Performance of Some Dawoud-Kibra Estimators for Logistic Regression

Model: Application to Pena data set

OLASUNKANMI JAMES OLADAPO1,*, OLUSEGUN O.ALABI2, KAYODE AYINDE2,3

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

1*Corresponding author

Abstract:A logistic regression model's parameters are usually estimated using the maximum likelihood (ML)

method. As a consequence of the problem of multicollinearity, unstable parameter estimates are obtained, and the

mean square error (MSE) obtained cannot also be relied upon. There have been several biased estimators proposed

to deal with multicollinearity, and the logistic Dawoud-Kibra (LDK) estimator is one of them, and research has

shown that biasing parameters have an effect on MSE, too. Our study proposed seven LDK biasing estimators and

all of them were subjected to Monte Carlo simulations, as well as using Pena data sets. According to the simulation

study, LDK estimators outperform Logistic Ridge Regression (LRR) and ML methods. Furthermore, application to

Pena real data set also align with the simulation results.

Key-words: Logistic regression, Multicollinearity, Biased estimators, Maximum likelihood, Simulation, MSE.

Received: January 5, 2023. Revised: October 15, 2023. Accepted: November 17, 2023. Published: December 31, 2023.

1 Introduction

It was Frisch [1] who first introduced the concept of

multicollinearity in multiple regression models. It is a

common occurrence in applied research. For linear

regression models and logit regression models, the use

of ordinary least squares (OLS) and maximum

likelihood (ML) leads to high variance and unstable

parameter estimates. Due to multicollinearity,

regression analysis's conclusions may be questioned.

As a correction measure for linear regression, ridge

regression (RR) has become increasingly popular in

recent decades [2]. Many studies have been conducted

to estimate the ridge parameter k, originally proposed

by Hoerl and Kennard [2, 3]. Such studies, performed

on ridge regression, include the works of Gibbons [4],

Lawless and Wang [5], and Dempster et al. [6] Hoerl

and Kennard; [2] Hoerl et al.; [7] McDonald and

Galarneau [8] Alkhamisi et al. [9] Alkhamisi and

Shukur [10] Muniz and Kibria [11] Muniz et al. [12]

and Månsson et al. [13] Lukman and Ayinde [14]

Ayinde et al. [15] and others too. However, logit

models have not received much attention, and only a

few researchers have tried to work on them. Those

who have studied them are the likes of Schaeffer et al.

[16, 17], Månsson and Shukur [18], Kibria et al. [19],

and a few others. Almost all the researchers working

on the logit models focused only on the RR and paid

little or no attention to the biasing parameter of all

other estimators proposed by other researchers to

handle the problem of multicollinearity in the logit

models.

This research paper will focus on proposing some

Logistic Dawoud and Kibra (LDK) estimators,

following the works of Kibra et al. [19, 20]. Research

has shown that the biasing parameters of an estimator

have an effect on the value of the mean square error

(MSE). Therefore, its anticipated LDK estimators will

have MSE values lower than those of the LRR and

ML. The MSE has been one of the criteria used in

judging the performances of estimators.

This research paper is structured as follows: Section 2

entails the materials and methodology adopted for the

study. Section 3 includes the results and discussion of

simulation and numerical results. Section 4 provides a

succinct overview and conclusions.

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Olasunkanmi James Oladapo,

Olusegun O. Alabi, Kayode Ayinde

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2. Materials and Methodology.

Adopting the concepts in the research works of Kibra

et al. and Kibra [19, 20], we will be proposing some

LDK estimators in this section for estimating the

biasing parameter k.

2.1 Logit Regression

The logit regression has always been one of the most

common statistical methods used whenever the i-th

value of the dependent variable (y) follows a Be (πi )

distribution with the following parameter value:

 





'

exp1

exp

i

ix

x





(1)

Where

'

i

x

is the ith row of X, and is a

)1(  pn

data

matrix, having p explanatory variables and such that



is a

 

11 p

vector of coefficients. Using the

Maximum Likelihood technique, which maximizes

the following log likelihood, is one of the most used

ways to estimate



and can be expressed as:

L=

     

i

n

i

ii

n

i

iyy



  

1log1log

11

(2)

To get this we set the first derivative of equation (2) to

be equal to zero, the ML estimates can now be derived

by solving the equation below

 

0

1









i

n

i

ii xy

l





(3)

The iterative weighted least square (IWLS) algorithm

is used:

 

zWXXWX

MLE ˆ

'

ˆ

'

ˆ1





(4)

Where the following are the expression of

W

ˆ

and

z

ˆ

respectively

 

ii

W



 1

ˆ

and

z

ˆ

is known to be a vector where

the ith element equals

   

ii

y

z





ˆ

1

ˆ

log 





Since equation (3) is nonlinear in



, we can express

the MSE of the ML estimator as:

 

   

 

1

2

1

ˆj

ML ML ML

ij

E L E E tr X WX

    







     

(5)

j



is said to be the jth eigenvalues of the

XWX ˆ



matrix.

The LRR estimator, proposed by Schaeffer et al. [16],

is a substitute for ML estimates that mitigates

multicollinearity problems. Instead of estimating the

regression model coefficients directly, it estimates the

inverse of the covariance matrix. In this way, the LRR

estimator effectively reduces small eigenvalues

caused by multicollinearity; therefore, the regression

coefficients are more reliable and robust.

The LRR estimator is express as:

 

MLELRR XWXkIXWX



ˆ

'

ˆ

'

ˆ1



(6)

With k has the biasing parameter,

W

ˆ

and

MLE



ˆ

is the

MLE



ˆ

estimates derived from equation (4). The LRR

estimator MSE is shown to be:

 

   

2

22

11

LRR LRR LRR

jj

E L E E

k

kk

   









  







(7)

The Logistic Dawoud and Kibra (LDK) estimator,

which is a special two parameter estimator of Kibra-

Lukman (KL) estimator that was proposed by Afzal et

al [21] and it also handle the problem of

multicollinearity effectively too. The estimator LDK

is defined as:

   

MLELDK IdkXWXIdkXWX



ˆ

)1(

ˆ

')1(

ˆ

'

ˆ1 

(8)

With k and d, the biasing parameters,

W

ˆ

and

MLE



ˆ

is

the

MLE



ˆ

estimates derived from equation (4).

The MSE of the LDK estimator is express to be:

 

  

       

 

  







p

i

p

ii

i

ii

i

LDK dk

dk

MSE

1 1

2

1

14

1

ˆ











(9)

where

2

j



is expressed as the jth element of



and



is known to be the eigenvector expressed as









XWX ˆ

, where

 

j

diag





.

2.2 The Dawoud-Kibra Estimators.

There are numerous approaches that have been

developed for the linear regression model and then

transferred to the logistic ridge regression model for

selecting a ridge parameter. A biasing parameter k

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Olasunkanmi James Oladapo,

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from Hoerl and Kennard's [2, 3] research in the

classical RR is represented as follows:

2

max

2

1

ˆ







HK

k

(10)

As shown below, the aforementioned biasing

parameter was also used to obtain the biasing

parameter that Kibria [19] suggested.

l

i

j

GM

k1

1

2

ˆ























(11)

Later on, equation (10) was adopted into the LRR by

Schaeffer et al. [16] as:

2

max

1

ˆ





SRW

k

(12)

However the biasing parameters k and d for LDK can

be gotten from the MSE:

 

  

       

 

  







p

i

p

ii

i

ii

i

LDK dk

dk

MSE

1 1

2

1

14

1

ˆ











Then, by differentiating

 

LDK

MSE



ˆ

w.r.t. d and

equating to 0, we have

However, d depends on

the unknown . For

practical purposes, it will be replaced by its unbiased

estimator . Hence, this will be expressed as:

 

1

21

ˆ

12



















p

iii

i

k

d





(13)

Then, by differentiating

 

LDK

MSE



ˆ

w.r.t. k and

equating to 0, we have

 















2

1

i

d

k





(14)

However, k depends on the unknown . For practical

purposes, it will be replaced by its unbiased estimator

. Hence, this will be expressed as:

 















2

ˆ

2

1

ˆ

i

d

k





(15)

Following the works of Schaeffer et al. [16] and Kibra

et al and Kibra [19, 20] the following biasing

parameter k for LDK are proposed as:

 















2

1ˆ

2

1

11

ˆ

i

p

i

AM

d

p

k





(16)

 



















p

i

HM

d

pk

12

ˆ

2

1

ˆ





(17)

 



























2

ˆ

2

1

ˆ

i

MAX

d

Maximumk





(18)

 



























2

ˆ

2

1

ˆ

i

MIN

d

Minimumk





(19)

 



























2

ˆ

2

1

ˆ

i

MED

d

Mediank





(20)

 

2

ˆˆ

ˆMINMAX

MR

kk

k



(21)

2.3 The Monte Carlo Simulation

As the main objective of this paper is to ascertain the

effects of multicollinearity on ML, LRR, and LDK

Estimators, the degree of correlation between the

regressors is the most significant variable in the

experiment. Accordingly, we generate the explanatory

variables using the following formula, which allows

us to adjust the correlation's strength:

 

ipijij zzx



 2/1

2

1

i=1, 2,…,

, j=1,2,…,p (22)

The term

2



describes the level of correlation

between the explanatory factors and

ij

z

is the usual

normal distribution's pseudorandom numbers as well.

The four different levels of correlation that are being

evaluated are 0.8, 0.9, 0.95, and 0.99 respectively.

i



ˆi



i



ˆi



 

1

21

1

2



















p

iii

i

k

d





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DOI: 10.37394/232021.2023.3.16

Olasunkanmi James Oladapo,

Olusegun O. Alabi, Kayode Ayinde

E-ISSN: 2732-9976

132

Volume 3, 2023

Similarly, the dependent variable comes from

 

i

Be



distribution where

 





'

exp1

exp

i

ix

x





(23)

We set

1





, then the MSE is minimized when this

coefficient is chosen, in accordance with Newhouse

and Oman's [22] assertion that if our MSE is a function

of



,

2



and k if all the explanatory variables utilized

are fixed. Sample sizes used are 50,

60,70,80,90,100,150,200 and 250, likewise the

number of explanatory variable considered are p=3,

4,5 and 6 .We will be able to determine which of the

Dawoud-Kibra biasing k parameters will be more

efficient using this experimental design.

We may find additional details on simulation

processes from the works of Kibria [20] Lukman et al

[23]; Oladapo et al [24,25]; Muniz and Kibria [11];

Idowu et al [26]; Owolabi et al [27]; Månsson and

Shukur [18] and others too.

3. Results and Discussion

3.1 Simulation Results

Results from both real-world data and Monte Carlo

simulations are presented in this section. Tables 1

through 4 display the MSE values of all the estimators

used in the Monte Carlo study, and Table 5 displays

the MSE values from a real-life data set. Additionally,

the effects of varying the many factors we employed

in this investigation on the ML, LRR, and LDK

estimators are also covered.

Table 1 shows the estimated MSE values when n = 50,

60, and 70 for explanatory variables, p = 3, 4, and 6. It

can be observed that the LDK with the biasing

parameter k of the Median (MED) version gives the

lowest MSE in all cases, with few exceptions.

Table 1: Estimated MSE for different estimator when p=3, 4, 5 and 6 when n=50, 60 and 70

n



P

MLE

RIDGE

DWD

DWDME

D

DWDA

M

DWDH

M

DWDMA

X

DWDMI

N

DWDM

R

50

0.8

P3

3.0134

1.1814

1.7391

1.0676

1.0149

5.3236

1.2515

1.6028

1.1761

P4

7.9549

2.2022

3.1145

1.8250

2.1445

11.5557

2.9825

3.5858

2.7601

P5

9.4676

2.4363

3.3850

1.7441

2.2337

15.0712

3.6191

4.1488

3.2496

0.9

P3

5.6308

1.8903

2.5500

1.4500

1.5099

5.9809

1.7870

2.5973

1.8724

P4

14.8361

3.7137

4.9707

2.7193

3.4177

16.0040

4.6564

6.0935

4.6380

P5

18.8392

4.4338

5.7219

3.3098

4.2719

22.0825

6.1699

7.7862

5.8966

0.95

P3

11.2610

3.4218

4.1419

2.6086

2.6929

8.4299

3.0232

4.7187

3.3710

P4

30.0187

7.1853

9.0706

4.7665

6.5271

27.4225

8.9713

12.0734

8.8419

P5

47.2517

9.3427

10.6448

6.1189

8.7187

46.7525

17.6080

19.4230

11.9529

0.99

P3

59.2260

16.5227

16.8080

14.1993

12.2145

34.6057

12.6600

23.1891

14.1815

P4

158.611

36.4383

41.0370

20.6063

34.2339

130.699

45.2742

64.5093

38.5315

P5

205.860

44.972

202.364

203.297

204.133

205.666

202.475

186.491

202.413

60

0.8

P3

2.4854

1.0930

1.4374

0.8925

0.7803

4.6463

0.9476

1.3806

0.9419

P4

5.2939

1.6122

2.3796

1.1421

1.3809

8.7297

1.9256

2.5126

1.9483

P5

5.8827

1.6936

2.4839

1.2193

1.3175

11.4159

2.1385

2.7495

1.9077

0.9

P3

4.9228

1.8428

2.3971

1.3138

1.3603

5.0615

1.5324

2.4047

1.7521

P4

10.4370

2.7783

3.8214

2.0090

2.6639

11.8563

3.1496

4.5404

3.7450

P5

13.4614

3.3901

4.6813

2.4497

2.8022

16.9669

4.0399

5.9529

4.2871

0.95

P3

9.7720

3.2493

3.9695

2.3307

2.5091

6.8777

2.5684

4.2936

3.3139

P4

21.5940

5.5589

6.9320

3.8352

5.9078

19.9931

6.0877

9.4089

7.6073

P5

28.7990

6.8667

8.9802

5.2633

6.9909

29.4111

8.4786

12.4831

9.8113

0.99

P3

51.1616

14.8469

16.0517

13.1154

13.5494

29.1057

10.4073

20.4184

14.7440

P4

112.583

27.4557

31.4111

18.2555

32.4423

93.4984

30.8212

49.4065

34.3072

P5

139.995

30.1438

37.5924

22.3143

36.3197

125.311

35.8197

59.6002

39.1094

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DOI: 10.37394/232021.2023.3.16

Olasunkanmi James Oladapo,

Olusegun O. Alabi, Kayode Ayinde

E-ISSN: 2732-9976

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Volume 3, 2023

70

0.8

P3

2.2642

0.9261

1.4152

0.8510

0.8589

4.4711

0.9870

1.2523

1.0312

P4

3.0729

1.1828

1.7313

0.7969

0.8413

7.0600

1.2098

1.6311

1.0258

P5

5.0440

1.6044

2.3690

1.1760

1.2152

10.1655

1.9391

2.5519

1.7429

P6

7.2446

1.9445

3.0431

1.2420

1.5566

13.5424

2.7503

3.4616

2.4517

0.9

P3

4.1361

1.3787

1.9861

1.0633

1.3002

4.8238

1.4420

1.9447

1.6783

P4

6.2659

2.0261

2.8236

1.3620

1.4329

8.2506

1.8382

2.9689

1.9375

P5

10.2050

2.7862

3.9426

2.2180

2.3544

13.5358

3.2415

4.7539

3.6131

P6

14.6311

3.4994

5.0923

2.1444

2.8953

19.0866

4.2545

6.6134

4.5341

0.95

P3

8.6273

2.5826

3.3307

1.9202

2.3896

6.8469

2.5479

3.6586

3.0848

P4

13.2264

3.8898

5.0430

2.6938

2.8508

12.4850

3.4892

5.8005

3.9922

P5

21.4089

5.3822

7.1031

4.2611

5.3253

22.2454

6.3741

9.8633

7.4715

P6

38.3056

7.7671

9.7615

5.0326

10.6950

40.2202

14.6099

13.9551

14.7786

0.99

P3

45.6046

11.9755

13.1080

10.5775

12.6906

28.5947

10.2081

17.2596

13.8085

P4

71.2235

18.9457

22.1628

12.6500

15.1726

55.6095

16.6507

30.9732

17.9173

P5

116.284

26.8902

32.2734

20.5964

37.8054

105.429

32.5831

54.4174

38.0870

P6

167.394

36.2466

46.0637

24.4990

42.8970

156.495

50.1987

80.0121

48.6794

Bold values show the smallest MSE

Table 2 shows the estimated MSE values when n is 80

and 90 for explanatory variables, p = 3, 4, 5 and 6. It

can be observed that the LDK with the biasing

parameter k of the Median (MED) version gives the

lowest MSE values in almost all the designs used. In

the only six cases where the biasing parameter k of the

MED version is not the lowest MSE, the Arithmetic

Mean (AM) or Maximum (MAX) version takes the

lowest.

Table 2: Estimated MSE for different estimator when p=3, 4, 5 and 6 when n= 80 and 90

n



P

MLE

RIDGE

DWD

DWDME

D

DWDA

M

DWDH

M

DWDMA

X

DWDMI

N

DWDM

R

80

0.8

P3

1.67084

0.76002

1.17827

0.68708

0.65797

3.92575

0.84528

0.98687

0.77798

P4

2.37561

0.94544

1.42684

0.61893

0.64444

6.42256

1.14272

1.30217

0.86454

P5

4.56249

1.48556

2.30453

1.13672

1.0484

9.54568

1.80814

2.3741

1.5589

P6

6.10864

1.7288

2.69452

1.23503

1.29881

12.0151

2.45743

2.9857

1.97318

0.9

P3

3.28215

1.25322

1.8403

0.90998

1.0815

4.1846

1.37606

1.70816

1.43604

P4

4.92192

1.6199

2.32145

1.01588

1.1329

7.20872

1.61942

2.39717

1.63462

P5

9.5979

2.78954

4.0644

2.12481

2.02427

12.8622

3.18842

4.77824

3.14244

P6

12.8205

3.17826

4.71354

2.28173

2.78894

17.0417

4.27894

6.1909

4.19413

0.95

P3

6.70392

2.24802

3.00094

1.49985

1.93836

5.57572

2.33735

3.13699

2.69762

P4

10.4078

3.01482

4.02146

2.03269

2.33279

10.3204

2.74061

4.65749

3.30158

P5

19.7217

5.24449

7.30642

4.02324

4.42273

20.4511

5.88366

9.55326

6.37405

P6

25.501

5.78902

8.27669

4.30208

6.32236

27.6567

8.2777

12.1418

9.05972

0.99

P3

34.9579

10.2086

11.3736

8.98314

9.06533

21.5403

8.74356

14.1798

11.4622

P4

59.8215

15.456

18.3358

9.92249

13.7838

46.8038

12.8531

25.2849

16.3161

P5

107.684

26.3674

33.759

19.8468

27.6923

96.1537

31.3508

52.3113

32.5601

EQUATIONS

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Olusegun O. Alabi, Kayode Ayinde

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Volume 3, 2023

P6

142.28

30.1585

39.2711

24.6125

43.358

133.81

46.9836

70.5871

45.2701

90

0.8

P3

1.49818

0.70953

1.09467

0.63742

0.61847

3.85657

0.79869

0.92377

0.71497

P4

2.26256

0.88341

1.40847

0.64595

0.69797

5.90918

1.01898

1.23439

0.84665

P5

3.47151

1.19339

1.88429

0.91177

0.84869

8.56464

1.43314

1.8101

1.25234

P6

4.93237

1.46581

2.30555

1.03609

1.1637

10.9625

1.99174

2.3975

1.64359

0.9

P3

2.95288

1.14763

1.70848

0.84454

0.94364

4.01652

1.19596

1.53553

1.22847

P4

4.68027

1.48584

2.23311

1.05472

1.21078

6.76912

1.63985

2.18753

1.61355

P5

7.20916

2.08718

3.16389

1.64908

1.73209

10.5566

2.25829

3.4854

2.50726

P6

10.3385

2.66288

3.98108

1.77071

2.0069

14.5726

3.31374

4.8147

3.23471

0.95

P3

5.93709

1.97881

2.66166

1.34554

1.61218

5.1497

1.95366

2.71272

2.23391

P4

9.93969

2.80057

3.79945

1.96502

2.34297

9.96151

2.87208

4.31996

3.23358

P5

15.2227

3.93143

5.61191

3.35878

3.75497

16.3295

4.21628

7.03583

5.11609

P6

22.2637

5.35722

7.58602

3.70923

4.75993

24.255

6.58373

10.2753

7.04778

0.99

P3

31.2156

9.06655

10.0829

7.87813

7.87485

18.6596

7.00095

12.5249

9.53534

P4

56.133

14.7771

16.8065

10.3468

13.4994

44.3914

14.1579

24.9554

15.3007

P5

86.3689

20.8267

26.1321

16.6688

24.3271

76.622

22.8095

41.2024

26.5591

P6

127.411

28.0873

36.6302

20.5124

33.104

119.176

40.3713

61.642

39.953

Bold values show the smallest MSE

Table 3 shows the estimated MSE values when n is

100 and 150 for explanatory variables, p = 3, 4, 5, and

6. It can be observed that the LDK with the biasing

parameter k of the Median (MED) version gives the

lowest MSE values in almost all the designs used. In

the only seven cases where the biasing parameter k of

the MED version does not have the lowest MSE, the

Arithmetic Mean (AM) or Maximum (MAX) version

takes the lowest MSE values.

Table 3: Estimated MSE for different estimator when p=3, 4, 5 and 6 when n=100 and 150

n



P

MLE

RIDGE

DWD

DWDME

D

DWDA

M

DWDH

M

DWDMA

X

DWDMI

N

DWDM

R

100

0.8

P3

1.10667

0.57641

0.92229

0.59703

0.51189

3.47516

0.67127

0.7264

0.54209

P4

2.04769

0.85039

1.36836

0.56686

0.65772

5.63264

0.99689

1.14864

0.84464

P5

3.18344

1.14312

1.82142

0.7652

0.73325

7.77339

1.24737

1.69627

1.16373

P6

3.38548

1.20059

1.93054

0.71406

0.68376

9.40583

1.46008

1.81556

1.05236

0.9

P3

2.13936

0.8779

1.39011

0.71609

0.71637

3.44448

0.91839

1.19915

0.89358

P4

4.04429

1.38248

2.09127

0.86116

1.00724

6.10775

1.46213

2.01459

1.4555

P5

6.42167

1.93073

2.97477

1.30502

1.36996

9.382

1.99169

3.15922

2.34011

P6

6.7475

1.99696

3.19688

1.17718

1.28795

11.2886

2.21201

3.45088

2.06967

0.95

P3

4.5131

1.58155

2.19985

1.13836

1.27518

4.14274

1.52954

2.17144

1.6557

P4

8.27104

2.51889

3.53271

1.73128

1.97484

8.40246

2.38231

3.86988

2.80587

P5

13.8134

3.77986

5.41429

2.86362

3.03591

14.515

3.59937

6.50331

4.75187

P6

14.1998

3.73794

5.65766

2.4425

2.83459

16.5814

4.01551

6.95609

4.42935

EQUATIONS

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Olusegun O. Alabi, Kayode Ayinde

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Volume 3, 2023

0.99

P3

23.4325

6.65367

7.43936

5.76364

5.00067

13.8058

5.23017

9.05736

6.18347

P4

43.6483

11.4788

14.0823

8.17882

10.6092

34.006

9.82906

19.3253

12.4966

P5

75.7412

19.0236

24.6698

14.576

19.3033

66.097

18.3414

35.6377

23.7271

P6

83.9213

20.5012

28.3234

14.2543

20.0539

78.0145

23.5789

42.9646

26.0933

150

0.8

P3

0.82151

0.47392

0.7799

0.48763

0.42367

2.74849

0.53576

0.59676

0.45538

P4

1.15912

0.59309

0.97813

0.40149

0.43629

4.52427

0.6575

0.75689

0.54479

P5

1.58944

0.66214

1.20284

0.59098

0.55942

6.25743

0.91058

0.91821

0.68144

P6

2.12487

0.79073

1.50726

0.6085

0.63646

7.61557

1.1329

1.16051

0.86887

0.9

P3

1.53221

0.68931

1.17824

0.61379

0.60818

2.75719

0.75969

0.92177

0.70542

P4

2.34108

0.94283

1.56592

0.58713

0.70447

4.55324

0.9752

1.30587

0.98549

P5

3.39555

1.1314

1.944

0.91522

0.92921

6.68573

1.3572

1.72528

1.28829

P6

4.5645

1.38477

2.46174

1.03214

1.20082

8.79154

1.83033

2.27762

1.76696

0.95

P3

3.04433

1.11379

1.70868

0.85217

0.96199

3.15954

1.13891

1.51311

1.21765

P4

4.72117

1.56457

2.49307

1.02429

1.23206

5.55093

1.6333

2.35545

1.78623

P5

7.15526

2.07029

3.26311

1.75067

1.80586

8.86719

2.3168

3.41488

2.53422

P6

9.47384

2.53167

4.03438

1.94517

2.32271

11.9861

3.07958

4.6558

3.38184

0.99

P3

16.6319

4.83053

5.59182

4.2063

3.57767

9.91765

3.8345

6.50948

4.31519

P4

26.9865

7.52902

9.9394

5.14879

6.64162

21.3561

6.87033

12.0961

8.34484

P5

41.1357

10.3477

13.7712

8.29434

10.3638

36.7184

11.9716

19.8814

12.6076

P6

51.6935

12.2463

17.4252

9.66809

14.0596

48.3635

15.0132

26.1452

17.3009

Bold values show the smallest MSE

Table 4 shows the estimated MSE values when n is

200 and 250 for explanatory variables, p = 3, 4, 5, and

6. it can be observed that as there is increase in sample

sizes with a low multicollinearity strength of 0.8, the

LDK with the biasing parameter k of the Arithmetic

mean (AM) has the minimum MSE values compared

with the rest of the estimators proposed and compared

with. Also, at other multicollinearity levels, the

median (MED) version gives the lowest MSE values

in almost all the designs used. Except in five cases, the

biasing parameter k of the arithmetic mean (AM) has

the lowest MSE values again.

Table 4: Estimated MSE for different estimator when p=3, 4, 5 and 6 when n=200 and 250

n



P

MLE

RIDGE

DWD

DWDMED

DWDAM

DWDHM

DWDMAX

DWDMIN

DWDMR

200

0.8

P3

0.5562

0.3562

0.5861

0.3475

0.3104

2.2064

0.4326

0.4236

0.3491

P4

0.7832

0.4369

0.7596

0.3465

0.3402

3.9929

0.5193

0.5475

0.4012

P5

1.2874

0.6164

1.076

0.4883

0.4447

5.6277

0.7891

0.8014

0.6237

P6

1.7794

0.7238

1.332

0.5272

0.5634

6.8464

1.0012

0.9999

0.799

0.9

P3

1.0757

0.5552

0.9481

0.4746

0.4417

2.1763

0.608

0.7094

0.5339

P4

1.5963

0.7012

1.2151

0.456

0.51

3.8229

0.7905

0.9297

0.6893

P5

2.6508

1.0189

1.7239

0.7305

0.7049

5.7017

1.1385

1.4642

1.0186

P6

3.7422

1.2098

2.1293

0.8633

0.9656

7.5765

1.5594

1.8758

1.4429

0.95

P3

2.2299

0.9329

1.4868

0.6802

0.7709

2.3972

0.9502

1.242

1.0196

EQUATIONS

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Olusegun O. Alabi, Kayode Ayinde

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P4

3.2122

1.1137

1.8735

0.7343

0.9427

4.4117

1.3039

1.5946

1.3526

P5

5.4369

1.7511

2.7792

1.3962

1.3156

7.0486

1.7782

2.7187

1.8951

P6

7.6774

2.1229

3.4816

1.5849

1.991

10.013

2.6419

3.7625

2.9002

0.99

P3

11.981

3.6901

4.5526

2.8014

3.1929

7.3608

3.2207

5.0733

3.9902

P4

17.616

4.7227

6.4307

3.2996

4.5012

14.479

4.4991

7.8549

5.6858

P5

28.752

7.7927

10.618

6.461

6.6636

25.17

7.0923

13.952

8.2015

P6

42.839

10.455

14.348

8.5635

11.77

39.935

12.521

21.937

14.331

250

0.8

P3

0.3876

0.2656

0.4527

0.2951

0.2438

1.823

0.337

0.3099

0.2717

P4

0.6778

0.387

0.7016

0.334

0.3228

3.4348

0.4507

0.4739

0.3794

P5

0.9369

0.4849

0.8739

0.4232

0.3702

4.8709

0.5763

0.6142

0.4546

P6

1.2133

0.5914

1.0345

0.4018

0.3943

6.4796

0.7983

0.752

0.5615

0.9

P3

0.7641

0.417

0.7802

0.4361

0.3629

1.9291

0.508

0.5275

0.4376

P4

1.3725

0.6266

1.126

0.4675

0.5291

3.4313

0.7332

0.8176

0.6747

P5

1.9906

0.8217

1.4847

0.6062

0.576

4.8904

0.9479

1.1346

0.8067

P6

2.4794

0.9498

1.6524

0.602

0.653

6.6238

1.146

1.3607

1.0132

0.95

P3

1.548

0.6611

1.1944

0.562

0.5761

2.0985

0.7935

0.8633

0.7454

P4

2.771

1.0205

1.7289

0.7352

0.8293

3.8792

1.1399

1.4375

1.1565

P5

3.9976

1.3629

2.3283

1.0193

1.0252

5.702

1.5567

2.0644

1.6168

P6

5.091

1.6225

2.6839

1.0383

1.1809

7.8837

1.8657

2.6139

1.9725

0.99

P3

8.5376

2.5856

3.4005

2.012

2.2543

5.5918

2.3966

3.5737

2.8497

P4

14

3.8519

5.3812

2.8961

3.9074

11.503

3.8124

6.2268

4.9171

P5

21.286

5.7531

8.2829

4.3694

5.525

19.132

6.1618

10.417

7.7491

P6

28.087

7.323

10.763

5.4302

7.6465

26.466

7.9873

14.237

10.057

Bold values show the smallest MSE

3.2 Numerical example

Pena et al. [28] examined the impact of temperature,

pH, and soluble solids concentration on the chance of

Alicyclobacillus development in apple juice using a

logistic model. The eigenvalues of the matrix are

13464.7990, 1715.9257, 56.5515, and 3.5445.

Consequently, multicollinearity is present in the

model, as shown by the condition index (C.I.) of

61.6342.

When there is multicollinearity, the ML estimator

performs the least well, as expected. The choice of the

biasing parameters k and d determines the efficiency

of biased estimators. All of the proposed estimators

performed admirably, and one of them has the

minimum mean square error, which corresponds to the

simulation outcome.

Table5: Regression coefficients and MSE

0

ˆ



1

ˆ



2

ˆ



3

ˆ



4

ˆ



SMSE

ML



ˆ

-7.24633

1.885951

-0.06628

0.110422

-0.31173

21.35138842

LRR



ˆ

-2.4E-06

0.008038

-0.02442

0.015783

-0.01186

0.28340673

LDK



ˆ

7.244206

-1.74152

0.005895

-0.042

0.160006

21.57368811

EQUATIONS

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Olusegun O. Alabi, Kayode Ayinde

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Volume 3, 2023

LDKMED



ˆ

3.801875

0.438059

-0.03186

0.006407

-0.39032

5.782491876

LDKMN



ˆ

-4.75037

1.571575

-0.05868

0.086835

-0.33312

9.513392712

LDKMX



ˆ

7.244195

-1.74077

0.005875

-0.04191

0.159524

21.57256675

LDKMR



ˆ

7.240503

-1.57812

0.002386

-0.03006

0.076265

21.35343462

LDKHM



ˆ

7.245876

-1.85107

0.01198

-0.06703

0.259096

21.89852536

LDKAM



ˆ

7.222206 -1.28265 -0.00251 -0.02318 -0.03271 20.95427864

4. Conclusion

Based on the work of Kibra et al. [20], where real-

world data and Monte Carlo simulation studies were

utilized to examine the estimator performance, we

were able to suggest a few LDK estimators in this

paper for estimating the biasing parameter k. The

estimators’ performances were assessed using the

Mean Squared Error (MSE) criteria. In the simulation

study, it was observed that nearly all sample sizes that

were taken into account and that the biasing parameter

k with the Arithmetic mean version exhibits the lowest

MSE values when the strength of multicollinearity is

at 0.8. Additionally, at the remaining design used in

this paper, the biasing parameter k with the Median

(MED) version has the least, with the exception in

some cases. In addition, from the numerical example,

the biasing parameter k with the Median (MED) and

the LRR have the two lowest MSEs, respectively.

Hence, based on our findings, both in simulation and

numerical examples, we thereby recommend to

practitioners, researchers, and scientists that when

faced with multicollinearity issues in using the logistic

model, they should use the LDK estimator with the

biasing k of the Arithmetic version when the

multicollinearity level is not severe, but in severe

multicollinearity cases, they should go with the

Median (MED) version.

References

[1] Frisch, R. Statistical Confluence Analysis by

Means of Complete Regression Systems;

University Institute of Economics: Oslo, Norway,

1934.

[2] Hoerl, A.E. & Kennard, R.W. Ridge regression:

biased estimation for nonorthogonal problems,

Technometrics, 12(1) 1970: 55–67.

[3] Hoerl, A. E., & Kennard, R. W. Ridge

regression: Application to non-orthogonal

problems. Technometrics, 12(1) 1970:69–82.

[4] Gibbons, D. G. A simulation study of some ridge

estimators. Journal of the American Statistical

Association, 76 ,1981:131–139.

[5] Lawless, J. F., & Wang, P. A simulation study of

ridge and other regression estimators.

Communications in Statistics: Theory and

Methods, 5, 1976:307–323

[6] Dempster, A. P., Schatzoff, M., & Wermuth, N.

A simulation study of alternatives to ordinary

least squares. Journal of the American Statistical

Association, 72,1977: 77–91.

[7] Hoerl, A.E., Kennard, R.W., & Baldwin, K.F.

Ridge regression: Some simulation.

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