there are
errors in ranking demonstrated by [3]. The RSS method of stratified ranked set sampling (SRSS) suggested by [4]. Some nonparametric tests for assessing the assumption of perfect ranking in RSS and powerful rank tests for perfect rankings proposed by [5-9], a Unified ranked sampling (URSS) suggested by [9-10]. RSS is called Double ranked set sampling (DRSS) developed by [1]. The RSS method is efficiency increasing the number of set and the number of cycles.
This study aims to propose the Stratified Double Unified Ranked Set Sampling (SDURSS) for estimating the population mean of asymmetric distributions and to study the efficiency of the empirical mean estimator based on SDURSS. Estimators in literature.
I. Simple Random Sampling (SRS)
SRS is a method of selecting units out of units such that every one of the distinct samples has an equal chance of being drawn.
II. Stratified Sampling
In the stratified sampling method, the population of units is divided into non-overlapping subgroups known as strata each stratum has units, respectively, such that For full benefit from stratification, the size of the hth strata, denoted by for, must be known. Then the samples are drawn independently from each stratum, producing sample sizes denoted by , such that the total sample size is If a simple random sample is taken from each stratum, the whole procedure is known as a stratified simple random sampling (SSRS).
RSS technique can be described as follows:
Step 1: Select samples of the SRS method from the population of interest. Step 2: Allocate the selected units as randomly as possible into sets, each of size. Step 3: Rank the units in each set with respect to the variable of interest.
Step 4: Choose a sample by taking the smallest ranked unit in the first set, the second smallest
ranked unit in the second set, continue the process until the largest ranked unit is selected from the
last set. Then the taken samples are measured the variable of interest.
Step 5: Repeat step 1 through step 4 for cycles to draw the RSS sample of size. [12]
Example 1: Let (m = 7, r = 1) be
II. Materials and methods
III. Ranked Set Sampling (RSS)
X21, X22 , X23 ,X24,X25 , X26 , X27
27
X51, X52, X53, X54,X55 ,X56,X57
57
X61,X62,X63,X64,X 65,X66,
67
Then the measured RSS units are Xn,X22,X33,X44,X55,X66,X7 The empirical mean estimator of RSS is given by
r m
mr
j=1 i=1
and variance can be estimated by
2 2 CT 1
Var ) = ----— - A ■
v ' mr m y ' '
Also
1 r m
1 r m mr *
mr ■ . , . , j=1 '=1
Considerations:
1. Note: m = set size, r = number of cycles )times(, n = sample of size(
2. The RSS use for Population infinite.
IV. Unified Ranked Set Sampling (URSS)
URSS technique can be described as follows:
Step 1: Use a SRS method to select m2 units from the population of interest and rank them with respect to the variable of interest.
Step 2: Select the sample units for measurement as follow If m is an odd number, the ranked
(if] \
--b(/-l)ff7 units will be selected for i = \2,---,m. On the other hands, if m is an even
number, divide the sample unit into 2 sections from size of the sample unit m2 . Where sections 1 select ^m + (/-1)mJ units and sections 2 select m + lj + (z- l)mJ units will be selected, for
i = 1,2, • • -,m
Step 3: Repeat steps 1 and 2 for r cycles ) for /= 1,2, ■■■,;•( to draw the URSS of size n = mr Define ^,]ybe the URSS sampled unit of the ith rank from the jth cycle, where j =1,2, ■■■,m and
/=1,2,- --,/•. [9-10] )even number(
Example 2: Consider the case of (m = 6, r = 1) . Draw a simple random sample of size m2 = 62 = 36 units as
X11, X12, X13, X14, X15, X16, X17,X18,X19,X10,X11,X12,X13,X14,X15,X16,X17,X18, X19, X20,X21,X22, X23,X24,X25,X26,X27,X28,X29, X30, X31,X32, X33,X34, X35, X36
From the size of the sample unit m2, divide the sample unit into 2 sections
( YYl i
Where section 1, We select the sample unit from I — + (i -l)m I for i = 1,2, • • •, m
X11 = X12 = X13 = X14
X,
X16 = X17 = X18 = X19 = X10 = X11
X12 ' X13 ' X14 ' X15 ' X16
Where section 2, We select the sample unit from | | + l |+ (' -!)»' | for 7=l,2,---,7
Xr
X,
X19,X20,X21,X22,X23
X-
24
X25,X26,X27,X28, X29
X
30
X31,X32,X33,X34,X35, X
36
Let X13,X19,X15,X22,X28,X34 is DURSS of size 6 )odd number(
Example 3: Consider the case of (m = 7,r = 1) . Draw a simple random sample of size m2 = 72 = 49 units as
X11, X12, X13, X14, X15, X16, X17, X18, X19, X10, X11, X12, X13, X14> X15, X16, X17, X18, X19,X20,X21,X22,X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35 , X36, X37, X38, X39,X40,X41, X42, X43, X44, X45, X46, X47, X48, X49
We select the sample unit from
I m +1
(i-l)m for 7=1,2,
X11, X12, X13, X14, X15 X16
X17, X18, X19, X10, Xn, X12 ,Xn ,Xi A , X,
L13
29
X
30
14? 15
X3
X19, X20, X21, X22, X23, X25, X26, X27, X28 , X X36, X37, X38, X39,X40,X41, X42, X43, X44, X45, X46, X47 , X48, X49
Let X14,X11,X18,X25,X32,X39,X46 is URSS of size 7
X
16
X17, X18,
31, X32, X33, X34,X
35
V. Double Unified Ranked Set Sampling (DURSS) In research, the DURSS method is applied from the [11] follows:
Step 1: Use a SRS method to identify m3 elements from the target population and divide these elements randomly into m sets each of size m2 elements.
Step 2: Use the usual URSS procedure on each set to obtain m ranked set samples of size m each. Step 3: Apply the URSS procedure again on step )2( to obtain a DURSS of size m. The procedure is illustrated for the case of even and odd in the following example. )even number(
Example 4: Consider the case of (m = 6,r = 1) . Draw a simple random sample of size m3 = 63 = 216 elements )6 sets of size 36 each(. Assume the elements are
z(1U(1).x(1).-,x(1).x(2).x(2).x(2).-,x(2).x(3).x(3).x(3).-,x(3).
(1) (2) (3) (36) (1) (2) (3) (36) (1) (2) (3) (36)
X(4) x(4) x(4) ••• x(4) x(5) x(5) x(5) ••• x(5) x(6) x(6) x(6) ••• x(6)
(1)" (2)" (3)" " (36)" (1)" (2)" (3)" " (36)" (1)" (2)" (3)" " (36)
After ranking the elements of each set obtain 6 ranked set samples of size 6 each ) m2 = 62 = 36 (.
X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1) X(1)
(2) ' (3)' (4) ' (5)' (6)' (7)' (8) ' (9)' (10) ' (11) ' (12) ' (13) ' (14) ' (15) ' (16) ' (17) ' (18) ' X(1) ,X(1) , X(1) ,X(1) ,X(1) ,X(1) , X(1) ,X(1) ,X(1) ,X(1) ,X(1) ,X(1) , X(1) , X(1) , X(1) , X(1) , X(1
(20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36
X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^
(2) ' (3) ' W ' (5) ' (6) ' M ' M ' (9) ' (10) ' (") ' M ' M ' M ' M ' (16) ' M ' (18) '
X(2) x^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X
(20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34)
(35)
(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)'
X<3),X<3),X<3),X<3),X<3),X<3), X<3),X<3),X<3),X<3),X<3),X<3),X<3),X<3),X<3),X<3),X'
(20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34)
(35)
X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <41
(2)
(3)
(4) (5)
(6)
(7)
(8)
(9)
(10) (11) (12) (13) (14) (15) (16) (17) (18)
X (4 \ X (4 ■>, X (4), X (4\ X (4 \ X (4\ X (4\ X (4), X (4X (4X (4X (4X (4 \ X (4X (4X (4X (4
(20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36 X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5) X(5)
(2^ (3^ M ' (5) ' (6)' (7)' (8) ' (9)' (10) ' (11)' (12^ (13^ (14^ (15)' (16^ (17) ' (18) ' X(5) , X(5) , X(5) , X(5) , X(5) ,X(5) , X(5) , X(5) , X(5) , X(5) , X(5) , X(5) , X(5) , X(5) , X(5) , X(5) , X(5)
(20^ (21) ' (22)' (23^ (124^ (25^ (26) ' (27^ (28^ (29^ (30^ (31) ' (32^ (33)' (34^ (35^ (36) _
and
X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X(6) X (6)
' (2) ' (3) ' (4) ' (5) ' (6) ' (7) ' (8) ' (9) ' (10^ (11^ (12^ (13^ (14^ (15)' (16^ (17) ' (18^ ,X(6\ X(6\ X(6\ X(6\ X(6\ X(6\ X(6\ X(6^ X(6\ X(6\ X(6\ X(6\ X(6\ X(6\ X(6\ X(6\ X(6 J
(20^ (21) ' (22^ (23^ (24^ (25^ (26) ' (27^ (28^ (29^ (30^ (M) ' (32^ (33^ (34^ (35^ (36) _
We select the sample unit from the elements of each set obtain 6 ranked set samples of size 6 each
X« X«
X X X X X X X X X
,X
, X
, X«, X
0) (21)
, x^ X
0) (21)
, XJX
0) (21)
(1) i3ll
X(1) X(1) X(1) X(1) X(1)
(4) ' (5) ' (6) ' (7) ' (8) '
"(1) !9ll
(1) (22) ;
(10)
X(1), x x x x w, X
<23) <24> <25> <26> <27>
x(1), x x x x x
-(1) m:
(11) (12) (13) (14)
X(1), X« x« x« x'
X(1) x(1) x(1)
<29) (30) (31) <32)'
(16)
(1), X
(33)
h(1)
(34)
(17) (18)
X(1), X1
(35) (36
xv2 \ x (2 \ x (2 \ x (2 \ x (2'
(4) (5) (6) (7)
(8)
xv2 \ x (2 \ x (2 \ x (2 \ x (2 \ X
(10) (11) (12) (13) (14)
(2) (15)'
x 2 \ x (2 \ x (2)
(16) (17) (18)
-(2) i2^
xv 2 \ x (2 \ x (2 \ x (2 \ x (2 \ X (23) (24> <25) <26) <27)
(2) i2^
xv2 \ x 12 x 12 k x 12 x 12x
(29) (30) (31) (32)
(33)
'(2) X34Xj
X1 xl
(35) (36
X(3\x« x<3),x<3),x1
(4) (5) (6) (7) (8) |J9]_
^ (3), X (3), X (3), X (3), X (3)
mo (23) (24) (25) (26) (27)
xv 3 \ x (3 \ x (3 j, x x'
(10)'
'(3) (28)
(12) (13)
<3 >, X'
(14)
to-
, X
x[j J, X[J J, X[J J, X[J \ X (29) (30) (31) (32) (33)
(3)
(17 r (18) 3
(3 \ XX'
(16)
X
(35) (36
xv4 \ x (4 i x (4 \ x (4 i x (41x
(4) (5)
(6)
(7)
(8)
xv 4 \ x (41 x (4 x (4 i x (4;, X
(10) (11) (12) (13)
(14)
-(4)
Ml
r4 \ x (4\ x (4)
(16) (17)
(18)
, xrX
0) (21)
-(4)
m.
r4\ X (4\ x (4\ x (4x (4),
(23) (24) (25) (26) (27)
-(4) 128!
14\ 14\ 14\ 14\ 14\
x14 j, x 14 x 14 x 14 x 14 x
<29) (30) (31) (32)
(33)
-(4) 134I—
(35) (36
-(5) (3) '
X(sJ X(sJ X(sJ X(sJ X(sJ X
(4) (5)
(6)
(7)
(8)
■(5)
x(5) X(5) X(5) X(5) X(5) X
(10) (11) (12) (13)
(14)
(5)
-(5) X(5) X(5)
(16) (17)
(18)
,x <5), X
0) (21)
■(5) m-
X(5J X(5J X(5J X(5J X(5J X
(23) (124) (25) (26)
(27)
and
-(5)
X(5 \ X (5 \ X (5 \ X (5 \ X (5 \ X
(29^ (30^ (32^ (33) '
-(5)
■i3£L
X(5) x(5J
(35) ' (36)
(6 ) (3) '
x(6) X(6) X(6) X(6) X(6) X
(4) (5)
(6)
(7)
(8)
K6)
X2X-1
X(6J X(6J X(6J X(6J X(6J X
(10^ (11)' (12)' (13)' (14)'
(6 )
-(6) X(6) X (' (16)' (17) '
,x <6), X
20) (21)
(6) (22c
X(6 \ X (6 \ X (6 \ X (6 \ X (6
(23^ (24^ (25^ (26^ (27)'
-(6) i2^
X(6 \ X (6 \ X (6 \ X (6 \ X (6 \ X
(29^ (30)' (31^ (32^ (33)'
(6)
(18)
X{6) X(6) (35^ (36) ,
So, we have 6 DURSS
X(1) X (1) X (1) X (1) X (1)
(3) (9) (15) (22)
(28)
X (2 J X (2) X (2) X (2 J X (2 J
(3) ' (9^ (15^ (22^ (28) X (3) X (3) X (3) X (3) X (3)
(3) (9)
(15) (22)
(28)
X (4 J X (4) X (4) X (4 J X (4 J
(3^ (9^ (15)' (22)' (28)
X (5) X (5 ) X (5) X (5) X (5 ) (3) ' (9^ (15^ (22^ (28) 1
X (6) X (6) X (6) X (6) X (6)
X
(34)
,X(2
(34)
,X(3),
(34)
(4)
(34)'
(5)
(34)' (6 )
X
X
(3) (9) (15)
We select the sample unit from 6 DURSS
x « X « X
X
(22) (28) (34>
(3) (9) X(2) X(2)
h(1) (15) '
(3)
(9)
-(2) (15)'
x« X(3),X
■(3)
(15)'
(3) (9) _
x<4) x(4) X(4) X
X(1), X ^.
<22^ <28)'
X(2 \ X (2 ^ <22^ (28)
X(3) X <3)
(3)
(9)
(15)
(22) (28) X (4 )
(4)
x« x<5) X(5) X
(3)
(9)
(15)
(5) ml
x (6\x (6\x (6\x
(3) (9)
(15)
(6) i22!
X
(34)
.X(2 >,
(34)
X <3),
(34)
.X(4),
(34)
(5)
(34)' (6 )
X
(28) X (5) (28)
X(6), X' (28) (34)
M. A. Alomair, C. Peanpailoon, Roohul A., Tundo and KUI Rather RT&A, No 1 (82) ESTIMATING THE POPULATION MEAN USING..._Volume 20, March 2025
Let X(1),X(2\X(3J,X(4J,X(5J,X(6J is DURSS of size 6
(15) (15) (15) (22) (22) (22)
)odd numberf
Example 5: Consider the case of (m = 7, r = i) . Draw a simple random sample of size m3 = 73 = 343 elements )7 sets of size 49 each(. Assume the elements are
(1) (2) (3) (49) (1) (2) (3) (49) (1) (2) (3) (49) (1) (2) (3) (49)
x(5) x(5) x(5) ••• x(5) x(6) x(6) x(6) ••• x(6) x(7) x(7) x(7) ••• x(7)
(1)" (2)" (3)" " (49)" (1)" (2)" (3)" " (49)" (1)" (2)" (3)" " (49)
After ranking the elements of each set obtain 7 ranked set samples of size 7 each) m2 = 72 = 49 (. X (1) X (1) X (1) X (1) X (1) X (1) X (1) X (1) X (1) X(1) X(1) X (1) X(1) X (1) X(1) X(1) X (1) X (1)
W ' (2) ' (3)' (4) ' (5) ' (6) ' (7) ' (8) ' (9) ' (10^ (") ' M ' (») ' M ' (15) ' M ' (17) ' (18) '
X i1
(19
X
(37
X <2
(1)
X <2
(19
X <2
(37
X <3
(1)
X (3
(19
X
(37
X <4
(1)
X <4
(19
X (4
(37
X <5
(1)
X (5
(19
X <5
(37
X <6
(1)
X <6 (1
X <6
(3
x(1) ,X (1), X (1) ,X(1) ,X (1) ,X (1) ,X (1) ,X (1) ,X (1), X (1) ,X (1), X (1), X (1), X (1), X (1), X (1), X (1),
(20^ (21^ (22^ p3)' (24^ (25^ (26^ (27^ p8)' (29^ (30^ (31^ (32^ (»J' (34^ (35^ (36)' , X (1), X (1), X (1), X (1), X (1), X (1), X (1), X (1), X (1), X (1), X (1), X (1)
(38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49)
X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^ X ^
(2) ' (3) ' M ' (5) ' (6)' (7) ' (8)' (9) ' (10) ' (11) ' (12) ' (13) ' (14) ' (15) ' (16) ' (17) ' (18) '
X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ X^ (20)' (21)' (22)' (23)' (24)' (25)' (26)' (27)' (28)' (29)' (30)' (31)' (32)' (33)' (34)' (35)' (36)'
(t ^ (T ^ (T ^ (T ^ (T ^ (T ^ (T ^ (T ^ (T ^ (T ^ (T ^ (T ^ ,X12 j, X12 j, X12 j, X12 j, X12 j, X12 j, X12 j, X12 j, X12 j, X12 j, X12 j, X12 j
(38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49)
X(3), X <3), X <3), X «, X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3),
(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)
X(3 >, X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), X <3), (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36)
, X(3), X(3J, X(3J, X(3J, X(3J, X(3), X(3), X(3), X(3), X(3), X(3), X(3)
(38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49)
X(4 >, X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4),
(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)
X(4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4), X <4),
(20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36)
^4^ ^4^ ^4^ ^4^ (4^ ^4^ ^4^ ^4^ ^4^ (4 \
, X14 j, X14 j, X14 j, X14 j, X14 j, X14 j, X14 j, X14 j, X14 j, X14 j, X14 j, X14 j
(38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49) X (5 J X (5 J X (5 J X (5) X (5 J X (5 J X (5 J X (5 J X (5 J X (5 J X (5 J X (5 J X (5 J X (5 J X (5 J X (5 J X (5 J
(2^ (3^ (4^ (5^ (6) ' (7^ (8^ (9^ (10^ (12^ (13^ (14^ (15) ' (16^ (17^ (18) '
X (5 \ X (5 \ X (5 \ X(5 \ X (5 ) ,X (5 \ X (5 \ X (5 \ X (5 \ X (5 \ X (5 \ X (5 \ X (5 \ X (5 \ X (5 \ X (5 \ X (5 \ (2°) ' (21^ (22^ (23^ (124^ (25) ' (26^ (27^ (28^ (29^ (3°) ' (31^ (32^ (33^ (34^ (35) ' (36) '
X(5J , X(5J , X(5J , X(5J , X(5J , X(5J , X(5J , X(5J , X(5J , X(5J , X(5J , X(5J
(38) ' (39^ (40^ (41^ (42^ (43^ (44^ (45^ (46^ (47^ (48) ' (49) X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J X (6 J
' (2^ (3) ' (4^ (5) ' (6) ' (7)' (8) ' (9) ' (10)' (11) ' (12^ (13^ (14^ (15)' (16^ (17)' (18) '
,X (6 ^ X (6 \ X (6 ^ X (6 \ X (6 \ X (6 \ X (6 \ X (6 ^ X (6 \ X (6 \ X (6 \ X (6 \ X (6 \ X (6 \ X (6 \ X (6 \ X (6 \ (20^ (21^ (22) ' (23^ (24^ (25^ (26^ (27) ' (28^ (29^ (30^ (31^ (32) ' (33^ (34^ (35^ (36) '
, X(6} , X(6} , X(6} , X(6} , X(6} , X(6} , X(6} , X(6} , X(6} , X(6} , X(6} , X(6} (38^ (39^ (40) ' (41^ (42^ (43^ (44^ (45) ' (46^ (47^ (48^ (49)
and
X(7J X{1 ] X(7J X{1 ] X{1 ] X(7J X{1 ] X(7J X{1 ] X{1 ] X(7J X{1 ] X(7J X{1 ] X{1 ] X{1 ] X{1 ] X{1 ]
(1) ' (2) ' (3) ' (4) ' (5) ' (6) ' M ' (8) ' (9) ' M ' (") ' M ' (13) ' M ' M ' M ' (") ' (18) ' X (7 J X {1 ] X {1 ] X {1 ] X(J ] X {1 ] X(J ] X {1 ] X {1 ] X {1 ] X (7 J X (7 J X (7 J X {1 ] X {1 ] X {1 ] X {1 ] X (7 J
(19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (7), X (7), X (7), X (7), X (7), X (7), X (7), X (7), X (7), X (7), X (7), X (7), X (7)
(37) (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (48) (49)
We select the sample unit from the elements of each set obtain 7 ranked set samples of size 7 each
X « X « X « X
(1)
(2) (3)
POO
(4) '
r(1),x« x« x« x« xX
(5) (6) (7) (8) (9) (10)
(1) (11)'
X(1), X« X « X « X « X « X
(12) (13) (14) (15) (16)
(17)
f(1)
(18)'
X« , X «, X« , X« , X «, X« , X
(19) (20) (21) X «, X« , ^
(22) (23) (24)
(1) .(25) '
X(1), X «, X «, X «, X «, X «, X
(26) (27)
(37) (38) '
(1) (39)'
X(1), X«, X «, X «, X «, X «, X (40) (41) (42) (43) (44) (45)
(28) (29) (30) (31) X(1) X (1) X (1)
(1)
X(1), X (1), X (1), X (1), (33)' (34)' (35)' (36)'
h(1) (46)
(47) (48) (49)
X (2X (2X (2\ X
(1) (2) (3)
X12 \ X (2\ X (2\ X (2 \ X (2\ X (2\ X
(5) (6)
(7)
(8)
(9)
(10)
(2 ) (11) '
Xv 2X (2X (2X (2X (2X (2X
(12) (13) (14) (15) (16) (17)
(2) .(18) '
X(2X(2',X(2X(2',X(2X(2X
(19) (20) (21) (22) (23) (24)
(2) (25)'
Xy 2 \ X (2 >, X (2X (2X (2 \ X (2 \ X
<26) <27) <28) <29)
(30)
X (2X (2 \ X
(37) (38)
(2) (39)'
^ , 2) 12) 12) 12) 12) 12) 2 J, X 12 X 12 j, X 12 j, X 12 j, X 12 j, X (40) (41) (42) (43) (44) (45)
(2) .(46) '
t , 2) 12 , r J,xl2;,X (47) (48)
(31) 2)
(49)
(2) i32!
Xv2X(2X(2X(2
(33) (34) (35) (36)
X <3), X X <3), X
(1) (2) (3)
■(3)
(4) '
X(1), X <3), X <3), X <3), X <3), X <3), X
(5) (6)
(7)
(8) (9)
(10)
(3 ) .(11) '
),X<3),X<3),X<3),X<3),X<3),X
(12) (13) (14) (15) (16)
(17)
(3) (18)'
X<3),X<3),X<3),X<3),X<3),X<3),X
(19)
X <3), X
(37) (38)
(20) 3)
(21) (22) (23) (24)
(3)
(25) :
r3), X <3), x <3), X <3), X <3), X <3), X
(26) (27) (28) (29) (30)
-(3)
1391
,3 ^ ¡3) ¡3)
XJ, X 13 j, X 13 j, X , X , X , X (40) (41) (42) (43) (44) (45)
(3) Ml
4- I ¡3 I
r J, X X
(47) (48)
(31)
3)
(49)
(3)
r3), X <3), X <3), X <3), (33) (34) (35) (36)
X <4), X <4), X <4), X
(1) (2) (3)
Xv 41 X (4 X (4 X (4 X (4 X (41
(5) (6)
(7)
(8)
(9)
(10)
)
X(4 X(4 X(4 \ X(4 X(4 X(4
(12) (13) (14) (15) (16) (17)
.M.
X(4>,X(4>,X(4>,X(4>,X(4>,XX'
(19) (20) (21) (22) (23) (24)
X (4 \ X (4X
(37) (38)
4)
.(25) '
(4X (4), X (4X (4X (4 \ X X
(26) (27)
(28)
(4)
t , 4 \ 14 \ 14 \ 14 \ 14 \ 14 \
Xv J, X14X14j, X14j, X14j, X14j, X (40) (41) (42) (43) (44) (45)
46)
(30)
(31) (4)
l(32l_
-(4X (4X (4X (4
(33) (34) (35) (36)
(47) (48) (49)
X <5), X <5), X <5), X
(1) (2) (3)
-(5)
(4) '
X(5) X(5J X(5J X(5J X(5J X(5J X
(5) ' (6^ (7^ (8^ (9) ' (10) '
f-(5) (11)'
X(5J X(5J X(5J X(5J X(5J X(5J X
(12)' (13)' (14)' (15)' (16)' (17)
-(5) (18)'
X (5 ) X (5 ) X (5 ) X (5 ) X(5 ) X (5 )
(19) (20) (21) (22) (23)
(124)
"(5)
(25)
X (5) X (5) X (5) X (5) X (5) X (5) X
(26) (27) (28) (29)
(30)
X(5\ X(5\ (37^ (38) '
-(5)
(39) '
X(5 J, X (5 J, X (5 J, X (5 J, X (5 J, X (5 J, X'5 J.
(40) (41) (42) (43) (44) (45) [(46) '
X(5 \ X <5), X' <47) <48)
(31)
(5)
(49)
■(5) ml
X(5) X(5) X(5) X(5)
(33) (34) (35)
(36)
x<6) X<6) X<6) X
(2)
(3)
(6 ) (4) '
X (6 ) X (6 ) X (6 ) X (6 ) X (6 ) X (6 ) X (6 )
(5)' (6)' (7)' (8)' (9)' (10
X(6) X<6) X<6) X<6) X<6) X<6) X
(12) (13) (14) (15) (16)
(17)
(6 ) (18)'
X (6 \ X (6 \ X (6 \ X (6 \ X (6 \ X (6 \ X
(19) (20) (21) (22) (23) (24)
(6) (25)'
X(6 J X (6 J X (6 J X (6 J X (6 J X (6 J X
(26) (27) (28) (29)
(30)
X <6), X <6), X
<37) <38)
)
(39)'
X(6 J X (6 J X (6 J X (6 J X (6 J X (6 J X
(40) (41) (42) (43) (44)
(45)
and
(6) (46)'
X(6 \ X <6), X
<47) <48)
(31) (6) (49)
K6)
Ml
X(6 J X (6 J X (6 J X (6 J
(33) (34) (35)
(36)
X <7 \ X <7 \ X <7 \ X
(1) (2) (3)
X <7 \ X <7), X <7), X <7), X <7), X <7), X
(19) (20)
X <7), X <7), X
<37) <38)
so we have 7 DURSS
We select the sample unit from 7
X>, X^>, x1 (11) (18)
(25) (32) (39)
, X <2), X <2), X
(11) (18)
X <3), X X
(11) (18)
, X <4), X <4), X
(11) (18)
X(5), X <5), X (11) (18)
, X <6), X <6), X
(11) (18)
, X <7), X <7), X
5) (32) (39)
5) (32) (39) X(6) X(6)
(11)
DURSS 1
(18)
, X'X ' 25) (32) (39)
X^, X^, X
(11) (18) X(2) X(2)
(11) ' (18) '
X(3), X X
(11) (18)
X(4), X <4), X
(11) (18)
X(5>, X <5), X (11) (18)
X(6>, X <6), X
(11) (18)
X(7>, X <7), X
(11) (18)
X(2) X(2'
(32) (3)
X (4), X (4'
(32) (5)
(32)
X
, X1 X , X1 X , X , X'
1)
(46)' 2)
(46)'
3)
(46)'
4)
(46)'
5)
(46)'
6)
(46)'
7)
(46)'
(25)
Xw xy ! (32)' (39)
X(2 J X (2 J
5)
T5I
(32) (3)
X , X '
(32) (39)
X(4) x<4'
(32) (5)
(32) (39)
X(6 J, X (6 J (32)' (39)
X (7), X <7)
(32)' (39)
X , X1 X , X' X , X , X'
1)
(46)' 2)
(46)'
3)
(46)'
4)
(46)'
5)
(46)'
6)
(46)'
7)
(46)'
Let X^,X'2\X'3\X'4\X'5\X(6
(25) (25) (25) (25) (25)
X ' is DURSS of size 7
(25) ' (25)
VI. Stratified Unified Ranked Set Sampling )SDURSS(
The population of N units is divided into L non-overlapping sub-groups known as strata each
stratum have Vj, A^ units, respectively, such that V, + N2 H-----\-NL =N. The size of the hth
strata denotes by Nh for h = 1.2.•••./, . Then the samples are drawn independently from each stratum, producing samples sizes denoted by «1,n2,...,nL, such that the total sample size is
L
n = ^ nh. If the DURSS technique is applied for each stratum then the whole procedure is called a
h=1
SDURSS. Define Xk{j be the SDURSS sampled unit of the i'h rank, the j'h cycle in the hth
stratum, where= 1,2,-• -jn; / = 1,2,-••,/■; k = 1.2.• • •.m. and // = 1.2,- • •./, . The mean of selected units is used as a population mean estimator.
Example 6: Suppose that we have two strata, i.e. L - 2 and h = 1,2.Let (m,r) Assume that from the
first stratum we select a sample of size mxr = 6x2 = 12 and from the second stratum we want a sample of size mxr = 6x2 = 12Then the process as illustrates as follows : Stratum 1: Now, select 12 samples as follows: Consider the case of (stratum!, m = 6, r = 1) and (stratum! m = 6, r = 2) . Draw a simple random sample of size m3 = 63 = 216 elements )6 sets of size 36 each(. Assume the elements are
(r 1 X1(1) XD X1(1). [1]1 ' [2]1 ' [3]1 ' .. X1(1) X2(1 X2(1 X2(1.. ' w' № ' I2!1 ' [3]1 ' . X2W X3(1 X3M X3M . ' № ' [1]1 ' № ' [3]1 ' X3(1), [36]1
Stratum lr -1) X X41 X . [1]1 ' [2]1 ' [3]1 ' •• x4(1) x5(1) x5(1) x5(1) • ' [36]1 ' [1]1 ' [-]! ' [3]1 ' .. x5(l) x6(l) x6(l) x6(l) ' [36]1 ' [1]1 ' [2]1 ' [3]1 ' X6«, [36]1
1 (r x1(1),x1(1),x1(1),--[l]2 ' [2]2' [3]2' X 1(1),X 2(1),X 2(1),X2(1),-[36]2' [l]2 ' [2]2 ' [3]2 ' -,x2(1),x3(1),x3(1),x3(1),- [36]2' [1]2 ' [2]2 ' [3]2 ' ••,X3(1), I36!2
(r - 2) X 4(1), X 4(1),X 4(1),-[I]2 ' [2]2 ' [3]2 ' ••,x4(1),x5(1),x5(1),x5(1),- ' [36]2 ' [1]2 ' [2]2 ' [3]2 ' ••,x5(1),x6(1),x6(1),x6(1), [36]2' [1]2 ' [2]2 ' [3]2 ' ■■■, X6(1), [36]2
For h = 1 we have: X15« X15« X15«,X22«,X22«,X22« X15«,X15«,X15«,X22«,X22« X22«
[1]1 [2]1 [3]1 [4]1 [5]1 [6]1 [1]2 [2]2 [3]2 [4]2 [5]2 [6]2
Stratum 2: Now, select 12 samples as follows:
Consider the case of (stratum2, m = 6, r = 1) and (stratum2, m = 6, r = 2). Draw a simple random sample of size m3 = 63 = 216 elements )6 sets of size 36 each(. Assume the elements are
(r ü X12) X12) X12) ... X!(2) X2(2) X2(2) X2(2) ... X2(2) X3(2) X3(2) X3(2) ... X3(2) [1 ' M ' [3]1 ' ' [36H ' [1} ' [2} ' [3} ' ' [36} ' [1]1 ' [2} ' [3} ' ' [36} '
Stratum 2 lr -1) X4(2) X4(2) X4(2) ... X4(2) X5(2) X5(2) X5(2) ... X5(2) X6(2) X6(2) X6(2) ... X6(2) [1]1 ' [2} ' [3} ' ' [36} ' [If ' [2} ' [3} ' ' [36} ' [1} ' [2} ' [3} ' ' [36} '
(r 2\ X 12) , X12), X12\- •, X X2{2\ X2{2\ X2{2\- •, X^, X3{2\X3{2\ X3{2\- •, X3(2), [l]2 [2]2 [3]2 [36]2 [l]2 [2 ]2 [3]2 [36]2 [l]2 [2 ]2 [3]2 [36]2
v -2) X 4(2),X4(2),X4(2),-,X4(2),X5(2),X5(2),X5(2),-,X5(2),X6(2),X6(2),X6(2),-,X6(2), [l]2 [2]2 [3]2 [36]2 [l]2 [2]2 [3]2 [36]2 [l]2 [2]2 [3]2 [36]2
For h = 2 we have' XXXX22^ XX22^ XXXXX22^ X22^
[1]1 ' [2]1 ' [3]1 ' [4]1 ' [5]1 ' [6]1 ' [1]2 ' [2]2 ' [3]2 ' [4]2 ' [5]2 ' [6]2
)Define: h , k = number of ranking the elements of each set, h = stratum size, i = number ofeach
r = number of cycles )times(( Therefore, the measured SDURSS units are
X15(1) X15(1) X15(1) X22(1) X22(1) X22(1) X15(1) X15(1) X15(1) X22(1) X22(1) X22(1)
[1]1 ' [2]1 ' [3]1 ' [4]1 ' [5]1 ' [6]1 ' [1]2 ' [2]2 ' [3]2 ' [4]2 ' [5]2 ' [6]2 X15(2) X15(2) X15(2) x22(2) X22(2) X22(2) X15(2) X15(2) X15(2) X22(2) X22(2) X22(2)
[1]1 ' [2]1 ' [3]1 ' [4]1 ' [5]1 ' [6]1 ' [1]2 ' [2]2 ' [3]2 ' [4]2 ' [5]2 ' [6]2
where their mean of these units is used as an estimator of the population mean.
To compare the efficiency of the empirical mean estimator based on SDURSS with their counterparts in SRS, SSRS, SRSS, and SMRSS via a simulation in R )Version 4.3.2( under the
population of 100,000 units divided into two strata each stratum has 50,000 units with the numbers of set in each stratum m = 2,4,6,10 and the number of cycles r = 2,5 . Using 5000 replications, estimates of the means, variances and mean square errors.
III. Results and Discussions
I. Estimation of Population Mean
Let Xj,X2,...,Xn be n independent random variables from a probability density function f (V), with mean (and variance j. The empirical mean estimator of DURSS is given by
1 m r
mr SZ XPMi-1m j (!)
X,
DURSS
i=1 j=1
m m +1
where l = — if i is an even number and l =- if i is an odd number )for / = 1,2,...,m(.
The DURSS variance can be estimated by
mr -
1 I m r / _
TZj iZZ \X[l+(i-l)m] j ~ XDURSS
i=1 j=1
>1
The SDURSS estimator of the population mean is given by
XSDURSS ~ ^ Wh (XDURSS ) h=1
Where Wh and XhRSS is the DURSS mean estimator in the h stratum.
The
variance
of X
SFRSS
is given by
Var (XSDURSS ) = Var
L W ( mh r
y у у X
fi mhr ¿-i^-i ['+НИ]j
h=1 h \ i=1 j=1
= У W- УУ Var fx
tT ml r2 [ £ У ['+C-1'mh
L W - mh r
h=1 L
=1
mh r
h=1 mhr У i=1 j=\ ['+(i"1)mh^-h
LWl ,2
" m, Г ['+('-1)mh]j-h h=1 h
(2)
(3)
)4(
IV. Simulation Study
The simulation study is designed to investigate the performance of SDURSS for estimating the population mean compared to their counterparts in SRS, SSRS, and SRSS under parent asymmetric distributions: Exp) 1(, Geo) 0.5(, Gamma) 0.5,1(, Gamma) 1,2(, Beta ) 3,3(, Beta ) 9,2(, Weibull) 0.5,1(, Weibull) 1,2(, Log N)0,1(, Logistic)0,l(, CHI)1(. The simulations are done based on the population of 100,000 units is divided into two strata each stratum has 50,000 units, which are conducted for the numbers of set in each stratum m = 2,4,6,10 and the number of cycles r = 2,5 on 5,000 replications. If the underlying distribution is asymmetric, the efficiencies of SDURSS relative to SRS, SSRS, SRSS, and SMRSS, respectively are given by
eff (XSDURSS ,XSRS ) = eff {XSDURSS 'XSSRS ) = eff {XSDURSS ,XSRSS ) = eff {XSDURSS 'XSMRSS ) =
where MSE is the mean square error )MSE(. The simulation results are shown in Tables 1-3.
MSE (Xsrs )
MSE (XSDURSS )
MSE (XSSRS )
MSE (XSDURSS )
MSE (Xsrss )
MSE (XSDURSS ) MSE (XSMRSS ) MSE (XSDURSS )
Table 1: The efficiency of SDURSS relative to SRS, SSRS, SRSS and SMRSS for estimating the population mean with
m = 2 and r = 2,5
Efficiency
Distribution r SDURSS vs. SDURSS vs. SDURSS vs. SDURSS vs.
SRS SSRS SRSS SMRSS
Exp)1( 2 0.7259 0.3542 0.6500 0.6256
5 0.0409 0.0206 0.6197 0.6194
Geo)0.5( 2 0.7141 0.3609 0.6721 0.6717
5 0.0426 0.0209 0.6467 0.6479
Gamma)0.5,1( 2 0.6433 0.3343 0.6179 0.6040
5 0.0383 0.0195 0.6139 0.6088
Gamma)1,2( 2 0.7026 0.3572 0.6709 0.6739
5 0.0401 0.0201 0.6086 0.6171
Beta )3,3( 2 0.8158 0.4088 0.7422 0.7366
5 0.0443 0.0219 0.6806 0.6852
Beta )9,2( 2 0.7868 0.3959 0.7208 0.7107
5 0.0429 0.0213 0.6623 0.6632
Weibull)0.5,1( 2 0.5839 0.2680 0.5658 0.5407
5 0.0374 0.0186 0.5836 0.5369
Weibull)1,2( 2 0.7080 0.3512 0.6523 0.6644
5 0.0404 0.0201 0.6165 0.6199
Log N)0,1( 2 0.5840 0.3203 0.5488 0.6177
5 0.0366 0.0178 0.6153 0.5559
Logistic)0,1( 2 0.7376 0.3687 0.6582 0.6721
5 0.0399 0.0201 0.6191 0.6138
CHI)1( 2 0.6550 0.3469 0.6387 0.6079
5 0.0393 0.0200 0.6222 0.6233
Based on Table 1, the numbers of set in each stratum m = 2 and numbers of cycle r = 2 , it indicates that the SDURSS estimator is less efficient than SRS, SSRS, SRSS and SMRSS estimators all asymmetric distributions.
Table 2: The efficiency of SDURSS relative to SRS, SSRS, SRSS and SMRSS for estimating the population
mean with m = 4 and r = 2,5
Efficiency
Distribution r SDURSS vs. SDURSS vs. SDURSS vs. SDURSS vs.
SRS SSRS SRSS SMRSS
Exp)1( 2 0.9332 0.4654 0.4024 0.3737
5 0.0617 0.0313 0.4720 0.1862
Geo)0.5( 2 0.9068 0.4551 0.3939 0.3768
5 0.0618 0.0312 0.4675 0.2088
Gamma)0.5,1( 2 0.8045 0.3935 0.3509 0.3304
5 0.0551 0.0278 0.4217 0.1454
Gamma)1,2( 2 0.9144 0.4566 0.3913 0.3603
5 0.0611 0.0307 0.4724 0.1862
Beta )3,3( 2 1.4000 0.6931 0.5862 0.5552
5 0.0808 0.0406 0.6180 0.3185
Efficiency
Distribution r SDURSS vs. SDURSS vs. SDURSS vs. SDURSS vs.
SRS SSRS SRSS SMRSS
Beta )9,2( 2 1.2778 0.6481 0.5556 0.5185
5 0.0730 0.0365 0.5587 0.2628
Weibull)0.5,1( 2 0.5858 0.3235 0.2795 0.2676
5 0.0440 0.0218 0.3354 0.0640
Weibull)1,2( 2 0.9209 0.4507 0.4038 0.3824
5 0.0610 0.0303 0.4637 0.1839
Log N)0,1( 2 0.6315 0.2918 0.2912 0.2554
5 0.0442 0.0237 0.3748 0.0823
Logistic)0,1( 2 1.1039 0.5582 0.4666 0.4395
5 0.0649 0.0323 0.5025 0.1992
CHI)1( 2 0.7453 0.3833 0.3540 0.3336
5 0.0543 0.0273 0.4172 0.1449
Based on Table 2, the numbers of set in each stratum m = 4, we can conclude that the SDURSS estimator is less efficient compared to SRS, SSRS, SRSS and SMRSS estimators for the numbers of cycle r = 2 underlying all asymmetric distributions.
Table 3: The efficiency of SDURSS relative to SRS, SSRS, SRSS and SMRSS for estimating the population mean with
m = 6 and r = 2,5
Efficiency
Distribution r SDURSS vs. SDURSS vs. SDURSS vs. SDURSS vs.
SRS SSRS SRSS SMRSS
Exp)1( 2 0.1401 0.7593 0.4259 0.2994
5 0.0856 0.0427 0.4305 0.1173
Geo)0.5( 2 1.4798 0.7489 0.4205 0.4103
5 0.0868 0.0430 0.4352 0.1515
Gamma)0.5,1( 2 1.2423 0.6121 0.3599 0.1710
5 0.0730 0.0362 0.3663 0.0909
Gamma)1,2( 2 1.5048 0.7504 0.4287 0.1875
5 0.0848 0.0424 0.4296 0.1164
Beta )3,3( 2 2.4839 1.2516 0.6903 0.2645
5 0.1263 0.0630 0.6389 0.2194
Beta )9,2( 2 2.0938 1.0469 0.5781 0.2344
5 0.1102 0.0547 0.5534 0.1742
Weibull)0.5,1( 2 0.8754 0.4244 0.2435 0.0809
5 0.0503 0.0245 0.2469 0.0344
Weibull)1,2( 2 1.5581 0.7616 0.4327 0.1887
5 0.0849 0.0430 0.4331 0.1194
Log N)0,1( 2 0.8643 0.4264 0.2589 0.0808
5 0.0557 0.0273 0.2816 0.0444
Logistic)0,1( 2 1.8218 0.9101 0.5095 0.1399
5 0.0929 0.0466 0.4730 0.1166
CHI)1( 2 1.2097 0.5977 0.3581 0.2171
5 0.0723 0.0367 0.3665 0.0985
Based on Table 3, the numbers of set in each stratum m = 6, it implies that the SDURSS estimator is less efficient than SRS, SSRS, SRSS and SMRSS estimators for the numbers of cycle r = 2 based on all asymmetric distributions.
Table 4: The efficiency of SDURSS relative to SRS, SSRS, SRSS and SMRSS for estimating the population mean
with m = 10 and r = 2,5
Efficiency
Distribution r SDURSS vs. SDURSS vs. SDURSS vs. SDURSS vs.
SRS SSRS SRSS SMRSS
Exp)1( 2 0.0860 0.0432 0.0139 0.0044
5 0.0843 0.0417 0.2568 0.0522
Geo)0.5( 2 0.0864 0.0430 0.0143 0.0065
5 0.0846 0.0417 0.2531 0.0799
Gamma)0.5,1( 2 0.0735 0.0371 0.0122 0.0043
5 0.0717 0.0356 0.2148 0.0462
Gamma)1,2( 2 0.0863 0.0430 0.0142 0.0042
5 0.0843 0.0414 0.2531 0.0513
Beta )3,3( 2 0.1287 0.0643 0.0205 0.0051
5 0.1246 0.0623 0.3758 0.0780
Beta )9,2( 2 0.1110 0.0555 0.0179 0.0043
5 0.1085 0.0538 0.3271 0.0649
Weibull)0.5,1( 2 0.0512 0.0255 0.0086 0.0023
5 0.0488 0.0244 0.1510 0.0203
Weibull)1,2( 2 0.0877 0.0438 0.0143 0.0044
5 0.0846 0.0425 0.2560 0.0523
Log N)0,1( 2 0.0540 0.0275 0.0092 0.0022
5 0.0530 0.0280 0.1627 0.0223
Logistic)0,1( 2 0.0944 0.0473 0.0153 0.0026
5 0.0919 0.0462 0.2773 0.0382
CHI)1( 2 0.0754 0.0369 0.0126 0.0058
5 0.0710 0.0360 0.2185 0.0595
Based on Table 4, the numbers of set in each stratum m = 10, it implies that the SDURSS estimator is less efficient than SRS, SSRS, SRSS and SMRSS estimators for the numbers of cycle r = 2 based on all asymmetric distributions.
V. Real Data example
In this section, the application of the proposed sampling method is shown by using a real data example. The researcher went to the area to collect data by himself. The data sets used in this example include: There are a total of 5 plots of False pakchoi, with a length of 20 meters and a width of 1 meter. Each plant will have a minimum number of flowers of 3 flowers per plant. If data is collected in batches, it will be 25 plants per batch with 75-150 flowers. Where 1 plot can store 20 data sets from a total of 5 plots, totaling 100 data sets. Figure 1-2 illustrate False pakchoi and Table 5
represent Number set and real data.
Figure 1 Figure 2
Table 5: Number set and real data for False pakchoi
Number set data Number set data Number set data Number set data Number set data
Set 1 103 Set 21 125 Set 41 109 Set 61 89 Set 81 97
Set 2 115 Set 22 123 Set 42 141 Set 62 131 Set 82 148
Set 3 103 Set 23 129 Set 43 133 Set 63 123 Set 83 118
Set 4 117 Set 24 118 Set 44 114 Set 64 144 Set 84 90
Set 5 150 Set 25 99 Set 45 138 Set 65 128 Set 85 125
Set 6 110 Set 26 97 Set 46 111 Set 66 149 Set 86 104
Set 7 123 Set 27 92 Set 47 116 Set 67 123 Set 87 90
Set 8 102 Set 28 146 Set 48 146 Set 68 148 Set 88 129
Set 9 143 Set 29 143 Set 49 145 Set 69 105 Set 89 108
Set 10 76 Set 30 115 Set 50 132 Set 70 120 Set 90 110
Set 11 97 Set 31 76 Set 51 129 Set 71 114 Set 91 118
Set 12 135 Set 32 99 Set 52 127 Set 72 125 Set 92 118
Set 13 140 Set 33 117 Set 53 108 Set 73 130 Set 93 126
Set 14 81 Set 34 112 Set 54 107 Set 74 120 Set 94 83
Set 15 99 Set 35 89 Set 55 136 Set 75 137 Set 95 114
Set 16 136 Set 36 130 Set 56 97 Set 76 139 Set 96 121
Set 17 93 Set 37 111 Set 57 99 Set 77 84 Set 97 97
Set 18 103 Set 38 142 Set 58 112 Set 78 115 Set 98 108
Set 19 83 Set 39 136 Set 59 97 Set 79 147 Set 99 94
Set 20 90 Set 40 96 Set 60 123 Set 80 142 Set 100 113
Total Population flower False pakchoi is 11,636, population mean X = 116.36 to collect a sample of size 8, using set size is m = 4 and number of cycles )times( is r = 2 in SRS, SSRS, SRSS, and SDURSS designs, DURSS technique can be described as follows:
I. Draw a simple random sample of size m3 = 43 = 64 elements )4 sets of size 16 each(.
II. Use the usual URSS procedure on each set to obtain m ranked set samples of size m each.
III. Apply the URSS procedure again on step )2( to obtain a DURSS of size 8.
The measured values in both SRS, SSRS, SRSS, SMRSS, and SDURSS and designs are presented in Table 6.
Table 6: Sampled units in SRS, SSRS, SRSS, SMRSS, and SDURSS designs.
SRS 117 90 97 123 146 145 108 94
SSRS Stratum 1 115 104 92 132 114 140 114 99
Stratum 2 111 146 148 97 118 107 84 97
SRSS Stratum 1 90 99 109 144 76 90 105 129
Stratum 2 90 114 143 145 81 97 111 120
SMRSS Stratum 1 76 108 142 148 81 96 114 115
Stratum 2 97 111 103 144 89 94 104 127
SDURSS Stratum 1 123 129 146 150 81 83 112 136
Stratum 2 104 109 125 148 83 97 102 139
XSRS - 115
XSSRS(stratum1) = 113 75, XSSRS^stratum2) =113.5 XSRSS(stratum1) = 105 25, XSRSS(stratum2) = 11263 XSMRSS(stratum1) = 110, XSMRSS(stratumT) = 108 63 XSDURSS ( stratum1) = 120,X SDURSS ( stratumT) =113.38
S2 = 481.15
SRS
Sssrs = 20.7731 S2rss = 29.2044 Ssmrss = 41.1914
SSDURSS = 37.23332
VI. Conclusion
In conclusion, the proposed estimator in SDURSS provide efficient their counterparts in SRS, SSRS, SRSS, and SMRSS for eleven parent asymmetric distributions in the case of a larger sample size and small size numbers of cycle. For the small sample size, the proposed estimator in SDURSS
still provide less efficient than four methods, but it gives more efficient than SRS and SSRS in some cases.
VII. Acknowledgement
The authors would like to express our thanks to Research and Department of Organization Curriculum and Teaching Program Mathematics. Faculty Education Sakon Nakhon Rajabhat University. We also thank the referees and the associate editor for their useful comments and suggestions on the earlier draft that led to this improved version.
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