A New Biased Two-Parameter Estimator in Linear Regression Model
1OLASUNKANMI JAMES OLADAPO, 1JANET IYABO IDOWU, 1*ABIOLA TIMOTHY
OWOLABI, 2KAYODE AYINDE
1,2Department of Statistics
1Ladoke Akintola University of Technology
2Federal University of Technology
1Ogbomoso, Oyo State
2Akure, Ondo State
1,2NIGERIA
*Corresponding Author: E-mail: atowolabi@lautech.edu.ng
Abstract: The most frequently used estimation technique in the linear regression model is the ordinary
least squares (OLS) estimator. The presence of multicollinearity makes the technique inefficient and
gives misleading results. This study proposed a new biased two-parameter estimator to deal with the
multicollinearity problem. Theory and simulation results show that this estimator outperforms
existing estimators considered under some conditions, according to the mean squares error (MSE)
criterion. Finally, the real-life dataset illustrates the paper's findings, which agree with the theoretical
and simulation results.
Keywords: Multicollinearity, Mean Square Error, Monte Carlo Simulation, Ordinary Least Squares,
Biased Estimator.
1. Introduction
In the multiple linear regression model, it is
assumed that there is no perfect correlation among
the explanatory variables. Violation of this
assumption is the term multicollinearity. OLS
estimator in the multicollinearity is the best linear
unbiased estimator (BLUE) [1], but it is inefficient
and cannot give the desired results when
explanatory variables are correlated. It provides a
misleading conclusion.
Several researchers have proposed alternative
estimators to the OLS to overcome these effects of
multicollinearity. The authors include Hoerl and
Kennard [2], Liu [3], Stein [4], Swindel [5],
Ahmad and Aslam [6], Lukman et al. [7], Kibria
and Lukman [8] , Efron et al. [9], Draper and Smith
[10], Mansson et al., [11], Dempster et al. [12],
Akdeniz and Roozbeh [13], Muniz et al., [14],
Arashi and Valizadeh [15], Ayinde et al., [16],
Yang and Chang [17], Owolabi et al., [18, 19],
Oladapo et al., [20], and Idowu et al. [21]
This paper aims to introduce a new class of
two-parameter estimator to estimate regression
parameters when there is a problem of
multicollinearity and compare the performance of
the new estimator with the existing estimators.
The article is organized as follows. In section
2, the existing and new proposed estimators are
introduced. Section 3 provides theoretical
comparisons among the estimators and the choice
of biasing parameters. A simulation study is
conducted in Section 4 to evaluate the performance
of the proposed estimator. A numerical example is
given in Section 5 to illustrate the findings in the
paper, and Section 6 is some concluding remarks.
2. Some Existing Biased Estimators
and the Proposed Estimator
2.1 Existing estimators
Consider a multiple linear regression
model of the form
uXy
(1)
where y is an n × 1 vector of observations, β is a
p × 1 vector of unknown regression parameters,
X is an n × p observed matrix of the regression,
and u is an n × 1 vector of random errors, which
is distributed as multivariate normal with mean 0
and covariance matrix σ2In, In being an identity
matrix of order n. The OLS estimator of β in (1)
is obtained as:
yXXX ''
ˆ1
(2)
The canonical form of (1) can be written as:
Received: December 15, 2022. Revised: August 11, 2023. Accepted: September 7, 2023. Published: October 3, 2023.
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DOI: 10.37394/232021.2023.3.10
Olasunkanmi James Oladapo, Janet Iyabo Idowu,
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uZy
(3)
Where Z=XQ,
'Q
and Q is the orthogonal
matrix such that
XQXQZZ '''
p
diag
,...,,21
. The OLS estimator of
is
given as follows:
yZ'
ˆ1
(4)
And the mean squared error matrix (MSEM) of
ˆ
is given by
12
ˆ
MSEM
(5)
The ordinary ridge regression (ORR) estimator of
by Hoerl and Kennard [2] is defined as:
yLX
k'
ˆ
(6)
where
1
kIL
and k is a biasing parameter
such that k > 0.
'''
ˆ2ILILLLkMSEM
(7)
The Liu estimator by Liu [3] is defined as:
ˆˆ F
d
(8)
where
dIIF 1
and d is a biasing
parameter of the Liu Estimator.
'''
ˆ12 IFIFFFMSEM d
(9)
The Modified One-Parameter Liu (ML) estimator
by Lukman et al. [22] is defined as:
ˆˆ T
ML
(10)
where
dIIT 1
and d is a biasing
parameter of the Liu Estimator.
'''
ˆ12 ITITTTMSEM ML
(11)
The Kibria-Lukman [8] (KL) estimator is defined
as:
ˆˆ G
KL
(12)
where
kIkIG 1
and
'''
ˆ12 IGIGGGMSEM KL
(13)
2.2 The proposed estimator
Following the same concept proposed by Liu [23]
and Lukman and Kibria [22], the new proposed
biased two-parameter estimator is named Modified
Liu Kibria-Lukman estimator (MLKL) and is
denoted as
MLKL
ˆ
is defined as follows:
ˆˆ TG
MLKL
(14)
where k > 0 and 0 < d < 1.
Properties of the new estimator
The bias, covariance, and mean squared error
matrix (MSEM) of the proposed estimator are
given as follows:
ITGBMLKL
ˆ
(15)
''
ˆ12 GTTGDMLKL
(16)
''
ˆ12 GTTGMSEM MLKL
'' ITGITG
(17)
The following lemmas are useful to prove the
statistical property of
MLKL
ˆ
.
Lemma 1: Let S be an n x n positive definite
matrix, that is S > 0, and
be some vector, then
0'
S
if and only if
1' 1
S
.
Farebrother [24]
Lemma 2: Let
yBii
ˆ
i = 1, 2 be two linear
estimators of
. Suppose that
0cov
ˆ
cov 21
D
, where
i
ˆ
cov
i =
1, 2 denotes the covariance matrix of
i
ˆ
and
IXBBiasb iii ˆ
, i =1, 2.
Consequently,
2121 ˆˆˆˆ
MSEMMSEM
0
'
22
'
11
2 bbbbD
(18)
If and only if
1
2
1
'
11
2'
2 bbbDb
, where
MSEM
'
ˆ
cov
ˆiiii bb
Trenkler and
Toutenburg [25]
3. Comparison and Choice of Biasing
Parameter
3.1 Comparison among the estimators
In this section, the MSEM of the proposed
estimator,
MLKL
ˆ
, is compared with others
theoretically.
Comparison between
ˆ
and
MLKL
ˆ
.
The difference between
MLKL
MSEMMSEM
ˆˆ
is given as follows:
MLKL
MSEMMESM
ˆˆ
''''
1212 ITGITGGTTG
(19)
Let k > 0 and 0 < d < 1. Thus, the following
theorem holds.
Theorem 3.1: The estimator
MLKL
ˆ
is superior to
the estimator
ˆ
using the MSEM criterion, that is,
MSEM (
ˆ
) − MSEM (
MLKL
ˆ
) > 0 if and only if
1'''' 112 ITGGTTGITG
(20)
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Olasunkanmi James Oladapo, Janet Iyabo Idowu,
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Proof
''
ˆˆ 112 GTTGDD MLKL
p
i
iii
ii
ik
kd
diag
1
22
22
2
1
1
(21)
''
11 GTTG
will be a pdf if and only if
.01 2222 kdk iiii
Comparison between
k
ˆ
and
MLKL
ˆ
.
The difference between
MLKLk MSEMMSEM
ˆˆ
is given as
follows:
MLKLk MSEMMESM
ˆˆ
'''
2ILILLL
''''
12 ITGITGGTTG
(22)
Let k > 0 and 0 < d < 1. Thus, the following
theorem holds.
Theorem 3.2: The estimator
MLKL
ˆ
is superior to
the estimator
k
ˆ
using the MSEM criterion, that is,
MSEM (
k
ˆ
) − MSEM (
MLKL
ˆ
) > 0 if and only if
''''''' 12 ILILGTTGLLITG
1 ITG
(23)
Proof
'''
ˆˆ 12 GTTGLLDD MLKLk
p
i
iii
ii
i
i
k
kd
k
diag
1
22
22
2
2
1
(24)
''
1GTTGLL
will be a pdf if and only if
.01 222
2 kd iiii
Comparison between
d
ˆ
and
MLKL
ˆ
.
The difference between
MLKLd MSEMMSEM
ˆˆ
is given as
follows:
MLKLd MSEMMESM
ˆˆ
'''
12 IFIFFF
''''
12 ITGITGGTTG
(25)
Let k > 0 and 0 < d < 1. Thus, the following
theorem holds.
Theorem 3.3: The estimator
MLKL
ˆ
is superior to
the estimator
d
ˆ
using the MSEM criterion, that is,
MSEM (
d
ˆ
) − MSEM (
MLKL
ˆ
) > 0 if and only if
'' ITG
1''''' 112 ITGIFIFGTTGFF
(26)
Proof
'''
ˆˆ 112 GTTGFFDD MLKLd
p
i
iii
ii
ii
i
k
kdd
diag
1
22
22
2
2
2
11
(27)
''' 11 GTTGFF
will be a pdf if and only if
.0
2222 kdkd iiii
Comparison between
ML
ˆ
and
MLKL
ˆ
.
The difference between
MLKLML MSEMMSEM
ˆˆ
is given as
follows:
MLKLML MSEMMESM
ˆˆ
'''
12 ITITTT
''''
12 ITGITGGTTG
(28)
Let k > 0 and 0 < d < 1. Thus, the following
theorem holds.
Theorem 3.4: The estimator
MLKL
ˆ
is superior to
the estimator
ML
ˆ
using the MSEM criterion, that
is, MSEM (
ML
ˆ
) − MSEM (
MLKL
ˆ
) > 0 if and only
if
'' ITG
1''''' 112 ITGITITGTTGTT
(29)
Proof
'''
ˆˆ 112 GTTGTTDD MLKLML
p
i
iii
ii
ii
i
k
kdd
diag
1
22
22
2
2
2
11
(30)
''' 11 GTTGTT
will be a pdf if and only if
.0
2222 kdkd iiii
Comparison between
KL
ˆ
and
MLKL
ˆ
.
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Olasunkanmi James Oladapo, Janet Iyabo Idowu,
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The difference between
MLKLKL MSEMMSEM
ˆˆ
is given as
follows:
MLKLKL MSEMMESM
ˆˆ
'''
12 IGIGGG
''''
12 ITGITGGTTG
(31)
Let k > 0 and 0 < d < 1. Thus, the following
theorem holds.
Theorem 3.5: The estimator
MLKL
ˆ
is superior to
the estimator
KL
ˆ
using the MSEM criterion, that
is, MSEM (
KL
ˆ
) − MSEM (
MLKL
ˆ
) > 0 if and only
if
'' ITG
1''''' 112 ITGIGIGGTTGGG
(32)
Proof
'''
ˆˆ 112 GTTGGGDD MLKLKL
p
i
iii
ii
ii
i
k
kd
k
k
diag
1
22
22
2
2
2
1
(33)
''' 11 GTTGGG
will be a pdf if and only if
.01 2222 kdk iiii
3.2 Choice of Biasing Parameters
Different authors such as Hoerl et al. [26],
Wencheko [27], Kibria and Banik [28], Khalaf and
Shukur [29], and Owolabi et al. [30], among
others, have proposed different estimators of k and
d. An optimal value of k is the value that minimizes
the MSE of the proposed estimator, that is
MLKL
ˆ
,
considering d to be fixed.
''
ˆ12 GTTGMSEM MLKL
'' ITGITG
MLKLMLKL MSEMtraceMSEMdkg
ˆˆ
,
p
iiii
ii
k
kd
dkg
122
22
2
1
,
p
ii
ii
iii
k
kkddk
1
2
22
2
1
2
(34)
Let
0
,
k
dkg
;
p
iiii
ii
p
iiii
ii
k
kd
k
kd
32
22
2
22
2
2
1
2
1
2
p
iii
iiiii
k
dkkddk
22
2
1
122
2
p
iii
iiii
k
kkddk
32
2
2
1
2
2
(35)
12
1
22
222
dd
dd
k
iiii
iiii
(36)
Also, differentiating
dkg ,
with respect to d
gives
d
dkg ,
p
iiii
ii
k
kd
22
2
2
1
2
p
iii
iiiii
k
kkkddk
22
2
1
2
2
(37)
Let
0
,
d
dkg
;
kk
kkk
d
iiii
iiiiii
22
22 2
(38)
For practical purposes,
2
and
2
i
are replaced
with
2
ˆ
and
2
ˆi
, in equations (36) and (38),
respectively. Consequently, (36) becomes
12
ˆ
ˆ
1
ˆ
ˆ
ˆ22
222
dd
dd
k
iiii
iiii
(39)
And (38) becomes
kk
kkk
d
iiii
iiiiii
22
22
ˆ
ˆ
2
ˆ
ˆ
ˆ
(40)
4. Simulation Study
In this section, the Monte Carlo simulation scheme
was conducted to study the performance of the
proposed estimator to validate the theoretical
comparison. We followed the similar simulation
scheme used by many authors in their studies, such
as Wichern and Churchill [31], Clark and Troskie
[32], Dempster et al. [12], McDonald and
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Galarneau [33], Gibbons [34], Ahmad and Aslam
[6] and Newhouse and Oman [35].
The following expression generates the regressors
1
22,1
1
ij ij i p
x z z
,
1,2,...,in
,
1,2,..., .jp
(41)
where
ij
z
is independent standard normal
pseudorandom numbers,
is the correlation
between two explanatory variables, and p is the
number of explanatory variables. This study
considers the values of ρ to be 0.8, 0.9, 0.95, and
0.99. Also, explanatory variables (p) will be taken
to be three (3) and seven (7), and sample sizes 50
and 100 for the simulation study. The response
variable is defined as:
ippi eXXXY
...
2211
(42)
where
i
e
is normally distributed with mean 0 and
variance
2
. We choose
such that
=1. The
chosen values of
are 3, 5, and 10, and the
replication for the study is 1000 times. The mean
square error is then obtained as:
iij
jiij
MSE
ˆˆ
1000
1
ˆ
1000
1
(43)
4.1 Simulation Result
From Table 1-4, the simulation results presented
show that the proposed estimator outperforms all
other estimators used in this study. The proposed
estimator,
MLKL
ˆ
, in this study performs best at the
two different sample sizes considered (n = 50 and
100), four sigma levels (
= 1, 3, 5, and 10), four
different levels of multicollinearity levels (
=
0.8. 0.9, 0.95 and 0.99) and the two different
number of parameters (p=3 and 7). It provides a
smaller MSE compared with other estimators in
the study when the number of parameters is three
and seven. The OLS estimator is the least
performed estimator, which is expected due to the
facts established in the literature. The following
observations were also deduced from the result:
i. An increase in the level of correlation
results in an increase in the MSE for
all the estimators.
ii. The MSE increases for each estimator
as the level of error variances
increases.
iii. An increase in the sample size, n,
decreases the MSE for all the
estimators.
5. Numerical Example
In this section, Portland cement data was used to
demonstrate the performance of the proposed
estimator. The Portland cement data was originally
adopted by Woods et al. [36] and was later
adopted by Li and Yang [37] and then by Ayinde
et al. [38]. The data set is widely known as the
Portland cement dataset. The regression model for
these data is defined as:
1 1 2 2 3 3 4 4ii
y X X X X
(44)
where
i
y
= heat evolved after 180 days of curing
measured in calories per gram of cement,
1
X
=
tricalcium aluminate,
2
X
= tricalcium silicate,
3
X
= tetra calcium aluminoferrite, and
4
X
=
-
dicalcium silicate. The variance inflation factors
(VIF) are VIF1 = 38.50, VIF2 = 254.42, VIF3 =
46.87, and VIF4 = 282.51. Eigenvalues of
XX
matrix are λ1 = 44676.206, λ2 = 5965.422, λ3 =
809.952, and λ4 = 105.419, and the condition
number of
XX
is approximately 424. The VIFs,
eigenvalues, and condition numbers indicate that
severe multicollinearity exists. The estimated
parameters and the MSE values of the estimators
are presented in Table 5.
From Table 5, the proposed estimator (MLKL)
performs the best among all other estimators
because it gives the smallest MSE value. The OLS
estimator did not perform well in the presence of
multicollinearity, as it has the highest MSE.
6. Conclusions
This paper proposed a new two-parameter
estimator to handle the multicollinearity problem
for the linear regression models. The proposed
estimator was theoretically compared with five
other existing estimators. A simulation study was
then conducted to compare the performance of the
proposed estimator and the five existing
estimators: the OLS, Liu estimator, Ridge
estimator, KL estimator, and Modified One-
Parameter Liu estimator. It is evident from the
theoretical comparison that the proposed estimator
performs best among the existing estimators
considered in this research work.
A simulation study also supports the theoretical
analysis as the proposed estimator performs best
among all the existing estimators used in the study.
A real-life dataset was also analyzed to bolster the
theoretical comparison and simulation study
results. The proposed estimator gives the best
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Olasunkanmi James Oladapo, Janet Iyabo Idowu,
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E-ISSN: 2732-9976
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Volume 3, 2023
result using the Mean Square Error criterion
among the existing estimators.
We recommend this new estimator for
practitioners and researchers to use whenever there
is a multicollinearity problem among their
explanatory variables when using linear regression
models. This estimator will surely give a suitable
and reliable result or output.
LIST OF ABBREVIATIONS
OLS: Ordinary Least Square
MSEM: Mean Square Error Matrix
ORR: Ordinary Ridge Regression
ML: Modified Liu
KL: Kibria-Lukman
MLKL: Modified Liu Kibria-Lukman
VIF: Variance Inflation Factors
DECLARATIONS
Availability of data and materials
The datasets used in the study are available upon
reasonable request.
The authors declare that there is no competing
interest.
This research received no external funding.
OJ, JI, and K conceived and designed the idea. OJ,
JI, and AT analyzed and interpreted the results. OJ,
JI, and AT wrote the first draft of the work. K
supervised the work. All the authors reviewed the
results and approved the final version of the
manuscript.
Acknowledgments
Not applicable
Table 1: Estimated MSE when n=50 and p=3.
K
d
Sigma
Rho
OLS
ORR
LIU
K-L
ML
MLKL
0.3
0.2
3
0.8
1.227
1.1802
1.1098
1.1344
1.0535
0.9748
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
Conflict of Interest
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
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Volume 3, 2023
0.9
2.2158
2.0587
1.8412
1.9076
1.6674
1.4383
0.95
4.2136
3.6579
2.9999
3.1428
2.4725
1.8549
0.99
20.175
10.956
6.5009
4.5913
2.5317
0.6497
5
0.8
3.4082
3.2782
3.0825
3.1509
2.9261
2.7072
0.9
6.1551
5.7185
5.1142
5.2987
4.6314
3.9947
0.95
11.704
10.161
8.333
8.73
6.868
5.1526
0.99
56.042
30.434
18.058
12.754
7.0326
1.8048
10
0.8
13.633
13.113
12.329
12.603
11.704
10.827
0.9
24.62
22.874
20.457
21.194
18.525
15.978
0.95
46.818
40.644
33.332
34.92
27.472
20.61
0.99
224.17
121.73
72.233
51.015
28.131
7.2194
0.5
3
0.8
1.227
1.1802
1.153
1.1344
1.0124
0.9369
0.9
2.2158
2.0587
1.9774
1.9076
1.543
1.3318
0.95
4.2136
3.6579
3.4302
3.1428
2.1117
1.5878
0.99
20.175
10.956
10.733
4.5913
0.8096
0.2479
5
0.8
3.4082
3.2782
3.2026
3.1509
2.8117
2.6019
0.9
6.1551
5.7185
5.4928
5.2987
4.2857
3.6988
0.95
11.704
10.161
9.5284
8.73
5.8658
4.4104
0.99
56.042
30.434
29.813
12.754
2.2488
0.6885
10
0.8
13.633
13.113
12.81
12.603
11.246
10.406
0.9
24.62
22.874
21.971
21.194
17.142
14.794
0.95
46.818
40.644
38.114
34.92
23.464
17.642
0.99
224.17
121.73
119.25
51.015
8.9955
2.7542
0.8
3
0.8
1.227
1.1802
1.1971
1.1344
0.9721
0.8999
0.9
2.2158
2.0587
2.1188
1.9076
1.4237
1.2297
0.95
4.2136
3.6579
3.8903
3.1428
1.7807
1.3426
0.99
20.175
10.956
16.04
4.5913
0.1629
0.0795
5
0.8
3.4082
3.2782
3.3251
3.1509
2.6997
2.4988
0.9
6.1551
5.7185
5.8855
5.2987
3.9542
3.415
0.95
11.704
10.161
10.806
8.73
4.9464
3.7292
0.99
56.042
30.434
44.554
12.754
0.4524
0.2206
10
0.8
13.633
13.113
13.3
12.603
10.797
9.993
0.9
24.62
22.874
23.542
21.194
15.816
13.658
0.95
46.818
40.644
43.226
34.92
19.786
14.917
0.99
224.17
121.73
178.22
51.015
1.8099
0.8825
0.6
0.2
3
0.8
1.227
1.1362
1.1098
1.0491
1.0535
0.9023
0.9
2.2158
1.9183
1.8412
1.6431
1.6674
1.2416
0.95
4.2136
3.208
2.9999
2.3436
2.4725
1.3927
0.99
20.175
6.9114
6.5009
0.721
2.5317
0.1499
5
0.8
3.4082
3.1559
3.0825
2.9139
2.9261
2.5055
0.9
6.1551
5.3284
5.1142
4.5638
4.6314
3.448
0.95
11.704
8.9111
8.333
6.5101
6.868
3.8686
0.99
56.042
19.198
18.058
2.0029
7.0326
0.4162
10
0.8
13.633
12.623
12.329
11.655
11.704
10.02
0.9
24.62
21.313
20.457
18.255
18.525
13.79
EQUATIONS
DOI: 10.37394/232021.2023.3.10
Olasunkanmi James Oladapo, Janet Iyabo Idowu,
Abiola Timothy Owolabi, Kayode Ayinde
E-ISSN: 2732-9976
79
Volume 3, 2023
0.95
46.818
35.645
33.332
26.041
27.472
15.474
0.99
224.17
76.794
72.233
8.0117
28.131
1.6647
0.5
3
0.8
1.227
1.1362
1.153
1.0491
1.0124
0.8675
0.9
2.2158
1.9183
1.9774
1.6431
1.543
1.1504
0.95
4.2136
3.208
3.4302
2.3436
2.1117
1.1953
0.99
20.175
6.9114
10.733
0.721
0.8096
0.086
5
0.8
3.4082
3.1559
3.2026
2.9139
2.8117
2.4086
0.9
6.1551
5.3284
5.4928
4.5638
4.2857
3.1948
0.95
11.704
8.9111
9.5284
6.5101
5.8658
3.3199
0.99
56.042
19.198
29.813
2.0029
2.2488
0.2387
10
0.8
13.633
12.623
12.81
11.655
11.246
9.6317
0.9
24.62
21.313
21.971
18.255
17.142
12.777
0.95
46.818
35.645
38.114
26.041
23.464
13.28
0.99
224.17
76.794
119.25
8.0117
8.9955
0.9548
0.8
3
0.8
3.3789
3.1035
3.2884
2.8414
2.6147
2.2138
0.9
6.3296
5.406
6.0293
4.5647
3.9195
2.8913
0.95
12.278
9.1533
11.277
6.5421
4.9438
2.8608
0.99
59.959
20.078
47.489
3.3132
1.8153
0.2175
5
0.8
3.4082
3.1559
3.3251
2.9139
2.6997
2.3137
0.9
6.1551
5.3284
5.8855
4.5638
3.9542
2.9518
0.95
11.704
8.9111
10.806
6.5101
4.9464
2.816
0.99
56.042
19.198
44.554
2.0029
0.4524
0.1489
10
0.8
13.633
12.623
13.3
11.655
10.797
9.2518
0.9
24.62
21.313
23.542
18.255
15.816
11.805
0.95
46.818
35.645
43.226
26.041
19.786
11.264
0.99
224.17
76.794
178.22
8.0117
1.8099
0.5956
0.9
0.2
3
0.8
1.227
1.0947
1.1098
0.9706
1.0535
0.8355
0.9
2.2158
1.7923
1.8412
1.4156
1.6674
1.0722
0.95
4.2136
2.8383
2.9999
1.7424
2.4725
1.044
0.99
20.175
4.7705
6.5009
0.2033
2.5317
0.064
5
0.8
3.4082
3.0407
3.0825
2.6954
2.9261
2.3195
0.9
6.1551
4.9784
5.1142
3.9317
4.6314
2.9773
0.95
11.704
7.8841
8.333
4.84
6.868
2.8997
0.99
56.042
13.251
18.058
0.5647
7.0326
0.1775
10
0.8
13.633
12.162
12.329
10.78
11.704
9.2753
0.9
24.62
19.913
20.457
15.726
18.525
11.907
0.95
46.818
31.537
33.332
19.36
27.472
11.599
0.99
224.17
53.006
72.233
2.2591
28.131
0.7097
0.5
3
0.8
1.227
1.0947
1.153
0.9706
1.0124
0.8035
0.9
2.2158
1.7923
1.9774
1.4156
1.543
0.9943
0.95
4.2136
2.8383
3.4302
1.7424
2.1117
0.8988
0.99
20.175
4.7705
10.733
0.2033
0.8096
0.0512
EQUATIONS
DOI: 10.37394/232021.2023.3.10
Olasunkanmi James Oladapo, Janet Iyabo Idowu,
Abiola Timothy Owolabi, Kayode Ayinde
E-ISSN: 2732-9976
80
Volume 3, 2023
5
0.8
3.4082
3.0407
3.2026
2.6954
2.8117
2.2303
0.9
6.1551
4.9784
5.4928
3.9317
4.2857
2.7607
0.95
11.704
7.8841
9.5284
4.84
5.8658
2.4962
0.99
56.042
13.251
29.813
0.5647
2.2488
0.1419
10
0.8
13.633
12.162
12.81
10.78
11.246
8.9179
0.9
24.62
19.913
21.971
15.726
17.142
11.04
0.95
46.818
31.537
38.114
19.36
23.464
9.9845
0.99
224.17
53.006
119.25
2.2591
8.9955
0.5672
0.8
3
0.8
1.227
1.0947
1.1971
0.9706
0.9721
0.7721
0.9
2.2158
1.7923
2.1188
1.4156
1.4237
0.9195
0.95
4.2136
2.8383
3.8903
1.7424
1.7807
0.7653
0.99
20.175
4.7705
16.04
0.2033
0.1629
0.0473
5
0.8
3.4082
3.0407
3.3251
2.6954
2.6997
2.143
0.9
6.1551
4.9784
5.8855
3.9317
3.9542
2.5528
0.95
11.704
7.8841
10.806
4.84
4.9464
2.1252
0.99
56.042
13.251
44.554
0.5647
0.4524
0.1308
10
0.8
13.633
12.162
13.3
10.78
10.797
8.5681
0.9
24.62
19.913
23.542
15.726
15.816
10.208
0.95
46.818
31.537
43.226
19.36
19.786
8.5006
0.99
224.17
53.006
178.22
2.2591
1.8099
0.5227
NOTE: Minimum MSE value is bolded in each row.
Table 2: Estimated MSE when n=100 and p=3
K
d
sigma
rho
OLS
ORR
LIU
K-L
ML
MLKL
0.3
0.2
3
0.8
0.5596
0.5499
0.5345
0.5403
0.5223
0.5044
0.9
1.0178
0.9839
0.9322
0.9506
0.8909
0.8326
0.95
1.9464
1.8211
1.6447
1.7002
1.5038
1.3157
0.99
9.4002
6.9
4.8107
4.7956
3.0985
1.6185
5
0.8
1.5543
1.5275
1.4848
1.5009
1.4507
1.4012
0.9
2.8273
2.7332
2.5896
2.6407
2.4748
2.3127
0.95
5.4066
5.0586
4.5687
4.7227
4.1773
3.6546
0.99
26.112
19.167
13.363
13.321
8.607
4.4957
10
0.8
6.2173
6.11
5.9393
6.0036
5.8028
5.6045
0.9
11.309
10.933
10.358
10.563
9.8992
9.2507
0.95
21.626
20.235
18.275
18.891
16.709
14.618
0.99
104.45
76.667
53.452
53.285
34.428
17.983
0.5
3
0.8
0.5596
0.5499
0.5438
0.5403
0.5132
0.4957
0.9
1.0178
0.9839
0.9639
0.9506
0.8606
0.8044
0.95
1.9464
1.8211
1.7548
1.7002
1.4025
1.2276
0.99
9.4002
6.9
6.3497
4.7956
2.0693
1.0957
5
0.8
1.5543
1.5275
1.5107
1.5009
1.4254
1.3768
0.9
2.8273
2.7332
2.6774
2.6407
2.3906
2.2343
0.95
5.4066
5.0586
4.8743
4.7227
3.8957
3.4099
EQUATIONS
DOI: 10.37394/232021.2023.3.10
Olasunkanmi James Oladapo, Janet Iyabo Idowu,
Abiola Timothy Owolabi, Kayode Ayinde
E-ISSN: 2732-9976
81
Volume 3, 2023
0.99
26.112
19.167
17.638
13.321
5.7479
3.0434
10
0.8
6.2173
6.11
6.0427
6.0036
5.7016
5.5071
0.9
11.309
10.933
10.71
10.563
9.5622
8.9371
0.95
21.626
20.235
19.497
18.891
15.583
13.639
0.99
104.45
76.667
70.552
53.285
22.991
12.173
0.8
3
0.8
0.5596
0.5499
0.5532
0.5403
0.5042
0.487
0.9
1.0178
0.9839
0.9961
0.9506
0.8309
0.7767
0.95
1.9464
1.8211
1.8685
1.7002
1.3049
1.1428
0.99
9.4002
6.9
8.1072
4.7956
1.2584
0.6814
5
0.8
1.5543
1.5275
1.5368
1.5009
1.4004
1.3527
0.9
2.8273
2.7332
2.7668
2.6407
2.3079
2.1574
0.95
5.4066
5.0586
5.1902
4.7227
3.6245
3.1741
0.99
26.112
19.167
22.52
13.321
3.4955
1.8927
10
0.8
6.2173
6.11
6.1472
6.0036
5.6014
5.4106
0.9
11.309
10.933
11.067
10.563
9.2314
8.6293
0.95
21.626
20.235
20.761
18.891
14.498
12.696
0.99
104.45
76.667
90.08
53.285
13.982
7.5703
0.6
0.2
3
0.8
0.5596
0.5405
0.5345
0.5218
0.5223
0.4873
0.9
1.0178
0.9518
0.9322
0.8881
0.8909
0.7783
0.95
1.9464
1.708
1.6447
1.4859
1.5038
1.1518
0.99
9.4002
5.2929
4.8107
2.3853
3.0985
0.8317
5
0.8
1.5543
1.5014
1.4848
1.4495
1.4507
1.3534
0.9
2.8273
2.6439
2.5896
2.4669
2.4748
2.1618
0.95
5.4066
4.7444
4.5687
4.1273
4.1773
3.1991
0.99
26.112
14.702
13.363
6.6258
8.607
2.3101
10
0.8
6.2173
6.0056
5.9393
5.7978
5.8028
5.4135
0.9
11.309
10.575
10.358
9.8677
9.8992
8.6468
0.95
21.626
18.978
18.275
16.509
16.709
12.796
0.99
104.45
58.81
53.452
26.503
34.428
9.2401
0.5
3
0.8
0.5596
0.5405
0.5438
0.5218
0.5132
0.4789
0.9
1.0178
0.9518
0.9639
0.8881
0.8606
0.7521
0.95
1.9464
1.708
1.7548
1.4859
1.4025
1.0752
0.99
9.4002
5.2929
6.3497
2.3853
2.0693
0.5737
5
0.8
1.5543
1.5014
1.5107
1.4495
1.4254
1.33
0.9
2.8273
2.6439
2.6774
2.4669
2.3906
2.0889
0.95
5.4066
4.7444
4.8743
4.1273
3.8957
2.9864
0.99
26.112
14.702
17.638
6.6258
5.7479
1.5934
10
0.8
6.2173
6.0056
6.0427
5.7978
5.7016
5.3197
EQUATIONS
DOI: 10.37394/232021.2023.3.10
Olasunkanmi James Oladapo, Janet Iyabo Idowu,
Abiola Timothy Owolabi, Kayode Ayinde
E-ISSN: 2732-9976
82
Volume 3, 2023
0.9
11.309
10.575
10.71
9.8677
9.5622
8.355
0.95
21.626
18.978
19.497
16.509
15.583
11.945
0.99
104.45
58.81
70.552
26.503
22.991
6.3732
0.8
3
0.8
0.5596
0.5405
0.5532
0.5218
0.5042
0.4706
0.9
1.0178
0.9518
0.9961
0.8881
0.8309
0.7263
0.95
1.9464
1.708
1.8685
1.4859
1.3049
1.0014
0.99
9.4002
5.2929
8.1072
2.3853
1.2584
0.3679
5
0.8
1.5543
1.5014
1.5368
1.4495
1.4004
1.3068
0.9
2.8273
2.6439
2.7668
2.4669
2.3079
2.0173
0.95
5.4066
4.7444
5.1902
4.1273
3.6245
2.7815
0.99
26.112
14.702
22.52
6.6258
3.4955
1.0216
10
0.8
6.2173
6.0056
6.1472
5.7978
5.6014
5.2268
0.9
11.309
10.575
11.067
9.8677
9.2314
8.0686
0.95
21.626
18.978
20.761
16.509
14.498
11.125
0.99
104.45
58.81
90.08
26.503
13.982
4.0859
0.9
0.2
3
0.8
0.5596
0.5314
0.5345
0.504
0.5223
0.4708
0.9
1.0178
0.9213
0.9322
0.8299
0.8909
0.7277
0.95
1.9464
1.6055
1.6447
1.2988
1.5038
1.0086
0.99
9.4002
4.1956
4.8107
1.1091
3.0985
0.408
5
0.8
1.5543
1.476
1.4848
1.3999
1.4507
1.3075
0.9
2.8273
2.5591
2.5896
2.3051
2.4748
2.0212
0.95
5.4066
4.4597
4.5687
3.6077
4.1773
2.8013
0.99
26.112
11.654
13.363
3.0808
8.607
1.1331
10
0.8
6.2173
5.9041
5.9393
5.5996
5.8028
5.2295
0.9
11.309
10.236
10.358
9.2202
9.8992
8.0842
0.95
21.626
17.839
18.275
14.431
16.709
11.205
0.99
104.45
46.617
53.452
12.323
34.428
4.5317
0.5
3
0.8
0.5596
0.5314
0.5438
0.504
0.5132
0.4627
0.9
1.0178
0.9213
0.9639
0.8299
0.8606
0.7033
0.95
1.9464
1.6055
1.7548
1.2988
1.4025
0.9421
0.99
9.4002
4.1956
6.3497
1.1091
2.0693
0.2901
5
0.8
1.5543
1.476
1.5107
1.3999
1.4254
1.2849
0.9
2.8273
2.5591
2.6774
2.3051
2.3906
1.9533
0.95
5.4066
4.4597
4.8743
3.6077
3.8957
2.6165
0.99
26.112
11.654
17.638
3.0808
5.7479
0.8054
10
0.8
6.2173
5.9041
6.0427
5.5996
5.7016
5.1392
0.9
11.309
10.236
10.71
9.2202
9.5622
7.8126
0.95
21.626
17.839
19.497
14.431
15.583
10.465
0.99
104.45
46.617
70.552
12.323
22.991
3.2209
0.8
3
0.8
0.5596
0.5314
0.5532
0.504
0.5042
0.4547
EQUATIONS
DOI: 10.37394/232021.2023.3.10
Olasunkanmi James Oladapo, Janet Iyabo Idowu,
Abiola Timothy Owolabi, Kayode Ayinde
E-ISSN: 2732-9976
83
Volume 3, 2023
0.9
1.0178
0.9213
0.9961
0.8299
0.8309
0.6794
0.95
1.9464
1.6055
1.8685
1.2988
1.3049
0.8779
0.99
9.4002
4.1956
8.1072
1.1091
1.2584
0.195
5
0.8
1.5543
1.476
1.5368
1.3999
1.4004
1.2626
0.9
2.8273
2.5591
2.7668
2.3051
2.3079
1.8867
0.95
5.4066
4.4597
5.1902
3.6077
3.6245
2.4384
0.99
26.112
11.654
22.52
3.0808
3.4955
0.5414
10
0.8
6.2173
5.9041
6.1472
5.5996
5.6014
5.0497
0.9
11.309
10.236
11.067
9.2202
9.2314
7.546
0.95
21.626
17.839
20.761
14.431
14.498
9.7527
0.99
104.45
46.617
90.08
12.323
13.982
2.1649
NOTE: Minimum MSE value is bolded in each row.
Table 3: Estimated MSE when n=50 and p=7
K
d
sigma
rho
OLS
ORR
LIU
K-L
ML
MLKL
0.3
0.2
3
0.8
3.3789
3.2362
3.0254
3.097
2.8569
2.6242
0.9
6.3296
5.8369
5.1772
5.3669
4.6497
3.9699
0.95
12.278
10.524
8.5534
8.9216
6.9701
5.1847
0.99
59.959
31.635
19.093
13.216
7.675
2.4063
5
0.8
9.3859
8.9894
8.4038
8.6027
7.9359
7.2892
0.9
17.582
16.214
14.381
14.908
12.916
11.028
0.95
34.106
29.232
23.759
24.782
19.361
14.402
0.99
166.55
87.876
53.037
36.712
21.319
6.684
10
0.8
37.544
35.958
33.615
34.411
31.744
29.157
0.9
70.329
64.855
57.524
59.633
51.664
44.11
0.95
136.42
116.93
95.037
99.129
77.444
57.607
0.99
666.21
351.5
212.15
146.85
85.278
26.736
0.5
3
0.8
3.3789
3.2362
3.1553
3.097
2.7343
2.513
0.9
6.3296
5.8369
5.5941
5.3669
4.2755
3.658
0.95
12.278
10.524
9.8629
8.9216
5.9046
4.4285
0.99
59.959
31.635
31.601
13.216
3.055
1.1837
5
0.8
9.3859
8.9894
8.7648
8.6027
7.5951
6.9804
0.9
17.582
16.214
15.539
14.908
11.876
10.161
0.95
34.106
29.232
27.397
24.782
16.402
12.301
0.99
166.55
87.876
87.781
36.712
8.486
3.2881
10
0.8
37.544
35.958
35.059
34.411
30.38
27.921
0.9
70.329
64.855
62.157
59.633
47.505
40.645
0.95
136.42
116.93
109.59
99.129
65.606
49.205
0.99
666.21
351.5
351.12
146.85
33.944
13.152
0.8
3
0.8
3.3789
3.2362
3.2884
3.097
2.6147
2.4047
0.9
6.3296
5.8369
6.0293
5.3669
3.9195
3.3611
0.95
12.278
10.524
11.277
8.9216
4.9438
3.7436
0.99
59.959
31.635
47.489
13.216
1.8153
0.5532
EQUATIONS
DOI: 10.37394/232021.2023.3.10
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E-ISSN: 2732-9976
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5
0.8
9.3859
8.9894
9.1346
8.6027
7.263
6.6794
0.9
17.582
16.214
16.748
14.908
10.887
9.3363
0.95
34.106
29.232
31.325
24.782
13.733
10.399
0.99
166.55
87.876
131.91
36.712
5.0426
1.5366
10
0.8
37.544
35.958
36.538
34.411
29.052
26.717
0.9
70.329
64.855
66.992
59.633
43.55
37.345
0.95
136.42
116.93
125.3
99.129
54.93
41.594
0.99
666.21
351.5
527.66
146.85
20.17
6.1463
0.6
0.2
3
0.8
3.3789
3.1035
3.0254
2.8414
2.8569
2.4129
0.9
6.3296
5.406
5.1772
4.5647
4.6497
3.4002
0.95
12.278
9.1533
8.5534
6.5421
6.9701
3.8933
0.99
59.959
20.078
19.093
3.3132
7.675
0.7876
5
0.8
9.3859
8.6207
8.4038
7.8928
7.9359
6.7022
0.9
17.582
15.017
14.381
12.68
12.916
9.4448
0.95
34.106
25.426
23.759
18.172
19.361
10.815
0.99
166.55
55.772
53.037
9.2035
21.319
2.1879
10
0.8
37.544
34.483
33.615
31.571
31.744
26.808
0.9
70.329
60.067
57.524
50.719
51.664
37.779
0.95
136.42
101.7
95.037
72.689
77.444
43.257
0.99
666.21
223.09
212.15
36.814
85.278
8.7515
0.5
3
0.8
3.3789
3.1035
3.1553
2.8414
2.7343
2.3121
0.9
6.3296
5.406
5.5941
4.5647
4.2755
3.1396
0.95
12.278
9.1533
9.8629
6.5421
5.9046
3.3527
0.99
59.959
20.078
31.601
3.3132
3.055
0.4449
5
0.8
9.3859
8.6207
8.7648
7.8928
7.5951
6.422
0.9
17.582
15.017
15.539
12.68
11.876
8.721
0.95
34.106
25.426
27.397
18.172
16.402
9.3128
0.99
166.55
55.772
87.781
9.2035
8.486
1.2356
10
0.8
37.544
34.483
35.059
31.571
30.38
25.688
0.9
70.329
60.067
62.157
50.719
47.505
34.884
0.95
136.42
101.7
109.59
72.689
65.606
37.251
0.99
666.21
223.09
351.12
36.814
33.944
4.9425
0.8
3
0.8
3.3789
3.1035
3.2884
2.8414
2.6147
2.2138
0.9
6.3296
5.406
6.0293
4.5647
3.9195
2.8913
0.95
12.278
9.1533
11.277
6.5421
4.9438
2.8608
0.99
59.959
20.078
47.489
3.3132
1.8153
0.2175
5
0.8
9.3859
8.6207
9.1346
7.8928
7.263
6.1489
0.9
17.582
15.017
16.748
12.68
10.887
8.0312
EQUATIONS
DOI: 10.37394/232021.2023.3.10
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E-ISSN: 2732-9976
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0.95
34.106
25.426
31.325
18.172
13.733
7.9462
0.99
166.55
55.772
131.91
9.2035
5.0426
0.6041
10
0.8
37.544
34.483
36.538
31.571
29.052
24.595
0.9
70.329
60.067
66.992
50.719
43.55
32.125
0.95
136.42
101.7
125.3
72.689
54.93
31.784
0.99
666.21
223.09
527.66
36.814
20.17
2.4163
0.9
0.2
3
0.8
3.3789
2.9798
3.0254
2.6094
2.8569
2.2207
0.9
6.3296
5.0261
5.1772
3.8921
4.6497
2.9197
0.95
12.278
8.0565
8.5534
4.8238
6.9701
2.942
0.99
59.959
14.03
19.093
2.4148
7.675
0.3235
5
0.8
9.3859
8.2772
8.4038
7.2481
7.9359
6.1681
0.9
17.582
13.962
14.381
10.811
12.916
8.11
0.95
34.106
22.379
23.759
13.399
19.361
8.1719
0.99
166.55
38.973
53.037
6.7076
21.319
0.8984
10
0.8
37.544
33.109
33.615
28.992
31.744
24.672
0.9
70.329
55.846
57.524
43.245
51.664
32.44
0.95
136.42
89.516
95.037
53.597
77.444
32.687
0.99
666.21
155.89
212.15
26.831
85.278
3.5936
0.5
3
0.8
3.3789
2.9798
3.1553
2.6094
2.7343
2.1292
0.9
6.3296
5.0261
5.5941
3.8921
4.2755
2.7016
0.95
12.278
8.0565
9.8629
4.8238
5.9046
2.5544
0.99
59.959
14.03
31.601
2.4148
3.055
0.1571
5
0.8
9.3859
8.2772
8.7648
7.2481
7.5951
5.9139
0.9
17.582
13.962
15.539
10.811
11.876
7.5043
0.95
34.106
22.379
27.397
13.399
16.402
7.095
0.99
166.55
38.973
87.781
6.7076
8.486
0.4363
10
0.8
37.544
33.109
35.059
28.992
30.38
23.655
0.9
70.329
55.846
62.157
43.245
47.505
30.017
0.95
136.42
89.516
109.59
53.597
65.606
28.379
0.99
666.21
155.89
351.12
26.831
33.944
1.7452
0.8
3
0.8
3.3789
2.9798
3.2884
2.6094
2.6147
2.0399
0.9
6.3296
5.0261
6.0293
3.8921
3.9195
2.4936
0.95
12.278
8.0565
11.277
4.8238
4.9438
2.1998
0.99
59.959
14.03
47.489
2.4148
1.8153
0.1334
5
0.8
9.3859
8.2772
9.1346
7.2481
7.263
5.6659
EQUATIONS
DOI: 10.37394/232021.2023.3.10
Olasunkanmi James Oladapo, Janet Iyabo Idowu,
Abiola Timothy Owolabi, Kayode Ayinde
E-ISSN: 2732-9976
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Volume 3, 2023
0.9
17.582
13.962
16.748
10.811
10.887
6.9264
0.95
34.106
22.379
31.325
13.399
13.733
6.1101
0.99
166.55
38.973
131.91
6.7076
5.0426
0.3703
10
0.8
37.544
33.109
36.538
28.992
29.052
22.663
0.9
70.329
55.846
66.992
43.245
43.55
27.705
0.95
136.42
89.516
125.3
53.597
54.93
24.439
0.99
666.21
155.89
527.66
26.831
20.17
1.4812
NOTE: Minimum MSE value is bolded in each row.
Table 4: Estimated MSE when n=100 and p=7
K
d
sigma
rho
OLS
ORR
LIU
K-L
ML
MLKL
0.3
0.2
3
0.8
1.5582
1.5296
1.4842
1.5013
1.448
1.3954
0.9
2.9335
2.8321
2.6781
2.7326
2.555
2.3815
0.95
5.7032
5.3277
4.8022
4.9658
4.3823
3.8239
0.99
27.908
20.44
14.275
14.191
9.2139
4.8628
5
0.8
4.3282
4.2488
4.1228
4.1701
4.0221
3.876
0.9
8.1487
7.867
7.4391
7.5906
7.0971
6.6152
0.95
15.842
14.799
13.34
13.794
12.173
10.622
0.99
77.524
56.777
39.653
39.418
25.594
13.508
10
0.8
17.313
16.995
16.491
16.681
16.088
15.504
0.9
32.595
31.468
29.756
30.363
28.389
26.461
0.95
63.369
59.197
53.358
55.175
48.693
42.488
0.99
310.09
227.11
158.61
157.67
102.38
54.031
0.5
3
0.8
1.5582
1.5296
1.5117
1.5013
1.4211
1.3695
0.9
2.9335
2.8321
2.7724
2.7326
2.4647
2.2977
0.95
5.7032
5.3277
5.1305
4.9658
4.0808
3.5631
0.99
27.908
20.44
18.839
14.191
6.1861
3.3254
5
0.8
4.3282
4.2488
4.1992
4.1701
3.9474
3.8042
0.9
8.1487
7.867
7.7012
7.5906
6.8463
6.3825
0.95
15.842
14.799
14.251
13.794
11.336
9.8975
0.99
77.524
56.777
52.33
39.418
17.184
9.2372
10
0.8
17.313
16.995
16.797
16.681
15.79
15.217
0.9
32.595
31.468
30.805
30.363
27.385
25.53
0.95
63.369
59.197
57.006
55.175
45.342
39.59
0.99
310.09
227.11
209.32
157.67
68.735
36.949
0.8
3
0.8
1.5582
1.5296
1.5395
1.5013
1.3945
1.344
0.9
2.9335
2.8321
2.8685
2.7326
2.3761
2.2156
EQUATIONS
DOI: 10.37394/232021.2023.3.10
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E-ISSN: 2732-9976
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0.95
5.7032
5.3277
5.4703
4.9658
3.7908
3.3121
0.99
27.908
20.44
24.061
14.191
3.8168
2.1058
5
0.8
4.3282
4.2488
4.2764
4.1701
3.8735
3.7332
0.9
8.1487
7.867
7.9681
7.5906
6.6003
6.1543
0.95
15.842
14.799
15.195
13.794
10.53
9.2002
0.99
77.524
56.777
66.837
39.418
10.602
5.8493
10
0.8
17.313
16.995
17.105
16.681
15.494
14.933
0.9
32.595
31.468
31.872
30.363
26.401
24.617
0.95
63.369
59.197
60.781
55.175
42.12
36.801
0.99
310.09
227.11
267.35
157.67
42.409
23.397
0.6
0.2
3
0.8
1.5582
1.5018
1.4842
1.4466
1.448
1.3448
0.9
2.9335
2.7362
2.6781
2.5462
2.555
2.2204
0.95
5.7032
4.99
4.8022
4.3272
4.3823
3.3394
0.99
27.908
15.687
14.275
7.1158
9.2139
2.5384
5
0.8
4.3282
4.1717
4.1228
4.0182
4.0221
3.7355
0.9
8.1487
7.6006
7.4391
7.0728
7.0971
6.1678
0.95
15.842
13.861
13.34
12.02
12.173
9.276
0.99
77.524
43.574
39.653
19.766
25.594
7.0511
10
0.8
17.313
16.687
16.491
16.073
16.088
14.942
0.9
32.595
30.402
29.756
28.291
28.389
24.671
0.95
63.369
55.444
53.358
48.079
48.693
37.104
0.99
310.09
174.3
158.61
79.065
102.38
28.204
0.5
3
0.8
1.5582
1.5018
1.5117
1.4466
1.4211
1.32
0.9
2.9335
2.7362
2.7724
2.5462
2.4647
2.1427
0.95
5.7032
4.99
5.1305
4.3272
4.0808
3.1135
0.99
27.908
15.687
18.839
7.1158
6.1861
1.7693
5
0.8
4.3282
4.1717
4.1992
4.0182
3.9474
3.6665
0.9
8.1487
7.6006
7.7012
7.0728
6.8463
5.9518
0.95
15.842
13.861
14.251
12.02
11.336
8.6487
0.99
77.524
43.574
52.33
19.766
17.184
4.9146
10
0.8
17.313
16.687
16.797
16.073
15.79
14.666
0.9
32.595
30.402
30.805
28.291
27.385
23.807
0.95
63.369
55.444
57.006
48.079
45.342
34.595
0.99
310.09
174.3
209.32
79.065
68.735
19.658
0.8
3
0.8
1.5582
1.5018
1.5395
1.4466
1.3945
1.2954
0.9
2.9335
2.7362
2.8685
2.5462
2.3761
2.0664
0.95
5.7032
4.99
5.4703
4.3272
3.7908
2.8962
0.99
27.908
15.687
24.061
7.1158
3.8168
1.1502
EQUATIONS
DOI: 10.37394/232021.2023.3.10
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E-ISSN: 2732-9976
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Volume 3, 2023
5
0.8
4.3282
4.1717
4.2764
4.0182
3.8735
3.5983
0.9
8.1487
7.6006
7.9681
7.0728
6.6003
5.74
0.95
15.842
13.861
15.195
12.02
10.53
8.0449
0.99
77.524
43.574
66.837
19.766
10.602
3.195
10
0.8
17.313
16.687
17.105
16.073
15.494
14.393
0.9
32.595
30.402
31.872
28.291
26.401
22.96
0.95
63.369
55.444
60.781
48.079
42.12
32.179
0.99
310.09
174.3
267.35
79.065
42.409
12.78
0.9
0.2
3
0.8
1.5582
1.4748
1.4842
1.394
1.448
1.2962
0.9
2.9335
2.6454
2.6781
2.373
2.555
2.0707
0.95
5.7032
4.6849
4.8022
3.7723
4.3823
2.9175
0.99
27.908
12.451
14.275
3.3905
9.2139
1.2694
5
0.8
4.3282
4.0968
4.1228
3.8721
4.0221
3.6004
0.9
8.1487
7.3482
7.4391
6.5917
7.0971
5.7518
0.95
15.842
13.014
13.34
10.479
12.173
8.1041
0.99
77.524
34.587
39.653
9.4181
25.594
3.5262
10
0.8
17.313
16.387
16.491
15.488
16.088
14.401
0.9
32.595
29.393
29.756
26.367
28.389
23.007
0.95
63.369
52.055
53.358
41.914
48.693
32.416
0.99
310.09
138.35
158.61
37.672
102.38
14.105
0.5
3
0.8
1.5582
1.4748
1.5117
1.394
1.4211
1.2723
0.9
2.9335
2.6454
2.7724
2.373
2.4647
1.9985
0.95
5.7032
4.6849
5.1305
3.7723
4.0808
2.7219
0.99
27.908
12.451
18.839
3.3905
6.1861
0.9046
5
0.8
4.3282
4.0968
4.1992
3.8721
3.9474
3.5341
0.9
8.1487
7.3482
7.7012
6.5917
6.8463
5.5514
0.95
15.842
13.014
14.251
10.479
11.336
7.5609
0.99
77.524
34.587
52.33
9.4181
17.184
2.5127
10
0.8
17.313
16.387
16.797
15.488
15.79
14.136
0.9
32.595
29.393
30.805
26.367
27.385
22.205
0.95
63.369
52.055
57.006
41.914
45.342
30.243
0.99
310.09
138.35
209.32
37.672
68.735
10.051
0.8
3
0.8
1.5582
1.4748
1.5395
1.394
1.3945
1.2487
0.9
2.9335
2.6454
2.8685
2.373
2.3761
1.9277
0.95
5.7032
4.6849
5.4703
3.7723
3.7908
2.5336
EQUATIONS
DOI: 10.37394/232021.2023.3.10
Olasunkanmi James Oladapo, Janet Iyabo Idowu,
Abiola Timothy Owolabi, Kayode Ayinde
E-ISSN: 2732-9976
89
Volume 3, 2023
0.99
27.908
12.451
24.061
3.3905
3.8168
0.6059
5
0.8
4.3282
4.0968
4.2764
3.8721
3.8735
3.4685
0.9
8.1487
7.3482
7.9681
6.5917
6.6003
5.3548
0.95
15.842
13.014
15.195
10.479
10.53
7.0378
0.99
77.524
34.587
66.837
9.4181
10.602
1.6829
10
0.8
17.313
16.387
17.105
15.488
15.494
13.874
0.9
32.595
29.393
31.872
26.367
26.401
21.419
0.95
63.369
52.055
60.781
41.914
42.12
28.151
0.99
310.09
138.35
267.35
37.672
42.409
6.7316
NOTE: The bolded MSE value is the minimum in each row.
Table 5: Results of regression coefficients and the corresponding MSE values
Coef
OLS
ORR
LIU
KL
ML
MLKL
1
ˆ
-52.9936
-51.7281
-49.7842
-50.4627
-48.1795
43.81268
2
ˆ
0.071073
0.070811
0.070406
0.070548
0.070072
48.46854
3
ˆ
-0.4142
-0.412
-0.4086
-0.4098
-0.40579
14.06693
4
ˆ
-0.42347
-0.42653
-0.43124
-0.42959
-0.43513
0.867867
5
ˆ
-0.57257
-0.57329
-0.57439
-0.57401
-0.5753
0.376839
6
ˆ
48.41787
48.27776
48.06159
48.13766
47.88346
-4.94995
MSE
17095.15
16356.98
15261.43
15638.54
14355.67
13192.21
k/d
0.3
0.2
0.3
0.2
0.3/0.2
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DOI: 10.37394/232021.2023.3.10
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Abiola Timothy Owolabi, Kayode Ayinde
E-ISSN: 2732-9976
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Volume 3, 2023
