Research Article | | Peer-Reviewed

On the Performance of Some Estimation Methods in Models with Heteroscedasticity and Autocorrelated Disturbances (A Monte-Carlo Approach)

Received: 3 December 2023     Accepted: 25 December 2023     Published: 2 April 2024
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Abstract

The proliferation of panel data studies has been greatly motivated by the availability of data and capacity for modelling the complexity of human behaviour than a single cross-section or time series data and these led to the rise of challenging methodologies for estimating the data set. It is pertinent that, in practice, panel data are bound to exhibit autocorrelation or heteroscedasticity or both. In view of the fact that the presence of heteroscedasticity and autocorrelated errors in panel data models biases the standard errors and leads to less efficient results. This study deemed it fit to search for estimator that can handle the presence of these twin problems when they co- exists in panel data. Therefore, robust inference in the presence of these problems needs to be simultaneously addressed. The Monte-Carlo simulation method was designed to investigate the finite sample properties of five estimation methods: Between Estimator (BE), Feasible Generalized Least Square (FGLS), Maximum Estimator (ME) and Modified Maximum Estimator (MME), including a new Proposed Estimator (PE) in the simulated data infected with heteroscedasticity and autocorrelated errors. The results of the root mean square error and absolute bias criteria, revealed that Proposed Estimator in the presence of these problems is asymptotically more efficient and consistent than other estimators in the class of the estimators in the study. This is experienced in all combinatorial level of autocorrelated errors in remainder error and fixed heteroscedastic individual effects. For this reason, PE has better performance among other estimators.

Published in Mathematical Modelling and Applications (Volume 9, Issue 1)
DOI 10.11648/j.mma.20240901.13
Page(s) 23-31
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Modified, Method, Panel, Estimator, Simulations

1. Introduction
Panel data model have become increasingly popular in the past decades with the increased availability of cross country data sets. , were the first set of scholars to work on panel data. Panel data describes the number of individuals across a sequence of time periods. There are several key advantages of using panel data over a single time series or cross-section data set. This combination of time series with cross-section can enhance the quality and quantity of data in ways that would be possible using only one of these two dimensions. To realize the potential value of the information contained in a panel data, see .
Panel data typically contains some form of heteroscedasticity, serial correlation and /or spatial correlation. Therefore, robust inference in the presence of heteroscedasticity and serial correlation is an important problem in spatial data analysis, .
The essence of these two problems is that, in many econometrics studies, including panel study, the assumption of constant variance for the disturbance term is unrealistic. This, then calls for a restrictive assumptions for panel, where, in practice, the cross-sectional units may be of varying size and as a result exhibit different variation.
It was observed that some scholars, consider cases where one accounts for heteroscedasticity and ignores the possibility of serial correlation problems in the model and vice-versa . Few authors that consider both problems include. Of these authors, Baltagi et al, as well as both derived test for joint occurrence of heteroscedasticity and autocorrelation in one way and two-way error components respectively . The difference between this work and that of is that it focuses on the development of a new estimator and estimation of the parameters not testing for the presence of autocorrelation and heteroscedasticity in the context of a panel data regression model.
Furthermore, this study explicitly decomposes error term as the sum of two elements; capturing individual heterogeneity and autocorrelation in the remainder error term. By so doing, the data is then generated from the error component model with serial correlation and heteroscedasticity with the help of Monte-Carlo Simulation method.
The study was mainly targeted at investigating finite sample properties of five estimation methods of panel data models in the presence of autocorrelation and heteroscedasticity in one-way error component model. The estimation methods are BE, FGLS, M, MM and PE.
The inclusion of M and MM estimators in this study is due to their performance in the various work .
2. Methodology

2.1. Model Specification

Consider the following panel data model:
yit=Xit1β+uit,    i=1, , N;   t=1, ,T.(1)
where: β is K × 1 vector of regression coefficient.
yit represent the response variable;
xit represent the regressors;
uit represent disturbance term
The disturbance follow one-way error component model
uit=μi+vit(2)
μi~IID(0,σμi2) and vit follows AR(1) i.e.
vit=ρvi,t-1+εit(3)
Where .

2.2. Monte-Carlo Experiment

This work present a Monte-Carlo experiment that studied the finite sample properties of the five estimation methods; Between estimator (BE), Feasible Generalized Least Square (FGLS), Maximum estimator (M), Modified Maximum estimator (MM) and Proposed estimator (PE) applied to panel data model.
The Root Mean Square Error (RMSE) were used as criterion to assess the performances of these estimators.
The set- up in the experiment in this study was based on both individual- specific effect and remainder disturbance simultaneously with joint assumptions of heteroscedasticity and AR(1); generating contaminated data.

2.3. Data Generating Scheme

The design of Monte- Carlo experiments in this study uses the following information to generate infected data.
Consider the following panel model:
(4)
where The parameters were assigned as.
The cross- sections and time periods was chosen as N= 25,100,200 and T= 5, 10. For each combinations of N and T; 5000 replications were considered.
The autocorrelation coefficient values and heteroscedasticity used are specified as ρ varies as 0.3, 0.5, 0.8 and σμ2(1+λx̅i). λ=1.
The variance of individual effects, remainder error and error term is one. i.e.

2.4. Derivation of a New Proposed Panel Data Estimator

Consider the likelihood function of the normal distribution as follows;
L=1(2π)n2|V|1/2e-12U`V-1U(5)
where U=Y-
Therefore
L =1(2π)n2|V|1/2e-12Y-`V-1(Y-)(6)
Assume that there exist, linear restriction binding the regression coefficients vector of , where R is a row vector of ones defined as 1K1,11
(6) is transformed, subject to constraint Rβ, as follows:
l=(2π)-n2 v-12 e-12Y-`V-1Y--(7)
Taking natural logarithm of (7), we have,
loge l= -n2loge 2π-12loge V-12Y-1V-1Y--.(8)
loge l= -n2loge 2π-12loge V-12[Y1 V-1Y-Y1V-1-β1X1 V-1Y+β1X1V-1X β]-
loge l= -n2loge 2π-12loge V-12[Y1 V-1Y-2X1V-1+β1X1V-1]-
loge l= -n2loge 2π-12loge V-12[Y1 V-1Y-2X1V-1+β1X1V-1+2]
By maximizing the log- likelihood function, therefore,
Let logel=A
Aβ=-12[-2X1V-1Y+2X1V-1Xβ+2R]
Aβ=X1V-1Y-X1V-1Xβ-R
Setting Aβ=0
It implies that:
0=X1V-1Y-X1V-1Xβ-R
Therefore,
X1V-1Xβ=X1V-1Y-R
Pre-multiply by X1V-1X-1, it results into:
βP=(X1V-1X)-1X1V-1Y-R(X1V-1X)-1(9)
Var(βp)=E[βP-ββP-β1]
From (9),
βP=(X1V-1X)-1X1V-1(+U)-R(X1V-1X)-1(10)
βP=(X1V-1X)-1X1V-1Xβ+(X1V-1X)-1X1V-1U- R(X1V-1X)-1
βP=β+(X1V-1X)-1X1V-1U- R(X1V-1X)-1(11)
βP-β=(X1V-1X)-1X1V-1U- R(X1V-1X)-1(12)
Therefore,
Var(βP)=E[(X1V-1X)-1X1V-1U- R(X1V-1X)-1][(X1V-1X)-1X1V-1U- R(X1V-1X)-1]1
Var(βP)= (X1V-1X)-1X1V-1EUU1X(X1V-1X)-1+R(X1V-1X)-1(X1V-1X)-1R1
Var(βP)=σ2(X1V-1X)-1+R(X1V-1X)-1(X1V-1X)-1R1(13)
Biasβ̂=Eβ̂-β=Eβ̂-β(14)
EβP-β=E[(X1V-1X)-1X1V-1U- R(X1V-1X)-1]
EβP-β=[(X1V-1X)-1X1V-1E(U)- R(X1V-1X)-1]
Recall: EU=0
EβP-β=- R(X1V-1X)-1(15)
ABias(β) =EβP-β= R(X1V-1X)-1.(16)
3. Results
In this analysis, attention is focused on the asymptotic behaviour of five estimators in panel data infected with autocorrelated error terms of low, moderate and high levels and heteroscedastic structure. Monte-Carlo experiment is used to generate infectious data. 5000 replications are performed on combination of cross section, N=25,100 and 200 and time periods, T=5 and 10 for different degrees of autocorrelation and heteroscedastic structure (0.3, 0.5, and 0.8). R- Package is used for the analysis.
Table 1. Root Mean Square Error in the Presence of Heteroscedasticity and Low Autocorrelation.

N

T=5

T=10

β0

β1

β2

β0

β1

β2

25

BE

0.00436

0.13246

0.11342

0.17371

0.03544

0.22084

FGLS

0.04683

0.22915

0.61354

0.11077

0.77631

0.01124

ME

0.00437

0.13246

0.11343

0.17371

0.03544

0.22084

MME

0.06349

0.0681

0.24825

0.0499

0.03942

0.22238

PE

0.00021

0.00031

0.00078

0.00399

0.00011

0.00356

100

BE

3.72E-06

3.82E-08

0.03066

0.00037

0.00011

0.01132

FGLS

0.00075

0.00662

0.04158

0.0002

0.00785

0.05017

ME

3.70E-06

0.00394

0.03066

0.00037

0.00011

0.01132

Ugkl;

MME

0.01193

0.00961

0.05442

0.00866

0.00712

0.03283

PE

5.29E-06

0.00019

0.00047

0.000017

1.02E-04

0.00039

200

BE

0.00018

5.37E-05

0.00566

0.00031

0.00244

0.00098

FGLS

0.0001

0.00393

0.02509

0.00422

0.01802

0.00233

ME

0.00018

5.37E-05

0.00566

0.00031

0.00244

0.00098

MME

0.00569

0.00457

0.02635

0.00604

0.00592

0.02736

PE

0.000002

6.1E-07

0.00019

5.15E-06

0.000042

0.000027

Proposed estimator outperforms other estimators.
Table 2. Root Mean Squared Error in the Presence of Heteroscedasticity and Moderate Autocorrelation.

N

T=5

T=10

β0

β1

β2

β0

β1

β2

25

BE

0.0013093

0.0397366

0.0340263

0.0521136

0.0106306

0.066251

FGLS

0.0140494

0.0687456

0.1840608

0.0332317

0.2328942

0.003373

ME

0.0013095

0.0397381

0.03403

0.0521136

0.0106306

0.066251

MME

0.0190483

0.0204296

0.0744744

0.0149688

0.0118272

0.066713

PE

0.000064

0.0000936

0.000235

0.0119883

0.0000358

0.001069

100

BE

1.11E-06

1.14E-08

0.0091979

0.0001101

3.22E-05

0.003396

FGLS

0.0002264

0.0019866

0.012474

6.11E-05

0.0023553

0.015052

ME

1.11E-06

0.0011827

0.0091981

0.0001101

3.22E-05

0.003396

MME

0.0035777

0.0028833

0.0163271

0.0025988

0.0021363

0.009848

PE

3.29E-06

0.0000084

0.0002032

5.32E-06

3.37E-05

0.000117

200

BE

5.51E-05

1.61E-05

0.0016978

9.25E-05

0.0007306

0.000294

FGLS

3.05E-05

0.0011776

0.007526

0.001265

0.0054075

0.000699

ME

5.51E-05

1.61E-05

0.0016978

9.25E-05

0.0007306

0.000294

MME

0.0017085

0.0013696

0.007906

0.0018133

0.0017767

0.008208

PE

2.66E-06

1.8E-07

0.0000086

1.55E-06

0.0000028

0.000008

Proposed estimator outperforms other estimators.
Table 3. Root Mean Squared Error in the Presence of Heteroscedasticity and High Autocorrelation.

N

T=5

T=10

β0

β1

β2

β0

β1

β2

25

BE

0.0034915

0.10596432

0.0907368

0.1389696

0.0283483

0.176669

FGLS

0.0374651

0.1833216

0.4908288

0.0886178

0.6210512

0.008994

ME

0.0034921

0.10596821

0.0907466

0.1389696

0.0283483

0.176669

MME

0.0507954

0.05447904

0.1985984

0.0399168

0.0315392

0.177901

PE

0.000234

0.00196221

0.0001235

0.0000068

0.0007244

0.000015

100

BE

2.97E-06

3.05E-08

0.0245277

0.0002937

8.59E-05

0.009055

FGLS

0.0006037

0.0052976

0.033264

0.0001629

0.0062807

0.040139

ME

2.96E-06

0.00315392

0.0245281

0.0002937

8.59E-05

0.009055

MME

0.0095406

0.00768891

0.0435389

0.00693

0.0056968

0.026261

PE

1.55E-04

0.00070211

0.0000962

0.0000029

3.12E-04

0.000055

200

BE

0.0001468

4.29E-05

0.0045276

0.0002466

0.0019482

0.000785

FGLS

8.147E-05

0.00314034

0.0200694

0.0033732

0.0144199

0.001863

ME

0.0001469

4.29E-05

0.0045274

0.0002466

0.0019484

0.000785

MME

0.0045559

0.00365226

0.0210826

0.0048356

0.0047378

0.021888

PE

4.9E-07

1.56E-04

0.00000412

3.8E-07

0.0000022

0.0000004

Proposed estimator outperforms other estimators.
Table 4. Absolute Bias in the Presence of Heteroscedasticity and Low Autocorrelation.

N

T=5

T=10

β0

β1

β2

β0

β1

β2

25

BE

0.052136

0.287336

0.265776

0.329084

0.148666

0.371028

FGLS

0.170912

0.378084

0.618576

0.262836

0.6958

0.084084

ME

0.0521752

0.2874144

0.265972

0.329182

0.1486856

0.371224

MME

0.235396

0.216188

0.415912

0.178752

0.161504

0.378672

PE

0.003269

0.003945

0.006256

0.014119

0.002439

0.013334

100

BE

0.0304388

0.308504

0.2765364

0.0302604

0.0163621

0.168033

FGLS

0.043708

0.129164

0.325556

0.02254

0.139944

0.35378

ME

0.003038

0.0003136

0.276556

0.0302624

0.016366

0.168031

MME

0.17248

0.15484

0.36848

0.147

0.13328

0.28616

PE

0.0002320

0.000061

0.013233

0.005950

0.001557

0.008829

200

BE

0.0302428

0.01636208

0.1680308

0.0392134

0.1102108

0.069952

FGLS

0.02254

0.139944

0.35378

0.14504

0.29988

0.1078

ME

0.0302624

0.016366

0.1680308

0.0392196

0.1102304

0.069972

MME

0.16856

0.15092

0.3626

0.173656

0.171892

0.36946

PE

0.000595

0.0003038

0.008829

0.001433

0.007553

0.00333

Proposed Estimator is asymptotically consistent.
Table 5. Absolute Bias in the Presence of Heteroscedasticity and Moderate Autocorrelation.

N

T=5

T=10

β0

β1

β2

β0

β1

β2

25

BE

0.0156408

0.0862008

0.0797328

0.0987252

0.0445998

0.111308

FGLS

0.0512736

0.1134252

0.1855728

0.0788508

0.20874

0.025225

ME

0.0156526

0.08622432

0.0797916

0.0987546

0.0446057

0.111367

MME

0.0706188

0.0648564

0.1247736

0.0536256

0.0484512

0.113602

PE

0.0009807

0.0011835

0.0068769

0.0042359

0.0007319

0.004000

100

BE

0.0091316

0.0925512

0.0829609

0.0090781

0.0049086

0.05041

FGLS

0.0131124

0.0387492

0.0976668

0.006762

0.0419832

0.106134

ME

0.0009114

0.00009408

0.0829668

0.0090787

0.0049098

0.050409

MME

0.051744

0.046452

0.110544

0.0441

0.039984

0.085848

PE

0.0000696

0.00001835

0.003970

0.001785

0.000467

0.002648

200

BE

0.0090728

0.00490862

0.0504092

0.011764

0.0330632

0.020986

FGLS

0.006762

0.0419832

0.106134

0.043512

0.089964

0.03234

ME

0.0090787

0.0049098

0.0504092

0.0117659

0.0330691

0.020992

MME

0.050568

0.045276

0.10878

0.0520968

0.0515676

0.110838

PE

0.000038

0.000091

0.002648

0.000430

0.0002659

0.000999

Proposed estimator is more asymptotically consistent than Modified Maximum Estimators.
Table 6. Absolute Bias in the Presence of Heteroscedasticity and High Autocorrelation.

N

T=5

T=10

β0

β1

β2

β0

β1

β2

25

BE

0.0417088

0.2298688

0.2126208

0.2632672

0.1189328

0.296822

FGLS

0.1367296

0.3024672

0.4948608

0.2102688

0.55664

0.067267

ME

0.0417402

0.22993152

0.2127776

0.2633456

0.1189485

0.296979

MME

0.1883168

0.1729504

0.3327296

0.1430016

0.1292032

0.302938

PE

0.0026154

0.00315607

0.005005

0.0112958

0.0019518

0.010667

100

BE

0.024351

0.2468032

0.2212291

0.0242084

0.0130897

0.134426

FGLS

0.0349664

0.1033312

0.2604448

0.018032

0.1119552

0.283024

ME

0.0024304

0.00025088

0.2212448

0.0242099

0.0130928

0.134425

MME

0.137984

0.123872

0.294784

0.1176

0.106624

0.228928

PE

0.000185

0.00004892

0.0048714

0.0476045

0.0012456

0.007063

200

BE

0.0241942

0.01308966

0.1344246

0.0313707

0.0881686

0.055962

FGLS

0.018032

0.1119552

0.283024

0.116032

0.239904

0.08624

ME

0.0242099

0.0130928

0.1344246

0.0313757

0.0881843

0.055978

MME

0.134848

0.120736

0.29008

0.1389248

0.1375136

0.295568

PE

0.000076

0.0002431

0.0020635

0.001146

0.0060424

0.002664

Proposed estimator is more asymptotically consistent than Modified Maximum Estimator.
4. Discussion
The following results are obtained when there exists combination of different level of autocorrelation and heteroscedasticity irrespective of cross-section and time period.
1) The study came up with a proposed estimator;
2) Proposed estimator (PE) is asymptotically efficient and consistent;
3) PE is a more suitable technique for both small and large sample size than other existing estimators in this study.
5. Implication to Research and Practice
The implication of this work is that it will assist the firms, government, social and behavioral scientists in their decision making in order to minimize the effect of one-way error component on the parameter estimates.
6. Conclusion
This study considers a case of heteroscedastic individual random when first order serial correlation is present in the context of a panel data regression model. This is in contrary to the usual econometrics literature that deals with heteroscedasticity ignoring serial correlation or vice versa. The results of the Monte-Carlo experiment showed that PE outperforms other methods of estimation for it is asymptotically efficient and consistent in the presence of autocorrelation and heteroscedasticity. Therefore, it is more robust than the existing estimators in this study.
Future Research
The study can further be extended to non-linear panel data of balanced and unbalanced type. Bayesian inference can also be looked into for possible robust estimation.
Conflicts of Interest
The authors declare no conflicts of interest.
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    Ayansola, O. A., Adejumo, A. O. (2024). On the Performance of Some Estimation Methods in Models with Heteroscedasticity and Autocorrelated Disturbances (A Monte-Carlo Approach). Mathematical Modelling and Applications, 9(1), 23-31. https://doi.org/10.11648/j.mma.20240901.13

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    Ayansola, O. A.; Adejumo, A. O. On the Performance of Some Estimation Methods in Models with Heteroscedasticity and Autocorrelated Disturbances (A Monte-Carlo Approach). Math. Model. Appl. 2024, 9(1), 23-31. doi: 10.11648/j.mma.20240901.13

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    AMA Style

    Ayansola OA, Adejumo AO. On the Performance of Some Estimation Methods in Models with Heteroscedasticity and Autocorrelated Disturbances (A Monte-Carlo Approach). Math Model Appl. 2024;9(1):23-31. doi: 10.11648/j.mma.20240901.13

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  • @article{10.11648/j.mma.20240901.13,
      author = {Olufemi Aderemi Ayansola and Adebowale Olusola Adejumo},
      title = {On the Performance of Some Estimation Methods in Models with Heteroscedasticity and Autocorrelated Disturbances (A Monte-Carlo Approach)},
      journal = {Mathematical Modelling and Applications},
      volume = {9},
      number = {1},
      pages = {23-31},
      doi = {10.11648/j.mma.20240901.13},
      url = {https://doi.org/10.11648/j.mma.20240901.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mma.20240901.13},
      abstract = {The proliferation of panel data studies has been greatly motivated by the availability of data and capacity for modelling the complexity of human behaviour than a single cross-section or time series data and these led to the rise of challenging methodologies for estimating the data set. It is pertinent that, in practice, panel data are bound to exhibit autocorrelation or heteroscedasticity or both. In view of the fact that the presence of heteroscedasticity and autocorrelated errors in panel data models biases the standard errors and leads to less efficient results. This study deemed it fit to search for estimator that can handle the presence of these twin problems when they co- exists in panel data. Therefore, robust inference in the presence of these problems needs to be simultaneously addressed. The Monte-Carlo simulation method was designed to investigate the finite sample properties of five estimation methods: Between Estimator (BE), Feasible Generalized Least Square (FGLS), Maximum Estimator (ME) and Modified Maximum Estimator (MME), including a new Proposed Estimator (PE) in the simulated data infected with heteroscedasticity and autocorrelated errors. The results of the root mean square error and absolute bias criteria, revealed that Proposed Estimator in the presence of these problems is asymptotically more efficient and consistent than other estimators in the class of the estimators in the study. This is experienced in all combinatorial level of autocorrelated errors in remainder error and fixed heteroscedastic individual effects. For this reason, PE has better performance among other estimators.},
     year = {2024}
    }
    

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  • TY  - JOUR
    T1  - On the Performance of Some Estimation Methods in Models with Heteroscedasticity and Autocorrelated Disturbances (A Monte-Carlo Approach)
    AU  - Olufemi Aderemi Ayansola
    AU  - Adebowale Olusola Adejumo
    Y1  - 2024/04/02
    PY  - 2024
    N1  - https://doi.org/10.11648/j.mma.20240901.13
    DO  - 10.11648/j.mma.20240901.13
    T2  - Mathematical Modelling and Applications
    JF  - Mathematical Modelling and Applications
    JO  - Mathematical Modelling and Applications
    SP  - 23
    EP  - 31
    PB  - Science Publishing Group
    SN  - 2575-1794
    UR  - https://doi.org/10.11648/j.mma.20240901.13
    AB  - The proliferation of panel data studies has been greatly motivated by the availability of data and capacity for modelling the complexity of human behaviour than a single cross-section or time series data and these led to the rise of challenging methodologies for estimating the data set. It is pertinent that, in practice, panel data are bound to exhibit autocorrelation or heteroscedasticity or both. In view of the fact that the presence of heteroscedasticity and autocorrelated errors in panel data models biases the standard errors and leads to less efficient results. This study deemed it fit to search for estimator that can handle the presence of these twin problems when they co- exists in panel data. Therefore, robust inference in the presence of these problems needs to be simultaneously addressed. The Monte-Carlo simulation method was designed to investigate the finite sample properties of five estimation methods: Between Estimator (BE), Feasible Generalized Least Square (FGLS), Maximum Estimator (ME) and Modified Maximum Estimator (MME), including a new Proposed Estimator (PE) in the simulated data infected with heteroscedasticity and autocorrelated errors. The results of the root mean square error and absolute bias criteria, revealed that Proposed Estimator in the presence of these problems is asymptotically more efficient and consistent than other estimators in the class of the estimators in the study. This is experienced in all combinatorial level of autocorrelated errors in remainder error and fixed heteroscedastic individual effects. For this reason, PE has better performance among other estimators.
    VL  - 9
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics and Statistics, The Polytechnic, Ibadan, Nigeria

  • Department of Statistics, University of Ilorin, Ilorin, Nigeria