Mitigating Multicollinearity in Linear Regression Model with Two
Parameter Kibria-Lukman Estimators
IDOWU J. I.1, OWOLABI A. T. 1, OLADAPO O. J.1,*, AYINDE K.2,3, OSHUOPORU O. A.1,
ALAO A. N.4
1Department of Statistics,
Ladoke Akintola University of Technology,
Ogbomoso, Oyo State,
NIGERIA
2Department of Statistics,
Federal University of Technology,
Akure, Ondo state,
NIGERIA
3Department of Mathematics and Statistics,
Northwest Missouri State University,
Maryville, Missouri,
USA
4Department of Statistics,
Kwara State Polytechnic,
Ilorin, Kwara state,
NIGERIA
*Corresponding Author
Abstract: - This study delves into the challenges faced by the ordinary least square (OLS) estimator, traditionally
regarded as the Best Linear Unbiased Estimator in classical linear regression models. Despite its reliability under
specific conditions, OLS falters in the face of multicollinearity, a problem frequently encountered in regression
analyses. To combat this issue, various ridge regression estimators have been developed, characterized as one-
parameter and two-parameter ridge-type estimators. In this context, our research introduces novel two-parameter
estimators, building upon a recently developed one-parameter ridge estimator to mitigate the impact of
multicollinearity in linear regression models. Theoretical analysis and simulation experiments were conducted to
assess the performance of the proposed estimators. Remarkably, our results reveal that, under certain conditions,
these new estimators outperform existing estimators, displaying a significantly reduced mean square error. To
validate these findings, real-life data was employed, aligning with the outcomes derived from theoretical analysis
and simulations.
Key-Words: - Kibria-Lukman Estimator, New Two-parameter Kibria-Lukman Estimator, Modified Two-
parameter Kibria-Lukman Estimator, Monte Carlo Simulation, Multicollinearity, Mean
Square Error.
Received: $SULO, 20. Revised:'HFHPEHU, 02. $ccepted: 'HFHPEHU. 3ublished:'HFHPEHU
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DOI: 10.37394/23203.2023.18.63
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Ayinde K., Oshuoporu O. A., Alao A. N.
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1 Introduction
Consider the general linear regression model defined
in matrix form as:
yX


, (1)
where
y
is a
1n
vector of the response variable,
is a known
np
full-rank matrix of predictor
variables,
is a
1p
vector of unknown regression
parameters to be estimated, and
is
1n
vector of
random error such that E(
) = 0 and Cov (
) =
2I
. Equation (1) can be written in a canonical form
as:
Zy
(2)
where
XQZ
,
Q
and Q is the orthogonal
matrix whose columns constitute the eigenvectors of
XX
. Then
p
diagXQXQZZ
,...,
1
,
where
p
...
21
> 0 are the ordered eigen
values
XX
. The ordinary least square estimator
(OLSE) of
in (1) can be defined as:
1
ˆOLS Xy

(3)
where
XX

is the design matrix.
The Ordinary Least Squares (OLS) estimator is
considered the Best Linear Unbiased Estimator
(BLUE) when all assumptions of the classical linear
regression model remain intact and unviolated. This
characteristic has established the OLS estimator as
the most powerful and popular tool for estimating
regression models, [1]. To utilize the OLS estimator
for estimating model parameters in linear regression,
one essential assumption is the non-correlation of
explanatory variables. However, the problem of
multicollinearity arises when explanatory variables
exhibit strong correlations or linear relationships, [2].
Multicollinearity, if present, severely affects the OLS
estimator, leading to unstable and imprecise
parameter estimates, questionable predictions, and
invalid statistical inferences about model parameters.
It also results in regression coefficients with
exaggerated absolute values and signs that can
reverse with minor changes in the data, [3], [4]. It
also impacts t-tests, the extra sum of squares, fitted
values, predictions, and coefficient of determination,
[5]. To address the issue of multicollinearity, various
biased estimators have been developed. Some of
these one-parameter estimators include the Stein
estimator [6], principal component estimator [7],
ordinary ridge regression estimator by [8], modified
ridge regression by [9], contraction estimator [10],
Liu estimator [11], and KL [12]. Additionally, certain
authors have introduced two-parameter estimators to
combat multicollinearity, such as [13], [14], [15],
[16], [17], [18], [19] and [20]. The ordinary ridge
regression estimator (ORRE) is one of the most
widely used biased estimators. It intends to overcome
the multicollinearity problem by adding a positive
value, biasing or shrinkage parameter k, to the ill-
conditioned matrix's diagonal elements
XX
. The
selection of k is a major problem of ORRE. This is
because the biasing parameter k plays a significant
role in controlling the regression's bias toward the
mean of the dependent variable, [8].
The Ordinary ridge regression estimator, as
proposed by [8], stands as one of the most widely
used among these estimators. It effectively addresses
the issue of multicollinearity by introducing a
positive value, denoted as k, to the diagonal elements
of the Z’Z matrix. This constant k serves as the
biasing parameter. However, a significant challenge
associated with ridge regression lies in selecting this
biasing parameter k, as it plays a crucial role in
controlling the regression's bias towards the mean of
the dependent variable. Various authors have put
forth proposals for the biasing parameter k. Some
notable contributors include, [21], [22], [23], [24],
[25], [26], [27], [28], [29], [30] and numerous others.
[11], proposed another estimator with d as the biasing
parameter. The Liu Estimator gains preference due to
its capacity for appropriate d selection, being a linear
function of the biasing parameter d. However,
selecting a suitable k remains challenging because
the ordinary ridge regression estimator relies on a
nonlinear function of the biasing parameter k [8].
This paper aims to introduce new two-parameter
estimators for regression parameter estimation in
scenarios where independent variables exhibit
correlation. The performance of the proposed
estimators is compared with the OLSE, ridge
regression [8], Liu estimator [11], two-parameter
estimator (TP) [13], Modified Ridge Type (MRT)
[17], and Kibria-Lukman (KL) [12].
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2 Some Alternative Biased Estimators
and the Proposed Estimator
2.1 Some Estimators as Alternative to OLS
Ridge-type estimators have been proposed as an
alternative to the OLSE. The canonical form of
OLSE is written in equation (3). Following this, the
ordinary ridge regression (RE) proposed by [8] is
given as:
yZHyZkIk
RE ''
ˆ1
0
1
(4)
where k is the non-negative constant known as the
biasing parameter.
The Liu estimator (LE) [11] is defined as:
OLSOLSLE DDdIId
ˆˆˆ 10
1
(5)
where d is the biasing parameter of the Liu
Estimator.
The KL estimator [12] is given as:
OLSOLSKL HHkIkIk
ˆˆˆ 1
1
0
1
(6)
TP estimator [13] is given as:
OLSOLSTP HHkdIkIdk
ˆˆ
,
ˆ2
1
0
1
(7)
where k and d are the biasing parameters of the ridge
and Liu Estimator, respectively.
The Modified Ridge Type (MRT) Parameters
proposed [17] is given as:
OLSKOLSMRT RIdkdk
ˆˆ
)1(,
ˆ1
(8)
where
kIH
0
,
kIH
1
,
kdIH
2
,
kdIH
3
,
1
0
ID
,
dID
1
and
1
1
IdkRk
.
2.2 New Two parameter Kibria-Lukman
(NTPKL) Estimator
In this article, the ridge estimator was grafted into the
Kibria-Lukman estimator, [12], to propose new
parameter estimators. The first proposed two-
parameter estimator is defined as follows:
RENTPKL IkdIkddk
ˆ
,
ˆ2
1
1
(9)
RENTPKL HHdk
ˆ
,
ˆ3
1
2
(10)
where k>0, 0<d<1 and
21 dd
The following are the properties of the NTPKL
Estimator
1
03
1
2
,
ˆ
HHHdkE NTPKL
(11)
IHHHdkB NTPKL 1
03
1
2
,
ˆ
(12)
1
03
1
2
1
03
1
2
2
,
ˆ HHHHHHdkD NTPKL
(13)
The Mean Square Error Matrix (MSEM) of
the proposed estimator is given as:
2 1 1 1 1
2 3 0 2 3 0
1 1 1 1
2 3 0 2 3 0
ˆ,
'
NTPKL
MSEM k d H H H H H H
H H H I H H H I




(14)
2.3 Modified Two Parameter Kibria-Lukman
(MTPKL) Estimators
The Modified Two Parameter Kibria-Lukman
Estimators are special cases of the New Two
Parameter Kibra-Lukman Estimator in equation (9).
The first Modified estimator (MTPKL1) is obtained
by equating
2
d
to one (i.e.,
2
d
=1) in equation (9).
REMTPKL kIkdIdk
ˆ
,
ˆ1
1
(15)
REMTPLK HHdk
ˆ
,
ˆ1
1
21
(16)
where k>0 and 0<d<1.
The following are the properties of the MTPKL1
Estimator:
1
21
1
01 ,
ˆ
HHHdkE MTPKL
(17)
IHHHdkB MTPKL 1
21
1
01 ,
ˆ
(18)
1
21
1
0
1
21
1
0
2
1, HHHHHHdkD MTPKL
(19)
The Mean Square Error Matrix (MSEM) of the
proposed MTPKL1 estimator is given as:
2 1 1 1 1
1 0 1 2 0 1 2
1 1 1 1
0 1 2 0 1 2
ˆ,
'
MTPKL
MSEM k d H H H H H H
H H H I H H H I





(20)
The second Modified estimator (MTPKL2) is
obtained by equating
1
d
to one (i.e.,
1
d
=1) in
equation (9). It was also obtained by modifying, [13],
following the idea of the (k – d) class estimator
proposed by [31]. Thus, the MTPKL2 estimator can
be defined as:
REMTPKL kdIkIdk
ˆ
,
ˆ1
2
(21)
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REMTPKL HHdk
ˆ
,
ˆ3
1
02
(22)
where k>0 and 0<d<1.
The following are the properties of the MTPKL2
Estimator:
1
03
1
02 ,
ˆ
HHHdkE MTPKL
(23)
IHHHdkB MTPKL 1
03
1
02 ,
ˆ
(24)
1
03
1
0
1
03
1
0
2
2,
ˆ HHHHHHdkD MTPKL
(25)
The Mean Square Error Matrix (MSEM) of the
proposed MTPKL2 estimator is given as:
2 1 1 1 1
2 0 3 0 0 3 0
1 1 1 1
0 3 0 0 3 0
ˆ,
'
MTPKL
MSEM k d H H H H H H
H H H I H H H I





(26)
Since the MTPKL1 and MTPKL2 estimators are
special cases of NTPKL, Lemmas 1, 2 and 3 will be
used to make some theoretical comparisons between
NTPKL and six existing estimators and to prove the
statistical properties of the NTPKL Estimator.
Lemma 1. Let n x n matrices M > 0 and N > 0 (or
N
0
), Then M > N if and only if
1
1
NM
i
where
1
NM
i
is the largest
eigenvalue of the matrix
1
NM
, [32].
Lemma 2. Let M be an n x n positive definite matrix,
that is, M > 0, and
be some vector, then,
0
M
if and only if
1
1
M
, [33].
Lemma 3. Let
yAii
ˆ
, i = 1, 2, be two linear
estimators of
. Suppose
that
0
ˆˆ 21
CovCovD
, where
2,1,
ˆiCov i
denotes the covariance matrix of
and
2,1,
ˆ iIXABiasb iii
.
Consequently,
1 1 1 2
21 2 2 2
ˆ ˆ ˆ ˆ
0
MSEM MSEM
D bb b b

(27)
if and only if
1
2
2 1 1 2 1
ˆˆ
i i i i
b D bb b
whereMSEM Cov bb






[34]
2.4 Comparison of NTPKL Estimator with
Existing Estimators
In this section, the theoretical comparison is carried
out among the estimators to examine the performance
of the proposed estimator,
dk
NTPKL ,
ˆ
over other
estimators;
OLS
ˆ
,
RE
ˆ
,
LE
ˆ
,
NTP
ˆ
,
MRT
ˆ
,
KL
ˆ
.
2.4.1 Comparison between
dkand NTPKLOLS ,
ˆˆ
The MSEM of the estimator
yZ
OLS '
ˆ1
is
as follows:
12
ˆ
OLS
MSEM
(28)
The difference between (14) and (28)
1
03
1
2
1
03
1
2
212
,
ˆˆ HHHHHHdkMSEMMESM NTPKLOLS
IHHHIHHH 1
03
1
2
1
03
1
2'

(29)
Let k > 0 and 0 < d < 1. Thus, the following theorem
holds.
Theorem: The proposed estimator
dk
NTPKL ,
ˆ
is
superior to
OLS
ˆ
if and only if
1
1 1 2 1 1 1 1 1
2 3 0 2 3 0 2 3 0
11
2 3 0
'
1
H H H I H H H H H H
H H H I




(30)
Proof
2 1 1 1 1 1
2 3 0 2 3 0
ˆˆ
,
OLS NTPKL
D D k d
H H H H H H


p
i
ii
ii
ikkd
kd
diag
1
22
2
21
(31)
1
03
1
2
1
03
1
2
1 HHHHHH
will be pdf if and
only if
0
2222 kdkkd iiii
. By
lemma 3, the proof is completed.
2.4.2 Comparison between
k
RE
ˆ
and
dk
NTPKL ,
ˆ
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The bias vector, covariance matrix, and MSEM of
the estimator
yZkIk
RE '
ˆ1
are as
follows:
1
ˆkIkE RE
(32)
11
2
ˆ kIkIkD RE
(33)
and the Mean Square Error Matrix is given
as:
1
0
1
0
21
0
1
0
2'
ˆ HHkHHkMSEM RE

(
34)
The difference between (14) and (34)
dkMSEMkMESM NTPKLRE ,
ˆˆ
2 1 1 2 1 1 1 1 2 1 1
0 0 2 3 0 2 3 0 0 0
1 1 1 1
2 3 0 2 3 0
'
'
H H H H H H H H k H H
H H H I H H H I


(35)
Let k > 0 and 0 < d < 1. Thus, the following theorem
holds.
Theorem: The proposed estimate
dk
NTPKL ,
ˆ
is
superior to
k
RE
ˆ
if and only if
0,
ˆ
)(
ˆ dkMSEMkMSEM NTPKLRE
if and
only if
11
2 3 0
1
2 1 1 1 1 1 1 2 1 1
0 0 2 3 0 2 3 0 0 0
11
2 3 0
'
'
1
H H H I
H H H H H H H H k H H
H H H I






(36)
Proof: Considering the dispersion matrix difference
between
)(
ˆkD RE
and
dkD NTPKL ,
ˆ
1
03
1
2
1
03
1
2
21
0
1
0
2 HHHHHHHHDd
1 1 1
22
1 1 1
d
D kI kI kdI kdI
kI kdI kdI kI

11
2
2 1 1
2
d
D kI kdI
kdI kdI kI kdI




11
2
11
2
4
d
D kI kdI
kdI kI kdI




(37)
It is observed that Dd is positive definite. By
lemma 3, the proof is completed.
2.4.3 Comparison between
d
LE
ˆ
and
dk
NTPKL ,
ˆ
The bias vector, covariance matrix, and MSEM of
the estimator
OLSLiu dII
ˆˆ 1
are as
follows:
1
1
ˆ
IddB LE
(38)
11
1
11
2
ˆdIIdIID LE
(39)
21
0 1 0 1
'
00
ˆ''
1 ' 1 ' '
LE
MSEM D D D D
d D d D



(40)
Theorem: The proposed estimator
dk
NTPKL ,
ˆ
is
superior to
d
LE
ˆ
if and only if
0,
ˆ
)(
ˆ dkMSEMdMSEM NTPKLLE
if and
only if
1
1
0 1 0 1
2
11 2 1 1 1 1
2 3 0 2 3 0 2 3 0
00
11
2 3 0
''
'
1 ' 1 ' '
1
D D D D
H H H I H H H H H H
d D d D
H H H I
















Proof: Considering the dispersion matrix difference
between
)(
ˆdD LE
and
dkD NTPKL ,
ˆ
1
03
1
2
1
03
1
2
2
10
1
10
2'' HHHHHHDDDDDd
1 1 1 1
21
11
2
11
d
D I dI I dI
kdI kdI kI
kdI kdI kI



p
i
ii
ii
ii
i
dkkd
kdd
diagD
22
2
2
2
2
1
(41)
will be pdf if and only if
01 22
2
222 iiiii kdkkdd
For 0 < d < 1 and k > 0, it was observed that
01 22
2
222 iiiii kdkkdd
By lemma 3, the proof is completed.
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2.4.4 Comparison between
k
KL
ˆ
and
dk
NTPKL ,
ˆ
The bias vector, covariance matrix, and MSEM of
the estimator
ˆˆ 1kIkIk
KL
are as
follows:
IHHkB KL 1
1
0
ˆ
(42)
'
1
1
0
1
1
1
0
2
ˆHHHHkD KL
(43)
2 1 1 1
0 1 0 1
11
0 1 0 1
ˆ
''
KL
MSEM k H H H H
H H I H H I






(44)
Theorem: The proposed estimator
dk
NMKL ,
ˆ2
is
superior to
k
KL
ˆ
if and only if:
0,
ˆ
)(
ˆ dkMSEMkMSEM NTPKLKL
if and
only if
1
1 1 1
0 1 0 1
2
11 1 1 1 1
2 3 0 2 3 0 2 3 0
11
0 1 0 1
11
2 3 0
'
'
1
H H H H
H H H I H H H H H H
H H H H
H H H I















(45)
Proof: Considering the dispersion matrix difference
between
)(
ˆkD KL
and
dkD NTPKL ,
ˆ
2 1 1 1
0 1 0 1
2 1 1 1 1
2 3 0 2 3 0
d
D H H H H
H H H H H H
11
21
1 1 1
2
1
d
D kI kI kI kI
kdI kdI kI kdI
kdI kI

p
i
ii
ii
ii
i
dkkd
kd
k
k
diagD
22
2
2
2
2
will be pdf if and only if
0
2
2
22 kdkdkiiii
. For 0 < d <
1 and k > 0, it was observed that
0
2
2
22 kdkdkiiii
. By lemma
3, the proof is completed.
2.4.5 Comparison between
dk
MRT ,
ˆ
and
dk
NTPKL ,
ˆ
The bias vector, covariance matrix, and
MSEM of the estimator
OLSKOLSMRT RIdkdk
ˆˆ
)1(,
ˆ1
are
as follows:
IRdkB kMRT ,
ˆ
(46)
kkMRT RRdkD ',
ˆ12
(47)
''',
ˆ12 IRIRRRdkMSEM kkkkMRT

(48)
Where
1
)1(
IdkRk
. Let k > 0 and 0 < d
< 1. Thus, the following theorem holds.
Theorem: The proposed estimator
dk
NTPKL ,
ˆ
is
superior to
dk
MRT ,
ˆ
if and only if
0,
ˆ
),(
ˆ dkMSEMdkMSEM NTPKLMRT
if
and only if
11
2 3 0
1
2 1 2 1 1 1 1
2 3 0 2 3 0
11
2 3 0
'
' ' '
1
k k k k
H H H I
R R H H H H H H R R
H H H I






(49)
Proof: Considering the dispersion matrix difference
between
),(
ˆdkD MRT
and
dkD NTPKL ,
ˆ
1
03
1
2
1
03
1
2
212 ' HHHHHHRRD kkd
11
2
1 1 1
2
1
(1 ) (1 )
d
D k d I k d I
kdI kdI kI kdI
kdI kI

2
22
22
1
(1 )
p
i
i
d
ii
ii
i
kd
D diag kd
kd k











(50)
will be pdf if and only if
.0)1( 2222 dkkdkkd iiiiii
For 0 < d < 1 and k > 0, it was observed that
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.0)1( 2222 dkkdkkd iiiiii
By lemma 3, the proof is completed.
2.4.6 Comparison between
dk
TP ,
ˆ
and
dk
NTPKL ,
ˆ
The bias vector, covariance matrix, and
MSEM of the estimator
OLSTP kdIkIdk
ˆ
,
ˆ1
are as follows:
IHHdkB TP 2
1
0
,
ˆ
(51)
2
1
0
1
2
1
0
2',
ˆHHHHdkD TP
(52)
2 1 1 1
0 2 0 2
11
0 2 0 2
ˆ,
''
TP
MSEM k d H H H H
H H I H H I






(53)
Let k > 0 and 0 < d < 1. Thus, the following theorem
holds.
Theorem: The proposed estimator
dk
NTPKL ,
ˆ
is
superior to
dk
TP ,
ˆ
if and only if
0,
ˆ
),(
ˆ dkMSEMdkMSEM NTPKLTP
.
That is, if and only if,
1
1 1 1
0 2 0 2
2
11 1 1 1 1
2 3 0 2 3 0 2 0
11
0 3 0 3
11
2 3 0
'
( ) '( )'
1
H H H H
H H H I H H H H H
H H I H H I
H H H I
















(54)
Proof: Considering the dispersion matrix difference
between
),(
ˆdkD TP
and
dkD NTPKL ,
ˆ
1
03
1
2
1
03
1
2
2
2
1
0
1
2
1
0
2 HHHHHHHHHHDd
11
21
11
2
11
d
D kdI kI kdI kI
kdI kdI kI
kdI kdI kI



p
i
ii
ii
i
i
dkkd
kd
k
kd
diagD
1
22
2
2
2
2
(55)
will be pdf if and only if
0
2
2
4 kdkd iii
. For 0 < d < 1 and k
> 0, it was observed that
0
2
2
4 kdkd iii
. By lemma 3, the
proof is completed.
2.5 Determination of Biasing Parameters k
and d
Finding the appropriate ridge and Liu parameters, k
and d, respectively, is a critical issue in the study of
the ridge and Liu regression. These parameters may
either be non-stochastic or may depend on the
observed data. The choice of values for these ridge
parameters has been one of the most difficult
problems confronting the study of the generalized
ridge regression, [35]. The biasing parameters k
proposed by [12] in equation (6) will be used to
determine and evaluate the performance of the
proposed estimators compared to the OLS estimator
and other estimators. It is given as:
ii
k
22
2
ˆ
ˆ
2
ˆ
(56)
In determining the optimal value of d for
dk
NTPKL ,
ˆ
, k is fixed. The optimal value of d can
be regarded as to be that d that minimizes
IHHHIHHHHHHHHHdkMSEM NTPKL 1
03
1
2
1
03
1
2
1
03
1
2
1
03
1
2
2',
ˆ

dkMSEMtrdkMSEMdkf NTPKLNTPKL ,
ˆ
,
ˆ
),(
2
222
2
2
222
,
2
pii
iii
pii
i
iii
kd
f k d kd k
kd k k d
kd k








(57)
Differentiating
dkf ,
with respect to d and equate
to zero gives
p
i
ikkk
k
d2222
22
2
(58)
For practical purposes,
2
and
2
i
are replaced with
2
ˆ
and
2
ˆi
, respectively. Consequently,
(58) becomes
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p
iii
ii kkk
k
d2222
22
ˆ
ˆˆ
2
ˆ
ˆ
ˆ
(59)
3 Simulation Study
To support the theoretical comparison of the
estimators, a Monte Carlo simulation study was
conducted using R 4.0.2 to examine the performance
of these estimators. Also, in this section, the results
of the simulation will be discussed.
3.1 Simulation Technique
The simulation procedure used by [22], [25], [35]
and [36], were used to generate the explanatory
variables in this study: This is given as:
1
22,1
1
ij ij i p
x z z

,
1,2,...,in
,
1,2,..., .jp
(60)
where
ij
z
is an independent standard normal pseudo-
random number,
is the correlation between any
two explanatory variables considered to be 0.8, 0.9,
0.95, and 0.99. Also, p is the number of explanatory
variables, and p=3 is considered for the simulation
study. The variables are standardized so that
XX
and
yX
are in correlation forms. The values of
1

, [37]. The standard deviations in this
simulation study were σ = 1, 3, 5, and 10. In
comparing the estimators, k was chosen to be 0.3,
0.6, and 0.9, which lies between 0 and 1, following
[35] and [38]. We also chose d = 0.2, 0.5, 0.8 for
sample sizes 50 and 100. The replication for the
study is 1000 times. The mean square error was
obtained using:
iij
jiij
MSE
ˆˆ
1000
1
ˆ
1000
1
(61)
Table 1 (Appendix) shows that when k=0.3 and
d=0.2,
d
LE
ˆ
and
dk
MTPKL ,
ˆ1
consistently
perform better than
dk
NTPKL ,
ˆ
and other existing
estimators across the four levels of sigma (1, 3, 5,
and 10) and at the two sample sizes (n = 50 and 100),
particularly when rho is 0.8, 0.9, 0.95. At sample size
50,
k
KL
ˆ
and
dk
MTPKL ,
ˆ1
dominates other
estimators, while at n= 100, the proposed estimators,
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
perform better than
other estimators when rho is 0.99.
From Table 2, Table 3, Table 9 and Table 12 in
Appendix (when k=0.3 and d=0.5, k=0.3 and d=0.8,
k=0.7 and d=0.8, k=0.9 and d=0.8), the simulation
results show that two of the proposed estimators;
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
consistently
outperform
dk
NTPKL ,
ˆ
and other existing
estimators in this study at sample size 50 and 100
across the four levels of sigma (1, 3, 5 and 10) and at
rho is 0.8, 0.9, 0.95, and 0.99
From Table 4 and Table 7 in Appendix (when k=0.6
and d=0.2, k=0.7 and d=0.2), the simulation results
show that two of the proposed estimators;
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
consistently
dominate
dk
NTPKL ,
ˆ
and other existing estimators
in this study except for when rho is 0.99, where
dk
MTPKL ,
ˆ1
and
k
KL
ˆ
dominates
dk
NTPKL ,
ˆ
and other existing estimators at
sample size 50 and 100.
From Table 5 and Table 8 in Appendix (when k=0.6
and d=0.5, k=0.7 and d=0.5), the simulation results
show that two of the proposed estimators;
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
consistently
dominate
dk
NTPKL ,
ˆ
and other existing estimators
in this study except for when rho is 0.99, where
dk
MTPKL ,
ˆ1
and
k
KL
ˆ
dominates
dk
NTPKL ,
ˆ
and other existing estimators at
sample size 50 only.
From Table 6 in Appendix (when k=0.6 and d=0.8),
the simulation results show that two of the proposed
estimators;
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
consistently dominate
dk
NTPKL ,
ˆ
and other existing estimators in this study except for
when rho is 0.99, sigma = 10 and n=50 where
dk
MTPKL ,
ˆ1
and
dk
NTPKL ,
ˆ
dominates
dk
MTPKL ,
ˆ2
and other existing estimators.
From Table 10 in Appendix (when k=0.9 and d=0.2),
the simulation results show that two of the proposed
estimators;
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
consistently dominate
dk
NTPKL ,
ˆ
and other existing estimators in this study except for
when rho is 0.95 and 0.99, where
dk
MTPKL ,
ˆ1
and
k
KL
ˆ
dominates
dk
MTPKL ,
ˆ2
and other existing
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estimators at sample size 50. However, when the
sample size is 100,
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
consistently dominates
dk
NTPKL ,
ˆ
and other existing estimators except for when rho
0.99, where
dk
MTPKL ,
ˆ1
and
k
KL
ˆ
dominates
dk
MTPKL ,
ˆ2
and other existing estimators.
From Table 11 in Appendix (when k=0.9 and
d=0.5), the simulation results show that two of the
proposed estimators;
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
consistently dominate
dk
NTPKL ,
ˆ
and other existing estimators in this study while
dk
MTPKL ,
ˆ1
has the least MSE value among the
three proposed estimators across the sample sizes,
sigma and rho levels used in this study.
4 Numerical Example
In this section, Longley data was used to demonstrate
the performance of the proposed estimator. Longley
data were used by [39] and [40]. The regression
model for these data is defined as:
662211 XXXy
(62)
For more details on the data set, [39]. The
variance inflation factors are VIF1 = 135.53, VIF2 =
1788.51, VIF3 = 33.62, VIF4 = 3.59, VIF5 = 399.15
and VIF6 = 758.98. Eigenvalues of
XX
matrix are
λ1 = 2.76779×1012, λ2 = 7,039,139,179, λ3 =
11,608,993.96, λ4 = 2,504,761.021, λ5 = 1738.356, λ6
= 13.309 and the condition number of
XX
is
456,070. The VIFs, the eigenvalues, and the
condition number all indicate that severe
multicollinearity exists. The estimated parameters
and the MSE values of the estimators are presented in
Table 13 (Appendix). Two of the proposed
estimators perform best among other estimators as
they given the smallest MSE value
Though,
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
which are
special cases of
dk
NTPKL ,
ˆ
perform better than
dk
NTPKL ,
ˆ
and other existing estimators, as they
give smaller MSE values compared with all the
existing estimators considered in this study,
dk
MTPKL ,
ˆ1
performs best in all. This is consistent
with the results of the simulation study.
5 Summary and Concluding Remarks
This paper proposes new two-parameter estimators to
solve the multicollinearity problem for linear
regression models. The proposed estimators were
theoretically compared with six other existing
estimators. A simulation study was then conducted to
compare the performance of the proposed
estimators;
dk
MTPKL ,
ˆ1
,
dk
MTPKL ,
ˆ2
and
dk
NTPKL ,
ˆ
with six existing estimators. From the
theoretical comparison, simulation study, and
application of life data, the proposed estimators,
especially the two special cases of
dk
NTPKL ,
ˆ
which are
dk
MTPKL ,
ˆ1
and
dk
MTPKL ,
ˆ2
give
better results in terms of MSE. Hopefully, this paper
will be helpful to researchers in different fields. The
proposed estimators are as a result of this,
recommended for use by researchers in various
fields.
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E-ISSN: 2224-2856
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Idowu, J.I. Ayinde, K., Oladapo, O. J. and Owolabi,
A.T. conceived and designed the study. Oladapo, O.
J., Alao, N. A., Idowu, J.I. and Oshuporu, O. A.
conducted the data analysis. Ayinde K. supervised
the study. Idowu, J. I., Oladapo, O. J., Owolabi, A.T.,
Ayinde K., and Alao, N. A. interpreted the study
results. Owolabi, A. T., Oladapo, O.J., Idowu, J. I.
and Oshuporu, O. A. wrote the first draft of the
manuscript. Ayinde K. and Owolabi, A. T. reviewed
and corrected the draft manuscript. All authors read
through and approved the final manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research received no external funding.
Conflicts of Interest
The authors declare that they have no conflicts 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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APPENDIX
Table 1. Estimated MSE when k=0.3, d=0.2
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1311
0.1234
0.1261
0.1301
0.1322
0.1291
0.1251
0.1252
0.9
0.2462
0.2288
0.2046
0.2120
0.2255
0.2322
0.2220
0.2089
0.2094
0.95
0.4682
0.4064
0.3333
0.3492
0.3956
0.4184
0.3833
0.3393
0.3429
0.99
2.2417
1.2173
0.7223
0.5101
1.0998
1.3968
0.9166
0.4469
0.5771
3
0.8
1.2270
1.1802
1.1098
1.1344
1.1712
1.1895
1.1618
1.1255
1.1264
0.9
2.2158
2.0587
1.8412
1.9076
2.0293
2.0896
1.9980
1.8794
1.8847
0.95
4.2136
3.6579
2.9999
3.1428
3.5604
3.7658
3.4501
3.0533
3.0857
50
0.99
20.1751
10.9561
6.5009
4.5913
9.8985
12.5716
8.2494
4.0221
5.1935
5
0.8
3.4082
3.2782
3.0825
3.1509
3.2532
3.3040
3.2272
3.1264
3.1288
0.9
6.1551
5.7185
5.1142
5.2987
5.6370
5.8045
5.5499
5.2205
5.2351
0.95
11.7045
10.1609
8.3330
8.7300
9.8901
10.4606
9.5836
8.4815
8.5716
0.99
56.0420
30.4336
18.0581
12.7536
27.4958
34.9210
22.9151
11.1726
14.4263
10
0.8
13.6328
13.1127
12.3294
12.6032
13.0124
13.2158
12.9084
12.5051
12.5148
0.9
24.6204
22.8740
20.4565
21.1943
22.5479
23.2179
22.1993
20.8816
20.9400
0.95
46.8180
40.6437
33.3323
34.9200
39.5605
41.8426
38.3346
33.9260
34.2864
0.99
224.1680
121.7344
72.2327
51.0147
109.9835
139.6840
91.6605
44.6905
57.7054
1
0.8
0.0622
0.0611
0.0594
0.0600
0.0609
0.0613
0.0607
0.0598
0.0598
0.9
0.1131
0.1093
0.1036
0.1056
0.1086
0.1101
0.1078
0.1049
0.1050
0.95
0.2163
0.2023
0.1828
0.1889
0.1997
0.2051
0.1970
0.1864
0.1868
0.99
1.0445
0.7667
0.5345
0.5329
0.7248
0.8187
0.6710
0.4995
0.5275
3
0.8
0.5596
0.5499
0.5345
0.5403
0.5480
0.5518
0.5461
0.5384
0.5385
0.9
1.0178
0.9839
0.9322
0.9506
0.9774
0.9907
0.9706
0.9442
0.9448
0.95
1.9464
1.8211
1.6447
1.7002
1.7976
1.8458
1.7726
1.6775
1.6813
100
0.99
9.4002
6.9000
4.8107
4.7956
6.5235
7.3684
6.0393
4.4950
4.7478
5
0.8
1.5543
1.5275
1.4848
1.5009
1.5222
1.5328
1.5168
1.4957
1.4959
0.9
2.8273
2.7332
2.5896
2.6407
2.7149
2.7519
2.6961
2.6228
2.6243
0.95
5.4066
5.0586
4.5687
4.7227
4.9932
5.1273
4.9238
4.6598
4.6704
0.99
26.1116
19.1667
13.3631
13.3212
18.1207
20.4678
16.7757
12.4861
13.1882
10
0.8
6.2173
6.1100
5.9393
6.0036
6.0889
6.1314
6.0674
5.9827
5.9837
0.9
11.3093
10.9326
10.3582
10.5627
10.8597
11.0074
10.7844
10.4911
10.4973
0.95
21.6263
20.2345
18.2747
18.8909
19.9729
20.5090
19.6951
18.6390
18.6815
0.99
104.4463
76.6669
53.4523
53.2847
72.4828
81.8710
67.1027
49.9442
52.7527
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
623
Volume 18, 2023
Table 2. Estimated MSE when k=0.3, d=0.5
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1311
0.1281
0.1261
0.1287
0.1337
0.1261
0.1237
0.1237
0.9
0.2462
0.2288
0.2197
0.2120
0.2207
0.2374
0.2123
0.2043
0.2048
0.95
0.4682
0.4064
0.3811
0.3492
0.3802
0.4367
0.3512
0.3252
0.3278
0.99
2.2417
1.2173
1.1925
0.5101
0.9535
1.6899
0.5978
0.3726
0.4549
3
0.8
1.2270
1.1802
1.1530
1.1344
1.1578
1.2034
1.1348
1.1125
1.1132
0.9
2.2158
2.0587
1.9774
1.9076
1.9865
2.1365
1.9104
1.8384
1.8424
0.95
4.2136
3.6579
3.4302
3.1428
3.4215
3.9307
3.1608
2.9264
2.9506
50
0.99
20.1751
10.9561
10.7325
4.5913
8.5816
15.2088
5.3805
3.3531
4.0942
5
0.8
3.4082
3.2782
3.2026
3.1509
3.2161
3.3429
3.1522
3.0902
3.0920
0.9
6.1551
5.7185
5.4928
5.2987
5.5181
5.9347
5.3065
5.1066
5.1177
0.95
11.7045
10.1609
9.5284
8.7300
9.5043
10.9186
8.7801
8.1288
8.1961
0.99
56.0420
30.4336
29.8126
12.7536
23.8378
42.2467
14.9459
9.3141
11.3728
10
0.8
13.6328
13.1127
12.8101
12.6032
12.8643
13.3714
12.6084
12.3602
12.3677
0.9
24.6204
22.8740
21.9710
21.1943
22.0722
23.7388
21.2258
20.4259
20.4704
0.95
46.8180
40.6437
38.1136
34.9200
38.0173
43.6745
35.1204
32.5154
32.7845
0.99
224.1680
121.7344
119.2504
51.0147
95.3513
168.9870
59.7836
37.2565
45.4912
1
0.8
0.0622
0.0611
0.0604
0.0600
0.0606
0.0616
0.0600
0.0595
0.0595
0.9
0.1131
0.1093
0.1071
0.1056
0.1075
0.1112
0.1057
0.1039
0.1039
0.95
0.2163
0.2023
0.1950
0.1889
0.1959
0.2092
0.1891
0.1827
0.1830
0.99
1.0445
0.7667
0.7055
0.5329
0.6683
0.9001
0.5497
0.4551
0.4749
3
0.8
0.5596
0.5499
0.5438
0.5403
0.5452
0.5547
0.5404
0.5356
0.5357
0.9
1.0178
0.9839
0.9639
0.9506
0.9677
1.0008
0.9509
0.9347
0.9351
0.95
1.9464
1.8211
1.7548
1.7002
1.7631
1.8832
1.7022
1.6444
1.6473
100
0.99
9.4002
6.9000
6.3497
4.7956
6.0150
8.1006
4.9469
4.0958
4.2742
5
0.8
1.5543
1.5275
1.5107
1.5009
1.5144
1.5409
1.5010
1.4879
1.4881
0.9
2.8273
2.7332
2.6774
2.6407
2.6879
2.7800
2.6415
2.5963
2.5975
0.95
5.4066
5.0586
4.8743
4.7227
4.8976
5.2311
4.7284
4.5677
4.5758
0.99
26.1116
19.1667
17.6381
13.3212
16.7084
22.5017
13.7414
11.3772
11.8727
10
0.8
6.2173
6.1100
6.0427
6.0036
6.0574
6.1635
6.0041
5.9516
5.9523
0.9
11.3093
10.9326
10.7097
10.5627
10.7517
11.1201
10.5660
10.3851
10.3899
0.95
21.6263
20.2345
19.4972
18.8909
19.5904
20.9244
18.9138
18.2708
18.3033
0.99
104.4463
76.6669
70.5524
53.2847
66.8336
90.0070
54.9657
45.5089
47.4907
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
624
Volume 18, 2023
Table 3. Estimated MSE when k=0.3, d=0.8
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1311
0.1330
0.1261
0.1272
0.1353
0.1232
0.1223
0.1223
0.9
0.2462
0.2288
0.2354
0.2120
0.2162
0.2427
0.2030
0.1999
0.2001
0.95
0.4682
0.4064
0.4323
0.3492
0.3657
0.4555
0.3218
0.3119
0.3132
0.99
2.2417
1.2173
1.7822
0.5101
0.8349
2.0114
0.3858
0.3158
0.3479
3
0.8
1.2270
1.1802
1.1971
1.1344
1.1448
1.2175
1.1085
1.0997
1.1000
0.9
2.2158
2.0587
2.1188
1.9076
1.9451
2.1839
1.8268
1.7988
1.8007
0.95
4.2136
3.6579
3.8903
3.1428
3.2909
4.0992
2.8962
2.8074
2.8185
50
0.99
20.1751
10.9561
16.0396
4.5913
7.5144
18.1030
3.4725
2.8421
3.1315
5
0.8
3.4082
3.2782
3.3251
3.1509
3.1798
3.3820
3.0790
3.0546
3.0555
0.9
6.1551
5.7185
5.8855
5.2987
5.4031
6.0665
5.0742
4.9965
5.0017
0.95
11.7045
10.1609
10.8064
8.7300
9.1413
11.3868
8.0450
7.7985
7.8293
0.99
56.0420
30.4336
44.5544
12.7536
20.8735
50.2860
9.6458
7.8948
8.6985
10
0.8
13.6328
13.1127
13.3005
12.6032
12.7188
13.5279
12.3156
12.2179
12.2214
0.9
24.6204
22.8740
23.5419
21.1943
21.6120
24.2658
20.2964
19.9855
20.0064
0.95
46.8180
40.6437
43.2258
34.9200
36.5654
45.5471
32.1800
31.1940
31.3174
0.99
224.1680
121.7344
178.2178
51.0147
83.4941
201.1442
38.5833
31.5795
34.7943
1
0.8
0.0622
0.0611
0.0615
0.0600
0.0603
0.0620
0.0594
0.0592
0.0592
0.9
0.1131
0.1093
0.1107
0.1056
0.1065
0.1123
0.1035
0.1028
0.1028
0.95
0.2163
0.2023
0.2076
0.1889
0.1922
0.2134
0.1817
0.1792
0.1793
0.99
1.0445
0.7667
0.9008
0.5329
0.6183
0.9854
0.4500
0.4166
0.4252
3
0.8
0.5596
0.5499
0.5532
0.5403
0.5424
0.5576
0.5347
0.5329
0.5329
0.9
1.0178
0.9839
0.9961
0.9506
0.9581
1.0110
0.9317
0.9253
0.9255
0.95
1.9464
1.8211
1.8685
1.7002
1.7297
1.9210
1.6348
1.6123
1.6136
100
0.99
9.4002
6.9000
8.1072
4.7956
5.5649
8.8685
4.0503
3.7489
3.8264
5
0.8
1.5543
1.5275
1.5368
1.5009
1.5066
1.5489
1.4854
1.4802
1.4803
0.9
2.8273
2.7332
2.7668
2.6407
2.6614
2.8084
2.5881
2.5702
2.5708
0.95
5.4066
5.0586
5.1902
4.7227
4.8048
5.3360
4.5411
4.4785
4.4823
0.99
26.1116
19.1667
22.5199
13.3212
15.4581
24.6347
11.2509
10.4136
10.6289
10
0.8
6.2173
6.1100
6.1472
6.0036
6.0262
6.1958
5.9416
5.9207
5.9210
0.9
11.3093
10.9326
11.0674
10.5627
10.6454
11.2334
10.3522
10.2809
10.2831
0.95
21.6263
20.2345
20.7609
18.8909
19.2192
21.3441
18.1645
17.9139
17.9292
0.99
104.4463
76.6669
90.0797
53.2847
61.8325
98.5387
45.0034
41.6544
42.5154
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
625
Volume 18, 2023
Table 4. Estimated MSE when k=0.6, d=0.2
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1263
0.1234
0.1167
0.1244
0.1283
0.1224
0.1149
0.1153
0.9
0.2462
0.2132
0.2046
0.1827
0.2074
0.2196
0.2009
0.1774
0.1794
0.95
0.4682
0.3564
0.3333
0.2604
0.3391
0.3775
0.3173
0.2461
0.2566
0.99
2.2417
0.7679
0.7223
0.0801
0.6563
0.9998
0.4381
0.0644
0.2020
3
0.8
1.2270
1.1362
1.1098
1.0491
1.1193
1.1540
1.1012
1.0330
1.0362
0.9
2.2158
1.9183
1.8412
1.6431
1.8663
1.9760
1.8072
1.5955
1.6133
0.95
4.2136
3.2080
2.9999
2.3436
3.0516
3.3978
2.8556
2.2146
2.3094
50
0.99
20.1751
6.9114
6.5009
0.7210
5.9070
8.9983
3.9425
0.5794
1.8174
5
0.8
3.4082
3.1559
3.0825
2.9139
3.1090
3.2055
3.0587
2.8690
2.8779
0.9
6.1551
5.3284
5.1142
4.5638
5.1839
5.4888
5.0199
4.4316
4.4810
0.95
11.7045
8.9111
8.3330
6.5101
8.4768
9.4384
7.9322
6.1518
6.4149
0.99
56.0420
19.1985
18.0581
2.0029
16.4084
24.9953
10.9513
1.6093
5.0484
10
0.8
13.6328
12.6233
12.3294
11.6547
12.4356
12.8219
12.2341
11.4752
11.5108
0.9
24.6204
21.3135
20.4565
18.2547
20.7355
21.9550
20.0791
17.7258
17.9234
0.95
46.8180
35.6446
33.3323
26.0407
33.9073
37.7537
31.7291
24.6072
25.6600
0.99
224.1680
76.7940
72.2327
8.0117
65.6337
99.9814
43.8053
6.4375
20.1937
1
0.8
0.0622
0.0601
0.0594
0.0580
0.0597
0.0605
0.0592
0.0576
0.0576
0.9
0.1131
0.1058
0.1036
0.0987
0.1044
0.1072
0.1029
0.0974
0.0976
0.95
0.2163
0.1898
0.1828
0.1651
0.1851
0.1949
0.1798
0.1608
0.1622
0.99
1.0445
0.5881
0.5345
0.2651
0.5341
0.6687
0.4515
0.2342
0.2920
3
0.8
0.5596
0.5405
0.5345
0.5218
0.5368
0.5443
0.5330
0.5182
0.5185
0.9
1.0178
0.9518
0.9322
0.8881
0.9394
0.9648
0.9262
0.8762
0.8782
0.95
1.9464
1.7080
1.6447
1.4859
1.6658
1.7544
1.6184
1.4469
1.4600
100
0.99
9.4002
5.2929
4.8107
2.3853
4.8065
6.0184
4.0635
2.1077
2.6277
5
0.8
1.5543
1.5014
1.4848
1.4495
1.4912
1.5119
1.4806
1.4394
1.4403
0.9
2.8273
2.6439
2.5896
2.4669
2.6094
2.6800
2.5728
2.4338
2.4395
0.95
5.4066
4.7444
4.5687
4.1273
4.6272
4.8733
4.4956
4.0191
4.0555
0.99
26.1116
14.7025
13.3631
6.6258
13.3514
16.7177
11.2874
5.8547
7.2991
10
0.8
6.2173
6.0056
5.9393
5.7978
5.9646
6.0476
5.9223
5.7577
5.7613
0.9
11.3093
10.5755
10.3582
9.8677
10.4378
10.7202
10.2913
9.7352
9.7580
0.95
21.6263
18.9777
18.2747
16.5093
18.5089
19.4930
17.9825
16.0762
16.2220
0.99
104.4463
58.8099
53.4523
26.5030
53.4054
66.8709
45.1496
23.4186
29.1961
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
626
Volume 18, 2023
Table 5. Estimated MSE when k=0.6, d=0.5
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1263
0.1281
0.1167
0.1217
0.1312
0.1169
0.1123
0.1126
0.9
0.2462
0.2132
0.2197
0.1827
0.1992
0.2294
0.1838
0.1699
0.1714
0.95
0.4682
0.3564
0.3811
0.2604
0.3154
0.4103
0.2666
0.2268
0.2343
0.99
2.2417
0.7679
1.1925
0.0801
0.5301
1.4066
0.1837
0.0488
0.1193
3
0.8
1.2270
1.1362
1.1530
1.0491
1.0947
1.1811
1.0508
1.0095
1.0120
0.9
2.2158
1.9183
1.9774
1.6431
1.7923
2.0643
1.6530
1.5281
1.5414
0.95
4.2136
3.2080
3.4302
2.3436
2.8383
3.6931
2.3995
2.0409
2.1086
50
0.99
20.1751
6.9114
10.7325
0.7210
4.7705
12.6591
1.6527
0.4386
1.0732
5
0.8
3.4082
3.1559
3.2026
2.9139
3.0407
3.2808
2.9187
2.8038
2.8106
0.9
6.1551
5.3284
5.4928
4.5638
4.9784
5.7340
4.5914
4.2443
4.2812
0.95
11.7045
8.9111
9.5284
6.5101
7.8841
10.2588
6.6653
5.6693
5.8573
0.99
56.0420
19.1985
29.8126
2.0029
13.2514
35.1642
4.5908
1.2183
2.9811
10
0.8
13.6328
12.6233
12.8101
11.6547
12.1622
13.1229
11.6739
11.2140
11.2411
0.9
24.6204
21.3135
21.9710
18.2547
19.9132
22.9359
18.3648
16.9765
17.1238
0.95
46.8180
35.6446
38.1136
26.0407
31.5367
41.0351
26.6614
22.6773
23.4292
0.99
224.1680
76.7940
119.2504
8.0117
53.0058
140.6570
18.3634
4.8734
11.9245
1
0.8
0.0622
0.0601
0.0604
0.0580
0.0591
0.0611
0.0580
0.0570
0.0571
0.9
0.1131
0.1058
0.1071
0.0987
0.1024
0.1094
0.0988
0.0955
0.0956
0.95
0.2163
0.1898
0.1950
0.1651
0.1784
0.2028
0.1659
0.1546
0.1557
0.99
1.0445
0.5881
0.7055
0.2651
0.4662
0.7996
0.3035
0.1974
0.2345
3
0.8
0.5596
0.5405
0.5438
0.5218
0.5314
0.5500
0.5220
0.5129
0.5131
0.9
1.0178
0.9518
0.9639
0.8881
0.9213
0.9845
0.8892
0.8588
0.8604
0.95
1.9464
1.7080
1.7548
1.4859
1.6055
1.8252
1.4931
1.3914
1.4012
100
0.99
9.4002
5.2929
6.3497
2.3853
4.1956
7.1966
2.7312
1.7767
2.1101
5
0.8
1.5543
1.5014
1.5107
1.4495
1.4760
1.5277
1.4499
1.4246
1.4253
0.9
2.8273
2.6439
2.6774
2.4669
2.5591
2.7348
2.4700
2.3854
2.3898
0.95
5.4066
4.7444
4.8743
4.1273
4.4597
5.0699
4.1475
3.8648
3.8921
0.99
26.1116
14.7025
17.6381
6.6258
11.6543
19.9905
7.5867
4.9351
5.8613
10
0.8
6.2173
6.0056
6.0427
5.7978
5.9041
6.1110
5.7997
5.6983
5.7011
0.9
11.3093
10.5755
10.7097
9.8677
10.2365
10.9391
9.8800
9.5416
9.5591
0.95
21.6263
18.9777
19.4972
16.5093
17.8388
20.2795
16.5901
15.4592
15.5684
0.99
104.4463
58.8099
70.5524
26.5030
46.6173
79.9619
30.3465
19.7401
23.4451
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
627
Volume 18, 2023
Table 6. Estimated MSE when k=0.6, d=0.8
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1263
0.1330
0.1167
0.1191
0.1343
0.1116
0.1098
0.1100
0.9
0.2462
0.2132
0.2354
0.1827
0.1915
0.2394
0.1682
0.1629
0.1636
0.95
0.4682
0.3564
0.4323
0.2604
0.2941
0.4446
0.2241
0.2098
0.2131
0.99
2.2417
0.7679
1.7822
0.0801
0.4374
1.8841
0.0698
0.0387
0.0601
3
0.8
1.2270
1.1362
1.1971
1.0491
1.0710
1.2085
1.0029
0.9869
0.9881
0.9
2.2158
1.9183
2.1188
1.6431
1.7228
2.1545
1.5123
1.4651
1.4712
0.95
4.2136
3.2080
3.8903
2.3436
2.6472
4.0012
2.0164
1.8877
1.9174
50
0.99
20.1751
6.9114
16.0396
0.7210
3.9366
16.9565
0.6281
0.3474
0.5407
5
0.8
3.4082
3.1559
3.3251
2.9139
2.9747
3.3569
2.7854
2.7409
2.7440
0.9
6.1551
5.3284
5.8855
4.5638
4.7853
5.9848
4.2004
4.0692
4.0861
0.95
11.7045
8.9111
10.8064
6.5101
7.3532
11.1144
5.6012
5.2435
5.3261
0.99
56.0420
19.1985
44.5544
2.0029
10.9351
47.1015
1.7448
0.9649
1.5019
10
0.8
13.6328
12.6233
13.3005
11.6547
11.8982
13.4276
11.1406
10.9622
10.9749
0.9
24.6204
21.3135
23.5419
18.2547
19.1408
23.9392
16.8005
16.2759
16.3434
0.95
46.8180
35.6446
43.2258
26.0407
29.4132
44.4577
22.4050
20.9741
16.3434
0.99
224.1680
76.7940
178.2178
8.0117
43.7407
188.4059
6.9793
3.8599
16.3434
1
0.8
0.0622
0.0601
0.0615
0.0580
0.0585
0.0617
0.0568
0.0565
0.0565
0.9
0.1131
0.1058
0.1107
0.0987
0.1004
0.1116
0.0949
0.0936
0.0937
0.95
0.2163
0.1898
0.2076
0.1651
0.1721
0.2108
0.1531
0.1488
0.1493
0.99
1.0445
0.5881
0.9008
0.2651
0.4107
0.9425
0.2025
0.1689
0.1836
3
0.8
0.5596
0.5405
0.5532
0.5218
0.5260
0.5557
0.5112
0.5076
0.5077
0.9
1.0178
0.9518
0.9961
0.8881
0.9037
1.0044
0.8538
0.8419
0.8427
0.95
1.9464
1.7080
1.8685
1.4859
1.5485
1.8974
1.3778
1.3391
1.3436
100
0.99
9.4002
5.2929
8.1072
2.3853
3.6959
8.4827
1.8218
1.5201
1.6520
5
0.8
1.5543
1.5014
1.5368
1.4495
1.4611
1.5437
1.4200
1.4100
1.4103
0.9
2.8273
2.6439
2.7668
2.4669
2.5103
2.7901
2.3715
2.3386
2.3406
0.95
5.4066
4.7444
5.1902
4.1273
4.3015
5.2705
3.8271
3.7197
3.7322
0.99
26.1116
14.7025
22.5199
6.6258
10.2664
23.5632
5.0604
4.2223
4.5888
10
0.8
6.2173
6.0056
6.1472
5.7978
5.8445
6.1747
5.6799
5.6399
5.6412
0.9
11.3093
10.5755
11.0674
9.8677
10.0413
11.1605
9.4859
9.3542
9.3624
0.95
21.6263
18.9777
20.7609
16.5093
17.2058
21.0822
15.3082
14.8786
14.9288
0.99
104.4463
58.8099
90.0797
26.5030
41.0653
94.2527
20.2414
16.8890
18.3550
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
628
Volume 18, 2023
Table 7. Estimated MSE when k=0.7, d=0.2
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1247
0.1234
0.1137
0.1226
0.1270
0.1203
0.1117
0.1122
0.9
0.2462
0.2083
0.2046
0.1738
0.2019
0.2157
0.1944
0.1680
0.1706
0.95
0.4682
0.3419
0.3333
0.2361
0.3230
0.3655
0.2986
0.2211
0.2340
0.99
2.2417
0.6731
0.7223
0.0378
0.5677
0.9122
0.3503
0.0304
0.1489
3
0.8
1.2270
1.1221
1.1098
1.0222
1.1028
1.1427
1.0819
1.0040
1.0082
0.9
2.2158
1.8748
1.8412
1.5634
1.8164
1.9406
1.7489
1.5109
1.5339
0.95
4.2136
3.0769
2.9999
2.1241
2.9068
3.2895
2.6869
1.9893
2.1058
50
0.99
20.1751
6.0583
6.5009
0.3396
5.1093
8.2098
3.1526
0.2729
1.3398
5
0.8
3.4082
3.1167
3.0825
2.8391
3.0632
3.1739
3.0051
2.7883
2.8000
0.9
6.1551
5.2076
5.1142
4.3425
5.0455
5.3905
4.8579
4.1967
4.2603
0.95
11.7045
8.5469
8.3330
5.9004
8.0744
9.1375
7.4636
5.5258
5.8494
0.99
56.0420
16.8287
18.0581
0.9434
14.1924
22.8051
8.7573
0.7582
3.7218
10
0.8
13.6328
12.4665
12.3294
11.3553
12.2522
12.6954
12.0197
11.1520
11.1990
0.9
24.6204
20.8301
20.4565
17.3694
20.1815
21.5618
19.4312
16.7859
17.0405
0.95
46.8180
34.1876
33.3323
23.6018
32.2976
36.5501
29.8547
22.1034
23.3978
0.99
224.1680
67.3149
72.2327
3.7737
56.7697
91.2205
35.0295
3.0328
14.8873
1
0.8
0.0622
0.0597
0.0594
0.0573
0.0593
0.0602
0.0588
0.0569
0.0569
0.9
0.1131
0.1046
0.1036
0.0965
0.1030
0.1063
0.1014
0.0950
0.0953
0.95
0.2163
0.1859
0.1828
0.1579
0.1806
0.1918
0.1746
0.1531
0.1550
0.99
1.0445
0.5425
0.5345
0.2074
0.4873
0.6296
0.3989
0.1800
0.2436
3
0.8
0.5596
0.5374
0.5345
0.5158
0.5332
0.5418
0.5288
0.5117
0.5121
0.9
1.0178
0.9415
0.9322
0.8682
0.9273
0.9565
0.9120
0.8547
0.8574
0.95
1.9464
1.6727
1.6447
1.4207
1.6252
1.7257
1.5710
1.3774
1.3944
100
0.99
9.4002
4.8827
4.8107
1.8662
4.3853
5.6661
3.5896
1.6196
2.1920
5
0.8
1.5543
1.4929
1.4848
1.4327
1.4810
1.5050
1.4687
1.4212
1.4224
0.9
2.8273
2.6151
2.5896
2.4117
2.5757
2.6569
2.5334
2.3741
2.3816
0.95
5.4066
4.6464
4.5687
3.9463
4.5145
4.7937
4.3637
3.8262
3.8734
0.99
26.1116
13.5630
13.3631
5.1838
12.1814
15.7392
9.9711
4.4987
6.0889
10
0.8
6.2173
5.9714
5.9393
5.7309
5.9241
6.0202
5.8750
5.6847
5.6896
0.9
11.3093
10.4605
10.3582
9.6467
10.3029
10.6275
10.1335
9.4961
9.5264
0.95
21.6263
18.5858
18.2747
15.7850
18.0579
19.1747
17.4549
15.3045
15.4934
0.99
104.4463
54.2520
53.4523
20.7350
48.7256
62.9567
39.8844
17.9946
24.3556
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
629
Volume 18, 2023
Table 8. Estimated MSE when k=0.7, d=0.5
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1247
0.1281
0.1137
0.1195
0.1304
0.1140
0.1088
0.1092
0.9
0.2462
0.2083
0.2197
0.1738
0.1927
0.2269
0.1753
0.1599
0.1617
0.95
0.4682
0.3419
0.3811
0.2361
0.2975
0.4025
0.2438
0.2013
0.2104
0.99
2.2417
0.6731
1.1925
0.0378
0.4511
1.3408
0.1257
0.0233
0.0790
3
0.8
1.2270
1.1221
1.1530
1.0222
1.0749
1.1739
1.0245
0.9775
0.9807
0.9
2.2158
1.8748
1.9774
1.5634
1.7341
2.0416
1.5763
1.4373
1.4542
0.95
4.2136
3.0769
3.4302
2.1241
2.6776
3.6222
2.1941
1.8108
1.8928
50
0.99
20.1751
6.0583
10.7325
0.3396
4.0594
12.0670
1.1309
0.2088
0.7102
5
0.8
3.4082
3.1167
3.2026
2.8391
2.9856
3.2607
2.8455
2.7148
2.7237
0.9
6.1551
5.2076
5.4928
4.3425
4.8167
5.6710
4.3784
3.9920
4.0391
0.95
11.7045
8.5469
9.5284
5.9004
7.4378
10.0618
6.0946
5.0299
5.2578
0.99
56.0420
16.8287
29.8126
0.9434
11.2761
33.5193
3.1414
0.5799
1.9726
10
0.8
13.6328
12.4665
12.8101
11.3553
11.9416
13.0428
11.3809
10.8577
10.8934
0.9
24.6204
20.8301
21.9710
17.3694
19.2663
22.6840
17.5127
15.9669
16.1552
0.95
46.8180
34.1876
38.1136
23.6018
29.7514
40.2472
24.3787
20.1200
21.0313
0.99
224.1680
67.3149
119.2504
3.7737
45.1045
134.0774
12.5656
2.3198
7.8906
1
0.8
0.0622
0.0597
0.0604
0.0573
0.0586
0.0609
0.0574
0.0562
0.0563
0.9
0.1131
0.1046
0.1071
0.0965
0.1008
0.1088
0.0967
0.0928
0.0931
0.95
0.2163
0.1859
0.1950
0.1579
0.1731
0.2008
0.1589
0.1463
0.1477
0.99
1.0445
0.5425
0.7055
0.2074
0.4192
0.7726
0.2509
0.1484
0.1879
3
0.8
0.5596
0.5374
0.5438
0.5158
0.5269
0.5484
0.5160
0.5055
0.5059
0.9
1.0178
0.9415
0.9639
0.8682
0.9066
0.9792
0.8697
0.8350
0.8371
0.95
1.9464
1.6727
1.7548
1.4207
1.5578
1.8068
1.4302
1.3164
1.3290
100
0.99
9.4002
4.8827
6.3497
1.8662
3.7726
6.9538
2.2581
1.3349
1.6907
5
0.8
1.5543
1.4929
1.5107
1.4327
1.4636
1.5234
1.4334
1.4041
1.4051
0.9
2.8273
2.6151
2.6774
2.4117
2.5184
2.7201
2.4158
2.3193
2.3251
0.95
5.4066
4.6464
4.8743
3.9463
4.3272
5.0190
3.9727
3.6565
3.6916
0.99
26.1116
13.5630
17.6381
5.1838
10.4795
19.3161
6.2725
3.7079
4.6962
10
0.8
6.2173
5.9714
6.0427
5.7309
5.8543
6.0937
5.7335
5.6165
5.6203
0.9
11.3093
10.4605
10.7097
9.6467
10.0734
10.8805
9.6631
9.2770
9.3001
0.95
21.6263
18.5858
19.4972
15.7850
17.3088
20.0761
15.8906
14.6258
14.7662
0.99
104.4463
54.2520
70.5524
20.7350
41.9180
77.2645
25.0898
14.8312
18.7847
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
630
Volume 18, 2023
Table 9. Estimated MSE when k=0.7, d=0.8
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1247
0.1330
0.1137
0.1166
0.1340
0.1080
0.1061
0.1062
0.9
0.2462
0.2083
0.2354
0.1738
0.1843
0.2384
0.1581
0.1523
0.1532
0.95
0.4682
0.3419
0.4323
0.2361
0.2751
0.4413
0.1991
0.1841
0.1881
0.99
2.2417
0.6731
1.7822
0.0378
0.3674
1.8533
0.0389
0.0188
0.0335
3
0.8
1.2270
1.1221
1.1971
1.0222
1.0481
1.2056
0.9703
0.9522
0.9537
0.9
2.2158
1.8748
2.1188
1.5634
1.6574
2.1453
1.4212
1.3691
1.3768
0.95
4.2136
3.0769
3.8903
2.1241
2.4752
3.9715
1.7915
1.6562
1.6918
50
0.99
20.1751
6.0583
16.0396
0.3396
3.3064
16.6799
0.3493
0.1685
0.3004
5
0.8
3.4082
3.1167
3.3251
2.8391
2.9110
3.3488
2.6947
2.6443
2.6484
0.9
6.1551
5.2076
5.8855
4.3425
4.6037
5.9590
3.9473
3.8026
3.8240
0.95
11.7045
8.5469
10.8064
5.9004
6.8757
11.0321
4.9764
4.6005
4.6993
0.99
56.0420
16.8287
44.5544
0.9434
9.1844
46.3331
0.9702
0.4681
0.8345
10
0.8
13.6328
12.4665
13.3005
11.3553
11.6432
13.3951
10.7776
10.5756
10.5922
0.9
24.6204
20.8301
23.5419
17.3694
18.4143
23.8360
15.7880
15.2092
15.2949
0.95
46.8180
34.1876
43.2258
23.6018
27.5030
44.1283
19.9058
18.4022
18.7972
0.99
224.1680
67.3149
178.2178
3.7737
36.7379
185.3324
3.8809
1.8724
3.3380
1
0.8
0.0622
0.0597
0.0615
0.0573
0.0579
0.0617
0.0560
0.0556
0.0556
0.9
0.1131
0.1046
0.1107
0.0965
0.0985
0.1114
0.0922
0.0907
0.0908
0.95
0.2163
0.1859
0.2076
0.1579
0.1661
0.2100
0.1448
0.1400
0.1406
0.99
1.0445
0.5425
0.9008
0.2074
0.3647
0.9307
0.1559
0.1247
0.1398
3
0.8
0.5596
0.5374
0.5532
0.5158
0.5207
0.5551
0.5037
0.4995
0.4997
0.9
1.0178
0.9415
0.9961
0.8682
0.8867
1.0023
0.8295
0.8160
0.8170
0.95
1.9464
1.6727
1.8685
1.4207
1.4947
1.8899
1.3023
1.2595
1.2652
100
0.99
9.4002
4.8827
8.1072
1.8662
3.2818
8.3766
1.4023
1.1214
1.2578
5
0.8
1.5543
1.4929
1.5368
1.4327
1.4465
1.5419
1.3989
1.3874
1.3879
0.9
2.8273
2.6151
2.7668
2.4117
2.4630
2.7842
2.3040
2.2665
2.2692
0.95
5.4066
4.6464
5.1902
3.9463
4.1518
5.2497
3.6175
3.4984
3.5144
0.99
26.1116
13.5630
22.5199
5.1838
9.1161
23.2683
3.8951
3.1149
3.4939
10
0.8
6.2173
5.9714
6.1472
5.7309
5.7859
6.1677
5.5957
5.5496
5.5514
0.9
11.3093
10.4605
11.0674
9.6467
9.8519
11.1367
9.2156
9.0659
9.0767
0.95
21.6263
18.5858
20.7609
15.7850
16.6072
20.9990
14.4698
13.9932
14.0574
0.99
104.4463
54.2520
90.0797
20.7350
36.4642
93.0733
15.5802
12.4593
13.9752
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
631
Volume 18, 2023
Table 10. Estimated MSE when k=0.9, d=0.2
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1217
0.1234
0.1081
0.1191
0.1246
0.1162
0.1056
0.1064
0.9
0.2462
0.1992
0.2046
0.1575
0.1915
0.2082
0.1823
0.1508
0.1546
0.95
0.4682
0.3154
0.3333
0.1937
0.2941
0.3434
0.2651
0.1783
0.1958
0.99
2.2417
0.5301
0.7223
0.0227
0.4374
0.7761
0.2297
0.0168
0.0854
3
0.8
1.2270
1.0947
1.1098
0.9706
1.0710
1.1205
1.0447
0.9484
0.9550
0.9
2.2158
1.7923
1.8412
1.4156
1.7228
1.8733
1.6395
1.3553
1.3892
0.95
4.2136
2.8383
2.9999
1.7424
2.6472
3.0910
2.3858
1.6035
1.7613
50
0.99
20.1751
4.7705
6.5009
0.2033
3.9366
6.9844
2.0674
0.1504
0.7679
5
0.8
3.4082
3.0407
3.0825
2.6954
2.9747
3.1124
2.9016
2.6339
2.6522
0.9
6.1551
4.9784
5.1142
3.9317
4.7853
5.2034
4.5538
3.7640
3.8584
0.95
11.7045
7.8841
8.3330
4.8400
7.3532
8.5861
6.6273
4.4542
4.8924
0.99
56.0420
13.2514
18.0581
0.5647
10.9351
19.4012
5.7429
0.4178
2.1329
10
0.8
13.6328
12.1622
12.3294
10.7802
11.8982
12.4492
11.6056
10.5338
10.6070
0.9
24.6204
19.9132
20.4565
15.7255
19.1408
20.8131
18.2145
15.0550
15.4324
0.95
46.8180
31.5367
33.3323
19.3600
29.4132
34.3446
26.5095
17.8169
19.5697
0.99
224.1680
53.0058
72.2327
2.2591
43.7407
77.6050
22.9718
1.6713
8.5317
1
0.8
0.0622
0.0591
0.0594
0.0561
0.0585
0.0597
0.0578
0.0555
0.0556
0.9
0.1131
0.1024
0.1036
0.0923
0.1004
0.1045
0.0983
0.0904
0.0909
0.95
0.2163
0.1784
0.1828
0.1444
0.1721
0.1857
0.1646
0.1388
0.1416
0.99
1.0445
0.4662
0.5345
0.1233
0.4107
0.5630
0.3144
0.1036
0.1730
3
0.8
0.5596
0.5314
0.5345
0.5040
0.5260
0.5369
0.5204
0.4988
0.4995
0.9
1.0178
0.9213
0.9322
0.8299
0.9037
0.9402
0.8845
0.8133
0.8176
0.95
1.9464
1.6055
1.6447
1.2988
1.5485
1.6709
1.4813
1.2486
1.2740
100
0.99
9.4002
4.1956
4.8107
1.1091
3.6959
5.0670
2.8297
0.9318
1.5563
5
0.8
1.5543
1.4760
1.4848
1.3999
1.4611
1.4915
1.4455
1.3855
1.3874
0.9
2.8273
2.5591
2.5896
2.3051
2.5103
2.6116
2.4569
2.2591
2.2710
0.95
5.4066
4.4597
4.5687
3.6077
4.3015
4.6415
4.1146
3.4681
3.5388
0.99
26.1116
11.6543
13.3631
3.0808
10.2664
14.0751
7.8603
2.5881
4.3229
10
0.8
6.2173
5.9041
5.9393
5.5996
5.8445
5.9660
5.7819
5.5417
5.5496
0.9
11.3093
10.2365
10.3582
9.2202
10.0413
10.4465
9.8275
9.0362
9.0838
0.95
21.6263
17.8388
18.2747
14.4306
17.2058
18.5659
16.4584
13.8722
14.1550
0.99
104.4463
46.6173
53.4523
12.3231
41.0653
56.3003
31.4412
10.3522
17.2912
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
632
Volume 18, 2023
Table 11. Estimated MSE when n=100, k=0.9, d=0.5
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1217
0.1281
0.1081
0.1153
0.1289
0.1085
0.1022
0.1027
0.9
0.2462
0.1992
0.2197
0.1575
0.1808
0.2220
0.1596
0.1416
0.1443
0.95
0.4682
0.3154
0.3811
0.1937
0.2662
0.3879
0.2046
0.1585
0.1705
0.99
2.2417
0.5301
1.1925
0.0227
0.3386
1.2353
0.0595
0.0124
0.0362
3
0.8
1.2270
1.0947
1.1530
0.9706
1.0369
1.1598
0.9742
0.9168
0.9217
0.9
2.2158
1.7923
1.9774
1.4156
1.6262
1.9982
1.4351
1.2721
1.2969
0.95
4.2136
2.8383
3.4302
1.7424
2.3957
3.4910
1.8403
1.4257
1.5336
50
0.99
20.1751
4.7705
10.7325
0.2033
3.0474
11.1181
0.5349
0.1103
0.3246
5
0.8
3.4082
3.0407
3.2026
2.6954
2.8800
3.2217
2.7055
2.5457
2.5594
0.9
6.1551
4.9784
5.4928
3.9317
4.5169
5.5505
3.9859
3.5330
3.6018
0.95
11.7045
7.8841
9.5284
4.8400
6.6547
9.6973
5.1120
3.9601
4.2600
0.99
56.0420
13.2514
29.8126
0.5647
8.4651
30.8837
1.4857
0.3064
0.9014
10
0.8
13.6328
12.1622
12.8101
10.7802
11.5189
12.8864
10.8204
10.1808
10.2359
0.9
24.6204
19.9132
21.9710
15.7255
18.0671
22.2019
15.9424
14.1306
14.4057
0.95
46.8180
31.5367
38.1136
19.3600
26.6191
38.7893
20.4483
15.8405
17.0400
0.99
224.1680
53.0058
119.2504
2.2591
33.8606
123.5349
5.9429
1.2255
3.6058
1
0.8
0.0622
0.0591
0.0604
0.0561
0.0576
0.0606
0.0561
0.0547
0.0547
0.9
0.1131
0.1024
0.1071
0.0923
0.0976
0.1077
0.0925
0.0878
0.0882
0.95
0.2163
0.1784
0.1950
0.1444
0.1632
0.1969
0.1460
0.1310
0.1331
0.99
1.0445
0.4662
0.7055
0.1233
0.3446
0.7259
0.1731
0.0822
0.1227
3
0.8
0.5596
0.5314
0.5438
0.5040
0.5181
0.5454
0.5044
0.4912
0.4917
0.9
1.0178
0.9213
0.9639
0.8299
0.8784
0.9689
0.8322
0.7895
0.7927
0.95
1.9464
1.6055
1.7548
1.2988
1.4688
1.7717
1.3133
1.1787
1.1974
100
0.99
9.4002
4.1956
6.3497
1.1091
3.1008
6.5331
1.5575
0.7389
1.1035
5
0.8
1.5543
1.4760
1.5107
1.3999
1.4392
1.5149
1.4010
1.3643
1.3658
0.9
2.8273
2.5591
2.6774
2.3051
2.4399
2.6915
2.3116
2.1928
2.2018
0.95
5.4066
4.4597
4.8743
3.6077
4.0800
4.9213
3.6480
3.2741
3.3260
0.99
26.1116
11.6543
17.6381
3.0808
8.6133
18.1475
4.3263
2.0524
3.0650
10
0.8
6.2173
5.9041
6.0427
5.5996
5.7569
6.0596
5.6038
5.4568
5.4628
0.9
11.3093
10.2365
10.7097
9.2202
9.7593
10.7658
9.2462
8.7711
8.8071
0.95
21.6263
17.8388
19.4972
14.4306
16.3200
19.6851
14.5917
13.0962
13.3036
0.99
104.4463
46.6173
70.5524
12.3231
34.4532
72.5899
17.3050
8.2091
12.2596
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
633
Volume 18, 2023
Table 12. Estimated MSE when n=100, k=0.9, d=0.8
Two estimators with Minimum MSE values are bolded in each row. The smaller of the two is italicized.
N
2
rho
OLS
RIDGE
LIU
K-L
MRT
TP
NTPKL
MTPKL1
MTPKL2
1
0.8
0.1363
0.1217
0.1330
0.1081
0.1118
0.1333
0.1013
0.0989
0.0992
0.9
0.2462
0.1992
0.2354
0.1575
0.1710
0.2364
0.1399
0.1333
0.1345
0.95
0.4682
0.3154
0.4323
0.1937
0.2422
0.4351
0.1578
0.1421
0.1471
0.99
2.2417
0.5301
1.7822
0.0227
0.2703
1.8030
0.0131
0.0101
0.0117
3
0.8
1.2270
1.0947
1.1971
0.9706
1.0046
1.1999
0.9087
0.8867
0.8890
0.9
2.2158
1.7923
2.1188
1.4156
1.5378
2.1274
1.2567
1.1968
1.2079
0.95
4.2136
2.8383
3.8903
1.7424
2.1794
3.9162
1.4186
1.2771
1.3226
50
0.99
20.1751
4.7705
16.0396
0.2033
2.4324
16.2272
0.1167
0.0898
0.1042
5
0.8
3.4082
3.0407
3.3251
2.6954
2.7899
3.3329
2.5232
2.4621
2.4685
0.9
6.1551
4.9784
5.8855
3.9317
4.2714
5.9094
3.4902
3.3237
3.3545
0.95
11.7045
7.8841
10.8064
4.8400
6.0539
10.8783
3.9406
3.5473
3.6738
0.99
56.0420
13.2514
44.5544
0.5647
6.7567
45.0756
0.3240
0.2492
0.2892
10
0.8
13.6328
12.1622
13.3005
10.7802
11.1585
13.3316
10.0908
9.8463
9.8718
0.9
24.6204
19.9132
23.5419
15.7255
17.0847
23.6374
13.9592
13.2930
13.4163
0.95
46.8180
31.5367
43.2258
19.3600
24.2157
43.5134
15.7624
14.1894
14.6951
0.99
224.1680
53.0058
178.2178
2.2591
27.0269
180.3023
1.2961
0.9966
1.1570
1
0.8
0.0622
0.0591
0.0615
0.0561
0.0568
0.0615
0.0544
0.0539
0.0539
0.9
0.1131
0.1024
0.1107
0.0923
0.0949
0.1109
0.0871
0.0853
0.0855
0.95
0.2163
0.1784
0.2076
0.1444
0.1551
0.2084
0.1295
0.1240
0.1249
0.99
1.0445
0.4662
0.9008
0.1233
0.2934
0.9100
0.0926
0.0670
0.0816
3
0.8
0.5596
0.5314
0.5532
0.5040
0.5105
0.5539
0.4889
0.4838
0.4840
0.9
1.0178
0.9213
0.9961
0.8299
0.8541
0.9981
0.7832
0.7667
0.7682
0.95
1.9464
1.6055
1.8685
1.2988
1.3954
1.8755
1.1648
1.1149
1.1233
100
0.99
9.4002
4.1956
8.1072
1.1091
2.6407
8.1898
0.8325
0.6026
0.7334
5
0.8
1.5543
1.4760
1.5368
1.3999
1.4179
1.5385
1.3579
1.3436
1.3443
0.9
2.827326 0
2.5591
2.7668
2.3051
2.3725
2.7726
2.1753
2.1296
2.1338
0.95
5.4066
4.4597
5.1902
3.6077
3.8759
5.2096
3.2353
3.0967
3.1201
0.99
26.1116
11.6543
22.5199
3.0808
7.3351
22.7494
2.3123
1.6738
2.0371
10
0.8
6.2173
5.9041
6.1472
5.5996
5.6715
6.1540
5.4315
5.3740
5.3768
0.9
11.3093
10.2365
11.0674
9.2202
9.4898
11.0902
8.7008
8.5182
8.5349
0.95
21.6263
17.8388
20.7609
14.4306
15.5034
20.8384
12.9409
12.3866
12.4801
0.99
104.4463
46.6173
90.0797
12.3231
29.3403
90.9977
9.2487
6.6946
8.1480
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
634
Volume 18, 2023
Table 13. The results of regression coefficients and the corresponding MSE values.
ˆ
RE
ˆ
LE
ˆ
KL
ˆ
NTP
ˆ
MRT
ˆ
NTPKL
ˆ
1
ˆMTPKL
2
ˆMTPKL
0
ˆ
-52.9936
-49.3576
-52.9936
-45.7216
41.6554
41.6554
-24.8522
-49.3576
-52.9936
0.0711
0.0703
0.0711
0.0696
47.4077
47.40767
47.42099
0.070317
0.071073
2
ˆ
-0.4142
-0.4079
-0.4142
-0.4015
13.7557
13.75574
13.64773
-0.40785
-0.4142
3
ˆ
-0.4235
-0.4323
-0.4235
-0.4411
0.8644
0.864403
1.018985
-0.43227
-0.42347
4
ˆ
-0.5726
-0.5746
-0.5726
-0.5767
0.3746
0.374561
0.415282
-0.57463
-0.57257
5
ˆ
48.4179
48.0144
48.4179
47.6109
-4.7292
-4.72919
2.27232
48.01439
48.41786
MSE
17095.15
15027.25
17095.14
13122.26
17095.14
13316.98
11510.3
2915.66
5150.228
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.63
Idowu J. I., Owolabi A. T., Oladapo O. J.,
Ayinde K., Oshuoporu O. A., Alao A. N.
E-ISSN: 2224-2856
635
Volume 18, 2023