Efficiency of Nearest Neighbour Balanced Block Designs for Second Order Correlated Error Structure

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 1 Ver. I (Jan. - Feb. 2017), PP 62-71
www.iosrjournals.org
DOI: 10.9790/5728-1301016271 www.iosrjournals.org 62 | Page
Efficiency of Nearest Neighbour Balanced Block Designs for
Second Order Correlated Error Structure
R. Senthil kumar1
and C. Santharam2
1
Assistant Professor, Department of Mathematics, Hindustan College of Arts and Science, Padur,
Chennai – 603103 (TN), INDIA
2
Rtd Professor, Department of Statistics, Loyola College, Chennai – 600034 (TN), INDIA
Abstract: The performance of a series of Complete and Incomplete Neighbour Balanced Block Designs for
Auto Regressive (AR), Moving Average (MA) and Nearest Neighbour (NN) error correlation structure is studied
when generalized least squares estimation is used. We have compared the efficiency of AR(2), MA(2) and NN
correlation structures. It is observed that efficiency for direct as well as neighbour effects is high, in case of
complete block designs for NN correlation structure. In case of incomplete block designs MA(2) correlation
structure turns out to be more efficient as compared to others models with  21,  in the interval 0.1 to 0.4.
It is therefore concluded that when block sizes are large and neighbouring plots are highly correlated,
generalized least squares for estimation of direct and neighbour effects can be used. The gain in efficiency of
NNBD and NNBIBD over regular block design is high under MA(2) models for direct and neighbour effects of
treatments.
Keywords: Neighbour Balanced Block Design; Correlated observations; Generalized least squares; Auto
Regressive; Moving Average; Nearest neighbour; Efficiency; Regular Block Design.
I. Introduction
In many experiments, especially in agriculture, the field on a given plot may be affected by treatments
on neighbouring plots as well as by the treatment applied to that plot. To diminish these undesirable effects,
Rees [15] first provided designs for the test and named such designs as neighbour designs. He used technique in
virus research which requires arrangement in circles of samples from a number of virus preparations in such a
way that over the whole set, a sample from each virus preparation appears next to a sample from every other
virus preparation. He defined a neighbour design as an arrangement of v antigens (called symbols) in b
circular plates (called blocks) such that: each block has k symbols, not necessarily distinct, each symbol
appears r times in the design and each symbol is a neighbour of every other symbol precisely  times. A
neighbour design with at least one block having less than v distinct symbols may called as incomplete block
neighbour designs. The designs in both series are neighbour balanced in the sense that every experimental
treatment has each other treatment once as a right neighbour and once as a left neighbour. When treatments are
varieties, neighbour effects may be caused by differences in height, root vigor, or germination date, especially
on small plots, which are used in plant breeding experiments. Treatments such as fertilizer, irrigation, or
pesticide may spread to adjacent plots causing neighbour effects. Such experiments exhibit neighbour effects,
because the effect of having no treatment as a neighbour is different from the neighbour effects of any treatment.
In case of block design setup if each block is a single line of plots and blocks are well separated, extra
parameters are needed for the effect of left and right neighbours. An alternative is to have border plots on both
ends of every block. Each border plots receives an experimental treatment, but it is not used for measuring the
response variable. These border plots do not add too much to the cost of one-dimensional experiments.
Neighbour balanced designs, where in the allocation of treatments is such that every treatment occurs equally
often with every other treatment as neighbours, are used for these situations and permit the estimation of direct
and neighbor effects of treatments. The neighbour effects are also called as interference effects, indirect effects
or remote treatment effects.
The effect of correlation on the usual two-way analysis of variance and on the power of usual tests has
been studied by Box [4]; Anderson et al. [1] and Aastveit [2]. Keedwell [9] considered 2-fold perfect circuit
designs, these being balanced circuit designs whose neighbour properties apply not only to immediate
neighbours but also to neighbours that are two places apart. In situations where the correlation structure among
the observations within a block is known, may be from the data of past similar experiments, it may be
advantageous to use this information in designing an experiment and analyzing the data so as to make more
precise inference about treatment effects Gill and Shukla [6]. Neighbour balanced block designs for correlated
errors by Kunert [10]. Lindner et al. [12] considered 2-perfect k-cycle systems of order v, i.e., balanced circuit
RND’s (Rees Neighbour Designs) whose neighbour properties hold both for immediate neighbours and for 2-

Efficiency of nearest neighbour balanced block designs for second order correlated error structure
places-apart neighbours. Optimal and highly efficient two dimensional designs have been constructed for
correlated errors on the torus and in the two dimensional plane by Morgan and Nizamuddin [13]. Azais et al.
[3] obtained a series of efficient neighbour designs with border plots that are balanced in 1v blocks of size v
and v blocks of size 1v , where v is the number of treatments. Bailey [5] has given some designs for
studying one-sided neighbour effects. These neighbour balanced block designs have been developed under the
assumption that the observations within a block are uncorrelated. Kunert et al. [11] considered two related
models for interference and have shown that optimal designs for one model can be obtained from optimal
designs for the other model. Martin and Eccelston [14] have given variance balanced designs under interference
and dependent observations. Santharam and Ponnuswamy [18] examined the optimality and efficiency of
nearest neighbor balanced block designs when error follows Auto Regressive (AR), Moving Average (MA) or
Auto Regressive Moving Average (ARMA) models. Senthil Kumar and Santharam [17] Efficiency of Nearest
Neighbour Balanced Block Designs using ARMA models. Senthil Kumar and Santharam [16] Efficiency of
NNBD over NNBIBD using First Order Correlated Models. Iqbal et al. [7] constructed second order neighbour
designs for 73  k in circular block using method of cyclic shifts.
In this paper, neighbour balanced block designs for observations correlated within a block have been
investigated for the estimation of direct as well as left and right neighbour effects of treatments. The
performance of these designs for AR(2), MA(2) and NN error correlation structure is studied when generalized
least squares estimation is used. We have also investigated the efficiency of Nearest Neighbour Balanced Block
Design (NNBD) and Nearest Neighbour Balanced Incomplete Block Design (NNBIBD) in comparison to
regular block design when the error follows second order correlated models with  21,  in the interval
-0.4 (-0.4) 0.4.
II. Model Structures and Information Matrix
Let  be a class of binary neighbour balanced block designs with bkn  units that form b blocks
each containing k units. ijY be the response from the
th
i plot in the
th
j block  bjki ,,2,1;,,2,1   .
The layout includes border plots at both ends of every block, i.e. at
th
0 and  th
k 1 position and observations
for these units are not modeled. The following fixed effects additive model is considered for analyzing a
neighbour balanced block design under correlated observations:
(2.1)
where  is the general mean,  ji, is the direct effect of the treatment in the
th
i plot of
th
j block, j is the
effect of the
th
j block,  jil ,1 is the left neighbour effect due to the treatment in the  th
i 1 plot of
th
j block,
 ji ,1 is the right neighbour effect due to the treatment in the  th
i 1 plot in
th
j block, ije are error terms
distributed with mean zero and a variance-covariance structure  bI ( bI is an identity matrix of order
b and  denotes the kronecker product). Assuming no correlation among the observations between the
blocks and correlation structure between plots within a block to be the same in each block,  is the correlation
matrix of k observations within a block. If the errors within a block follow an Second Order Auto Regressive
model (AR(2)) then kb MI  where kM is a kk  matrix given by
















0321
2101
1210
rrrr
rrrr
rrrr
M
kkk
k
k
k




The elements of kM are
      2
1
2
2
2
20 111  r
   02
2
11 1 rr  
   022
2
12 1 rr  
     ,1,1, elY ijjjijijiij

 

For 3k ,   02211 rrrr kkk   
If the errors within a block follow Second Order Moving Average model (MA(2)) the kb NI  where
kN is a kk  matrix given by
























2
2
2
1
2211
2
2
2
1211
2211
2
2
2
1
10000
01
001









kN
The NN correlation structure, the  is a matrix with diagonal entries as 1 and off-diagonal entries as  .
Model (2.1) can be rewritten in the matrix notation as follows:
(2.2)
where Y is 1n vector of observations, 1 is 1n vector of ones,
'
 is an vn incidence matrix
of observations versus direct treatments,  is 1v vector of direct treatment effects,
'
1 is a vn matrix of
observations versus left neighbour treatment,
'
2 is a vn matrix of observations versus right neighbour
treatment, l is 1v vector of left neighbour effects,  is 1v vector of right neighbour effects,
'
D is an
bn  incidence matrix of observations versus blocks,  is 1b vector of block effects and e is 1n vector
of errors. The joint information matrix for estimating the direct and neighbour (left and right) effects under
correlated observations estimated by generalized least squares is obtained as follows:
(2.3)
with
The above vv 33  information matrix  C for estimating the direct effects and neighbour effects of
treatments in a block design setting is symmetric, non-negative definite with row and column sums equal to
zero. The information matrix for estimating the direct effects of treatments from (2.3) is as follows:
(2.4)
where
and
Similarly, the information matrix for estimating the left neighbour effect of treatments  lC and right
neighbour effect of treatments  C can be obtained.
  1'111'1
1111 
 kkkk
21
1
221211 CCCCC 

'
11 )(  
bIC
 '
2
'
112 )()(  
bb IIC








 

'
22
'
12
'
21
'
11
22
)()(
)()(
bb
bb
II
II
C

















'
22
'
12
'
2
'
21
'
11
'
1
'
2
'
1
'
)()()(
)()()(
)()()(
bbb
bbb
bbb
III
III
III
C
eDlY   ''
2
'
1
'
1

2.1 Construction of Design
Tomer et al. [19] has constructed neighbour balanced block design with parameters v (prime or prime
power),  1 vvb ,   mvvr  1 ,  mvk  , 4,,2,1  vm  and  mv  using
Mutually Orthogonal Latin Squares (MOLS) of order v. This series of design bas been investigated under the
correlated error structure. It is seen that the design turns out to be pair-wise uniform with 1 and also
variance balanced for estimating direct  1V and neighbour effects  32 VV  .
Example:
Let 5v and 0m . The following is a neighbour balanced pair-wise uniform complete block design with
parameters 5v , 20b , 20r , 5k , 5 and 1 :
2 3 4 5 1 2 3
3 4 5 1 2 3 4
4 5 1 2 3 4 5
5 1 2 3 4 5 1
1 2 3 4 5 1 2
3 4 5 1 2 3 4
4 5 1 2 3 4 5
5 1 2 3 4 5 1
1 2 3 4 5 1 2
2 3 4 5 1 2 3
4 5 1 2 3 4 5
5 1 2 3 4 5 1
1 2 3 4 5 1 2
2 3 4 5 1 2 3
3 4 5 1 2 3 4
5 1 2 3 4 5 1
1 2 3 4 5 1 2
2 3 4 5 1 2 3
3 4 5 1 2 3 4
4 5 1 2 3 4 5
1 2 3 4 5 1 2
2 3 4 5 1 2 3
3 4 5 1 2 3 4
4 5 1 2 3 4 5
5 1 2 3 4 5 1
The information matrices for estimating the direct and neighbouring effects of treatments for AR(2) with
 = 0.1    21 is obtained given as below:







5
47944.15
J
IC
and





5
45525.15
J
ICCl 
Similarly for MA(2) & NN structures,







5
36396.15
J
IC
and





5
81641.16
J
ICCl 







5
12109.9
J
IC
and





5
44325.10
J
ICCl 
These matrices have been worked out using R package. For, 1m the resulting design will be a neighbour
balanced pair-wise uniform incomplete block design with parameters 5v , 20b , 16r , 4k , 4
and 1 .

III. Efficiency of Neighbour Balanced Pair-Wise Uniform Block Designs:
In this section, a quantitative measure of efficiency of the NNBD has been derived when error structure
follows AR(2) and MA(2) models. The comparison of universally optimal neighbour balanced design for v
treatments in  1v complete blocks of Azais et al. [3] considering observations to be correlated within the
blocks. We compare the average variance of an elementary treatment contrast

 'ss  in both cases. The
average variance of an elementary treatment contrast Kempthorne, [8] for direct effects of the neighbour
balanced design of Azais et al. [3] estimated by generalized least squares methods, is given by






1
1
1
2
1
2 v
s
sA
v
V 

where s ’s are the  1v non-zero eigen values of C for Azais et al. [3],
2
 is the variance of an
observation. The efficiency factor  E for direct effects of the neighbour balanced pair-wise uniform block
design is thus given as:
s ’s are the  1v non-zero eigen values of C . Similarly the efficiency  lE and  E for
neighbour effects (left and right) of treatments is obtained. The ranges of correlation coefficient   for
different correlation structures investigated are 40.0 for AR(2), MA(2) and NN correlation structures. For
these ranges, the matrix of correlation coefficients among observations within a block is positive definite. For
0 , the efficiency is that of totally balanced designs obtained by Tomer et al. [19].
In Tables 1, 2 and 3 the parameters of neighbour balanced pair-wise uniform block design for 5v
 0m and 6v  2,1,0m along with the efficiency for direct and neighbour effects (left and right)
has been shown. The efficiency values have been reported under the AR(2), MA(2) and NN correlation
structures with  21,  in the interval -0.4 (-0.4) 0.4. It is seen that efficiency for direct as well as neighbour
effects is high, in case of complete block designs i.e.,  0m for NN correlation structure. In case of
incomplete block designs  4,,2,1  vm  MA(2) correlation structure turns out to be more efficient as
compared to others models with  21,  in the interval 0.1 to 0.4. It is therefore concluded that when block
sizes are large and neighbouring plots are highly correlated, generalized least squares for estimation of direct
and neighbour effects can be used.
IV. Efficiency of NNBD and NNBIBD in comparison to Regular Block Design
In this section, a quantitative measure of efficiency of NNBD is also derived when the errors follow AR(2) and
MA(2) models. If errors within a block follow an AR(2) then
   
  







 



1
10
0
21
)2(
1
2
1,2
t
i
ijiAR rit
ttr
rrV

 

where       2
1
2
2220 111  r
   0211 1 rr  
    022
2
12 1 rr  
for 3k ,   02211 rrrr kkk   
 
 









 1
1
1
1
1
1
1
v
s
s
v
s
s
mv
v
E




If errors follow second order moving average model MA(2), then
   
  







 



1
10
0
21
)2(
1
2
1,2
t
i
ijiMA rit
ttr
rrV

 

where,
2
2
2
10 1  r
2111  r
22 r
for 3k , 0kr where
22
  
For NNBD the variance of an elementary contrast is given by
  




1
1
11
12
t
i
diN tV 
Where di ’s are non-zero values of dC . We define the efficiency of a design d relative to a regular block
design as
N
AR
V
V )2(
and
N
MA
V
V )2(
respectively for AR(2) and MA(2) models.
The Tables 4, 5, 6 and 7 shows the efficiencies of NNBD with 20,5  rt and 30,6  rt ,
 21,  = -0.4 (-0.4) 0.4 and 1 . The values in the tables show that as  increases from -0.4 to -0.1 the
gain in efficiency also increases and  decreases from 0.1 to 0.4 the gain in efficiency also decreases under
AR(2) and MA(2) models. The gain in efficiency of NNBD over regular block design is high under MA(2)
models ( 5t , 20r and 1 ) for direct and neighbour effects of treatments.
The Tables 8, 9, 10, 11, 12 and 13 show the efficiencies of NNBIBD with 16,5  rt , 25,6  rt and
20,6  rt ,  21,  = -0.4 (-0.4) 0.4 and 1 . The values in the tables show that as  increases from
-0.4 to -0.1, the gain in efficiency also increases and  decreases from 0.1 to 0.4, the gain in efficiency also
decreases under AR(2) and MA(2) models. The gain in efficiency of NNBIBD over regular block design is high
under MA(2) models ( 6t , 20r and 1 ) for direct and neighbour effects of treatments.
V. Conclusion
We have concluded that the efficiency for direct as well as neighbouring effects is high in the case of
complete block designs for NN correlation structure. In the case of incomplete block designs, MA(2)
correlation structure turned out to be more efficient as compared to other models with  21,  in the interval
0.1 to 0.4. The gain in efficiency of NNBD and NNBIBD over regular block design is high under MA(2)
models for direct and neighbour effects of treatments.
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[17]. R. Senthil Kumar & C. Santharam.: Efficiency of Nearest Neighbour Balanced Block Designs using ARMA models, International
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87-97 (2005).
[20]. Tomar, J.S.Jaggi, S.: Efficient neighbour balanced block designs for correlated observations, METRON – International Journal of
Statistics, Vol. LXV, n. 2, pp. 229-238 (2007).
Tables:
Table 1. Efficiency of neighbour balanced pair-wise uniform block designs AR(2) model
Parameters Correlation Structure
AR(2)
v b m r k ),( 21  E lE E
5 20 0 20 5 (-0.4,-0.4) 0.56427 0.56808 0.54146
(-0.3,-0.3) 0.64066 0.64761 0.64341
(-0.2,-0.2) 0.78467 0.71141 0.60989
(-0.1,-0.1) 0.76701 0.76929 0.76893
(0,0) 0.80000 0.80000 0.80000
(0.1,0.1) 0.80582 0.81691 0.79696
(0.2,0.2) 0.79178 0.80068 0.80757
(0.3,0.3) 0.72334 0.79433 0.73051
(0.4,0.4) 0.88824 0.69126 0.70063
6 30 0 30 6 (-0.4,-0.4) 0.60734 0.59666 0.58634
(-0.3,-0.3) 0.68867 0.68532 0.73889
(-0.2,-0.2) 0.74974 0.75752 0.76040
(-0.1,-0.1) 0.80637 0.79854 0.80464
(0,0) 0.83333 0.83333 0.83333
(0.1,0.1) 0.85754 0.84921 0.84619
(0.2,0.2) 0.83448 0.82871 0.88098
(0.3,0.3) 0.81760 0.80621 0.76668
(0.4,0.4) 0.72861 0.71027 0.68055
6 30 1 25 5 (-0.4,-0.4) 0.83447 0.82390 0.81067
(-0.3,-0.3) 0.87554 0.87287 0.87532
(-0.2,-0.2) 0.92940 0.92201 0.93455
(-0.1,-0.1) 0.97469 0.97316 0.98455
(0,0) 1.00000 1.00000 1.00000
(0.1,0.1) 1.03364 1.03548 1.04214
(0.2,0.2) 1.04870 1.05409 1.04523
(0.3,0.3) 1.03975 1.01356 1.01522
(0.4,0.4) 0.94793 1.15515 1.70573
6 30 2 20 4 (-0.4,-0.4) 0.93088 0.49412 0.87094
(-0.3,-0.3) 1.03841 1.00006 1.00628
(-0.2,-0.2) 1.13443 1.13032 1.13469
(-0.1,-0.1) 1.21476 1.24003 1.23816
(0,0) 1.25000 1.25000 1.25000
(0.1,0.1) 1.32514 1.34697 1.32511
(0.2,0.2) 1.38279 1.40407 1.41581
(0.3,0.3) 1.38420 1.41755 1.40362
(0.4,0.4) 1.43294 1.44810 1.42910
Table 2. Efficiency of neighbour balanced pair-wise uniform block designs MA(2) model
MA(2)
v b m r k ),( 21  E lE E
5 20 0 20 5 (-0.4,-0.4) 0.50455 0.50991 0.49682
(-0.3,-0.3) 0.57578 0.59003 0.58525
(-0.2,-0.2) 0.71456 0.69540 0.65266
(-0.1,-0.1) 0.73357 0.65061 0.74019
(0,0) 0.80000 0.80000 0.80000
(0.1,0.1) 0.85091 0.86773 0.84908
(0.2,0.2) 0.86011 0.85519 0.83159
(0.3,0.3) 0.83086 0.85669 0.82009

(0.4,0.4) 0.75814 0.70742 0.74459
6 30 0 30 6 (-0.4,-0.4) 0.53221 0.52322 0.50447
(-0.3,-0.3) 0.62195 0.61417 0.61648
(-0.2,-0.2) 0.69486 0.66319 0.68810
(-0.1,-0.1) 0.76892 0.76855 0.76305
(0,0) 0.83333 0.83333 0.83333
(0.1,0.1) 0.85323 0.86888 0.86780
(0.2,0.2) 0.87748 0.87339 0.87662
(0.3,0.3) 0.84813 0.81497 0.84059
(0.4,0.4) 0.77341 0.62246 0.77547
6 30 1 25 5 (-0.4,-0.4) 0.66551 0.62619 0.65638
(-0.3,-0.3) 0.74862 0.74420 0.75138
(-0.2,-0.2) 0.84034 0.83530 0.83880
(-0.1,-0.1) 0.92152 0.93806 0.93681
(0,0) 1.00000 1.00000 1.00000
(0.1,0.1) 1.06487 1.10040 1.07081
(0.2,0.2) 1.12050 1.08714 1.11359
(0.3,0.3) 1.13502 1.12539 1.12761
(0.4,0.4) 1.12955 1.11200 1.10951
6 30 2 20 4 (-0.4,-0.4) 0.79845 0.78892 0.79581
(-0.3,-0.3) 0.86223 0.93426 0.88878
(-0.2,-0.2) 1.05227 1.11215 1.06864
(-0.1,-0.1) 1.15628 1.17838 1.23751
(0,0) 1.25000 1.25000 1.25000
(0.1,0.1) 1.35875 1.35286 1.41124
(0.2,0.2) 1.42639 1.44671 1.47277
(0.3,0.3) 1.46134 1.44887 1.43499
(0.4,0.4) 1.49548 1.42926 1.50805
Table 3. Efficiency of neighbour balanced pair-wise uniform block designs NN model
NN
v b m r k 
E lE E
5 20 0 20 5 -0.4 0.57777 0.54370 0.55414
-0.3 0.77230 0.76003 0.71921
-0.2 1.12702 1.11899 1.12748
-0.1 1.32548 1.27062 1.26528
0 0.80000 0.80000 0.80000
0.1 1.33665 1.39663 1.26119
0.2 1.16767 1.20560 1.13899
0.3 0.77862 0.78772 0.76156
0.4 0.58218 0.63672 0.56695
6 30 0 30 6 -0.4 0.33727 0.36735 0.32629
-0.3 0.46641 0.50243 0.45668
-0.2 0.68408 0.68807 0.67801
-0.1 1.37697 1.39017 1.34549
0 0.83333 0.83333 0.83333
0.1 1.44357 1.40101 1.40017
0.2 0.93128 0.90471 0.87864
0.3 0.62177 0.62254 0.58294
0.4 0.47132 0.45645 0.45121
6 30 1 25 5 -0.4 0.15597 0.21636 0.44046
-0.3 0.24370 0.21980 0.49732
-0.2 0.48998 0.41370 0.82479
-0.1 0.90261 0.95073 0.95956
0 1.00000 1.00000 1.00000
0.1 1.01487 1.02528 1.13842
0.2 0.46956 0.55623 0.68783
0.3 0.38600 0.38782 0.34714
0.4 0.30592 0.31637 0.40601
6 30 2 20 4 -0.4 0.85435 0.90279 0.92465
-0.3 1.08256 1.28288 1.18901
-0.2 1.77180 1.65902 1.91532
-0.1 1.59519 1.85066 1.90607
0 1.25000 1.25000 1.25000
0.1 1.54050 1.83199 1.90737
0.2 1.85560 1.81908 1.86553
0.3 1.24573 1.30222 1.21305
0.4 0.94025 1.03260 0.77395

Table 4. Efficiency of NNBD using AR(2) Model ( 5t , 20r and 1 )
),( 21  E lE E
(-0.4, -0.4) 0.99707 0.94537 1.02110
(-0.3, -0.3) 1.10683 1.09303 1.05162
(-0.2, -0.2) 1.26307 1.40307 1.66570
(-0.1, -0.1) 2.58826 2.55587 2.57687
(0,0) 0.83954 0.85159 0.86020
(0.1,0.1) 2.42797 2.49175 2.43415
(0.2,0.2) 1.25211 1.29677 1.22696
(0.3,0.3) 0.93993 0.87836 0.93270
(0.4,0.4) 0.49401 0.30393 0.62504
Table 5. Efficiency of NNBD using AR(2) Model ( 6t , 30r and 1 )
),( 21  E lE E
(-0.4, -0.4) 0.53382 0.54339 0.54519
(-0.3, -0.3) 0.61517 0.61818 0.57215
(-0.2, -0.2) 0.79475 0.78659 0.79152
(-0.1, -0.1) 1.44493 1.43084 1.44203
(0,0) 0.83238 0.83238 0.84832
(0.1,0.1) 1.78390 1.80140 1.77907
(0.2,0.2) 0.93644 0.94297 0.85290
(0.3,0.3) 0.64644 0.65558 0.65871
(0.4,0.4) 0.35792 0.36716 0.37387
Table 6. Efficiency of NNBD using MA(2) Model ( 5t , 20r and 1 )
),( 21  E lE E
(-0.4, -0.4) 1.21164 1.14440 1.20920
(-0.3, -0.3) 1.30915 1.27530 1.22896
(-0.2, -0.2) 1.44002 1.49026 1.61606
(-0.1, -0.1) 2.15184 2.07303 2.12199
(0,0) 0.83954 0.85159 0.86020
(0.1,0.1) 2.26616 2.31200 2.25180
(0.2,0.2) 1.17302 1.17594 1.15405
(0.3,0.3) 0.77541 0.77173 0.78728
(0.4,0.4) 0.64231 0.76367 0.65257
Table 7. Efficiency of NNBD using MA(2) Model ( 6t , 30r and 1 )
),( 21  E lE E
(-0.4, -0.4) 0.66202 0.74614 0.68862
(-0.3, -0.3) 0.72312 0.80247 0.72799
(-0.2, -0.2) 0.88883 0.95291 0.90663
(-0.1, -0.1) 1.53897 1.58135 1.54437
(0,0) 0.83238 0.84678 0.84832
(0.1,0.1) 1.77126 1.72860 1.73183
(0.2,0.2) 0.86914 0.86297 0.83654
(0.3,0.3) 0.62122 0.64246 0.58433
(0.4,0.4) 0.51164 0.62631 0.49786
Table 8. Efficiency of NNBIBD using AR(2) Model ( 5t , 16r and 1 )
),( 21  E lE E
(-0.4, -0.4) 0.18527 0.28995 0.26233
(-0.3, -0.3) 0.28763 0.12989 0.16658
(-0.2, -0.2) 0.37600 0.37649 0.18700
(-0.1, -0.1) 0.71467 0.70473 0.62399
(0,0) 0.35740 0.35740 0.35740
(0.1,0.1) 0.72382 0.66743 0.75656
(0.2,0.2) 0.34895 0.35068 0.35801
(0.3,0.3) 0.29492 0.30014 0.29630
(0.4,0.4) 0.16895 0.12241 0.11859

),( 21  E lE E
(-0.4, -0.4) 0.07397 0.01262 0.13114
(-0.3, -0.3) 0.10149 0.11443 0.13766
(-0.2, -0.2) 0.18434 0.16134 0.07707
(-0.1, -0.1) 0.31455 0.23519 0.27502
(0,0) 0.33414 0.33164 0.33772
(0.1,0.1) 0.32705 0.31905 0.31155
(0.2,0.2) 0.15082 0.15102 0.15221
(0.3,0.3) 0.12668 0.12051 0.13172
(0.4,0.4) 0.07168 0.05185 0.03884
),( 21  E lE E
(-0.4, -0.4) 0.52996 0.71555 0.59596
(-0.3, -0.3) 0.56881 0.67374 0.62672
(-0.2, -0.2) 0.81717 0.56405 0.85855
(-0.1, -0.1) 1.50430 1.51538 0.36159
(0,0) 0.48303 0.48130 0.48606
(0.1,0.1) 1.33171 0.99505 1.42879
(0.2,0.2) 0.67638 0.69771 0.64563
(0.3,0.3) 0.45953 0.45152 0.42899
(0.4,0.4) 0.11869 0.08814 0.09162
Table 11. Efficiency of NNBIBD using MA(2) Model ( 5t , 16r and 1 )
),( 21  E lE E
(-0.4, -0.4) 0.25723 0.38273 0.35743
(-0.3, -0.3) 0.41218 0.16191 0.23084
(-0.2, -0.2) 0.43682 0.43746 0.22084
(-0.1, -0.1) 0.80030 0.74586 0.70076
(0,0) 0.35746 0.35746 0.35746
(0.1,0.1) 0.67062 0.65268 0.70637
(0.2,0.2) 0.31053 0.30814 0.31343
(0.3,0.3) 0.25911 0.24575 0.24123
(0.4,0.4) 0.14553 0.14651 0.14770
),( 21  E lE E
(-0.4, -0.4) 0.10080 0.01805 0.17602
(-0.3, -0.3) 0.12601 0.14248 0.17024
(-0.2, -0.2) 0.21132 0.18459 0.08900
(-0.1, -0.1) 0.33790 0.24780 0.29348
(0,0) 0.33414 0.33414 0.33414
(0.1,0.1) 0.31362 0.29660 0.29955
(0.2,0.2) 0.13776 0.14291 0.13943
(0.3,0.3) 0.11287 0.10556 0.11534
(0.4,0.4) 0.09128 0.08172 0.09062
),( 21  E lE E
(-0.4, -0.4) 0.67145 0.69124 0.70874
(-0.3, -0.3) 0.72723 0.76561 0.75328
(-0.2, -0.2) 0.91314 0.59420 0.94491
(-0.1, -0.1) 1.60506 1.61956 0.36742
(0,0) 0.48303 0.48130 0.48606
(0.1,0.1) 1.28310 0.78202 1.32538
(0.2,0.2) 0.63993 0.66086 0.60573
(0.3,0.3) 0.42334 0.42966 0.40811
(0.4,0.4) 0.33980 0.37711 0.25161

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