Adewale F. Lukman, B. M. Golam Kibria, Kayode Ayinde, Segun L. Jegede, "Modified One-Parameter Liu Estimator for the Linear Regression Model", Modelling and Simulation in Engineering, vol. 2020, Article ID 9574304, 17 pages, 2020. https://doi.org/10.1155/2020/9574304
Modified One-Parameter Liu Estimator for the Linear Regression Model
Adewale F. Lukman,^{1,2}B. M. Golam Kibria,^{3}Kayode Ayinde,^{4} and Segun L. Jegede^{1}
^{1}Department of Physical Sciences, Landmark University, Omu-Aran, Nigeria
^{2}Institut Henri Poincaré, France
^{3}Department of Mathematics and Statistics, Florida International University, USA
^{4}Department of Statistics, Federal University of Technology, Akure, Nigeria
Academic Editor: Mohamed B. Trabia
Received18 Dec 2019
Revised06 Jul 2020
Accepted11 Jul 2020
Published19 Aug 2020
Abstract
Motivated by the ridge regression (Hoerl and Kennard, 1970) and Liu (1993) estimators, this paper proposes a modified Liu estimator to solve the multicollinearity problem for the linear regression model. This modification places this estimator in the class of the ridge and Liu estimators with a single biasing parameter. Theoretical comparisons, real-life application, and simulation results show that it consistently dominates the usual Liu estimator. Under some conditions, it performs better than the ridge regression estimators in the smaller MSE sense. Two real-life data are analyzed to illustrate the findings of the paper and the performances of the estimators assessed by MSE and the mean squared prediction error. The application result agrees with the theoretical and simulation results.
1. Introduction
The linear regression model (LRM) is
where is a vector of the predictand, is a known matrix of predictor variables, is a vector of unknown regression parameters, is a vector of errors such that and , and is an identity matrix. The parameters in (1) are mostly estimated by the ordinary least square (OLS) estimator defined in (2)
The performance of the estimator is conditional on the non-violation of the assumption of model (1) that the predictor variables are independent. However, in most real-life applications, we observed that the predictor variables grow together, which result in the problem termed multicollinearity. The consequence of this on the OLS estimator is that it reduces its efficiency and it became unstable (for examples, [1, 2]). Many methods exist in literature to combat the multicollinearity problem. Biased estimators with one biasing parameter include the ridge regression estimator by Hoerl and Kennard [1] and the Liu estimator by Kejian [3], among others.
The objective of this paper is to propose a new one-parameter Liu-type estimator for the regression parameter when the predictor variables of the model are linearly related. Since we want to compare the performance of the proposed estimator with the usual Liu and ridge regression estimators, we will give a brief description of each of them as follows.
1.1. Ridge Regression Estimator (RRE)
Hoerl and Kennard [1] proposed by augmenting to the linear regression model (1). The ridge regression estimator is defined as
1.2. Liu Estimator
Since the ridge regression estimator is a complicated function of , Liu [3] derived by augmenting to the linear regression model (1) [4]. The Liu estimator of is
This estimator is a linear function of the shrinkage parameter .
1.3. Proposed One-Parameter Liu Estimator
One of the limitations of the shrinkage parameter by Kejian [3] is that it returns a negative value most of the time which affects the performance of the estimator [4, 5]. In this study, we augment to the LRM. This is done by minimizing subject to where is a constant. The modified Liu estimator of is
This modification provides a substantial improvement in the performance of the modified Liu estimator and will give a positive value of the shrinkage parameter . The estimator will always produce a smaller mean square error compared to the OLS estimator.
The bias, variance, and MSE of the proposed estimator are, respectively, given as follows:
To compare the performance of the estimators, we will consider the linear regression model in canonical form, which is given as follows:
where ,,. and are the eigenvalues and eigenvectors of . The ordinary, ridge, Liu, and modified Liu estimators of are
The following notations and lemmas will be used to prove the statistical property of .
Lemma 1. Given that matrix and is a vector, if and only if [6].
Lemma 2. Given two estimators of , and . Suppose that , where and represent the covariance matrix of and , respectively. Therefore, if and only if where where and represent the mean squared error matrix and bias vector of [7].
According to Özkale and Kaçiranlar [4], if , then where is the scalar mean square error.
The rest of the paper is as follows. The theoretical comparison among the estimators and the estimation of the biasing parameter of the proposed estimator are given in Section 2. A simulation study and numerical examples are conducted in Sections 3 and 4, respectively. This paper ends up with some concluding remarks in Section 5.
2. Comparison among Estimators
In this section, we will show theoretical comparisons among the estimators. First, we will compare between the proposed estimator and OLSE.
2.1. The Proposed Estimator and OLSE
Theorem 3. Given two estimators of ,, and , if , then the estimator is better than , that is, if and only if, where .
Proof. Recall that
Then,
We observed from equation (16) that which shows that.
2.2. The Proposed Estimator and RRE
Theorem 4. Given two estimators of ,, and , if , then the estimator is better than , that is, if and only if, where and .
Proof. Recall that
Therefore,
From equation (18), we observed that . Hence, this shows that .
2.3. The Proposed Estimator and Liu Estimator
Theorem 5. Given two estimators of ,, and , if , then the estimator is better than , that is, if and only if, where and .
Proof. Note that
Therefore,
From equation (20), we observed that . Hence, this shows that.
2.4. Determination of
The shrinkage parameter is selected according to Kejian [3], Kibria [8], Kibria and Banik [9], Lukman and Ayinde [10], and Qasim et al. [11].
The partial derivative of (21) with respect to and setting it to zero, we obtained
In eqn. (22), we replace and with its unbiased estimate and obtained:
Taking a critical look at equation (23), the estimate of the shrinkage parameter will often return a positive value since and will always be greater than zero. For practical purposes, we obtained the minimum value of (24) as
3. Simulation Study
As theoretical comparison among the estimators in Section 2 gives the conditional dominance among the estimators, a simulation study conducted using the R 3.4.1 programming languages is considered in this section to grasp the better picture about the performance of the estimators.
3.1. Simulation Technique
We generated the explanatory variables by the following references of Gibbons [12] and Qasim et al. [11]:
where are independent standard normal pseudorandom numbers, and represents the correlation between any two predictor variables. The number of predictor variables is three and seven. The predictands for the regression models are generated as follows:
where . is constrained to unity, according to Newhouse and Oman [13], Lukman et al. [14], and Lukman et al. [15]. We examined the performances of the estimators, using mean square error criteria. The simulation is performed using the following condition:
(1)Sample sizes: , 50, 100, and 200(2)Number of replications: 1000(3)The variances: are 1, 25, and 100(4)The multicollinearity levels: , 0.8, 0.9, and 0.99(5), 0.2,…, 0.9, and 1
The simulated results for and , 0.80, 0.90, and 0.99 are presented in Tables 1–3 for , 50, and 100, respectively. and , 0.80, 0.90, and 0.99 are presented in Tables 4–6 for , 50, and 100, respectively. For a better picture, we have plotted MSE vs. for ,, and , 0.90, and 0.99 in Figures 1 and 2, respectively. We also plotted MSE vs. and MSE vs. in Figure 3.
0.7
0.8
0.9
0.99
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
1
0.1
0.3621
0.3517
0.2909
0.2782
0.5473
0.5186
0.3750
0.3461
1.1540
1.0117
0.5323
0.4458
12.774
4.1554
0.8567
0.3894
0.2
0.3621
0.3420
0.2977
0.2722
0.5473
0.4929
0.3908
0.3329
1.1540
0.8995
0.5826
0.4096
12.774
2.3395
1.3877
0.4531
0.3
0.3621
0.3329
0.3047
0.2666
0.5473
0.4699
0.4073
0.3206
1.1540
0.8094
0.6376
0.3781
12.774
1.6033
2.1170
0.7151
0.4
0.3621
0.3245
0.3121
0.2612
0.5473
0.4493
0.4248
0.3092
1.1540
0.736
0.6973
0.3513
12.774
1.2143
3.0446
1.1753
0.5
0.3621
0.3167
0.3197
0.2561
0.5473
0.4306
0.4431
0.2985
1.1540
0.6754
0.7617
0.3292
12.774
0.9779
4.1705
1.8339
0.6
0.3621
0.3093
0.3276
0.2513
0.5473
0.4137
0.4622
0.2888
1.1540
0.6247
0.8307
0.3117
12.774
0.8214
5.4946
2.6906
0.7
0.3621
0.3025
0.3358
0.2468
0.5473
0.3991
0.4822
0.2799
1.1540
0.5819
0.9045
0.2990
12.774
0.7117
7.0169
3.7457
0.8
0.3621
0.2961
0.3443
0.2425
0.5473
0.3845
0.5031
0.2718
1.1540
0.5454
0.9830
0.2910
12.774
0.6314
8.7375
4.9990
0.9
0.3621
0.29
0.353
0.2385
0.5473
0.3718
0.5248
0.2646
1.1540
0.5139
1.0661
0.2876
12.774
0.5708
10.656
6.4505
1.0
0.3621
0.2844
0.3621
0.2348
0.5473
0.3601
0.5473
0.2583
1.1540
0.4867
1.1540
0.2890
12.774
0.5239
12.774
8.1003
5
0.1
8.0209
7.7585
6.1372
5.7600
12.967
12.232
8.3642
7.5029
28.462
24.840
12.067
9.5146
319.35
102.39
17.451
4.1230
0.2
8.0209
7.5109
6.3314
5.5771
12.967
11.567
8.8169
7.0943
28.462
21.945
13.492
8.3869
319.35
56.008
31.445
4.7897
0.3
8.0209
7.2770
6.5294
5.3980
12.967
10.962
9.2843
6.7003
28.462
19.588
15.017
7.3584
319.35
36.978
50.327
10.344
0.4
8.0209
7.0557
6.7311
5.2226
12.967
10.411
9.7664
6.321
28.462
17.641
16.640
6.4290
319.35
26.816
74.095
20.784
0.5
8.0209
6.8461
6.9367
5.0510
12.967
9.9071
10.263
5.9564
28.462
16.010
18.362
5.5988
319.35
20.580
102.75
36.112
0.6
8.0209
6.6474
7.146
4.8832
12.967
9.4445
10.775
5.6065
28.462
14.627
20.184
4.8677
319.35
16.415
136.29
56.327
0.7
8.0209
6.4588
7.3591
4.7192
12.967
9.0189
11.301
5.2713
28.462
13.442
22.105
4.2357
319.35
13.467
174.72
81.429
0.8
8.0209
6.2795
7.5759
4.5589
12.967
8.6262
11.842
4.9508
28.462
12.418
24.124
3.7028
319.35
11.293
218.04
111.42
0.9
8.0209
6.1090
7.7953
4.4024
12.967
8.2630
12.397
4.6450
28.462
11.526
26.243
3.2691
319.35
9.6369
266.24
146.30
1.0
8.0209
5.9467
8.0209
4.2497
12.967
7.9262
12.967
4.3538
28.462
10.741
28.462
2.9345
319.35
8.3433
319.33
186.06
10
0.1
31.993
30.939
24.421
22.901
51.819
48.871
33.333
29.865
113.84
99.331
48.088
37.822
1277.4
409.25
69.149
15.713
0.2
31.993
29.945
25.203
22.163
51.819
46.201
35.155
28.218
113.84
87.726
53.814
33.282
1277.4
223.57
125.20
18.323
0.3
31.993
29.005
26.000
21.440
51.819
43.775
37.034
26.629
113.84
78.278
59.935
29.137
1277.4
147.37
200.79
40.484
0.4
31.993
28.116
26.812
20.731
51.819
41.561
38.972
25.098
113.84
70.466
66.450
25.387
1277.4
106.67
295.94
82.197
0.5
31.993
27.274
27.639
20.038
51.819
39.536
40.968
23.626
113.84
63.919
73.361
22.031
1277.4
81.687
410.64
143.46
0.6
31.993
26.474
28.480
19.359
51.819
37.677
43.022
22.211
113.84
58.368
80.667
19.071
1277.4
64.998
544.90
224.28
0.7
31.993
25.715
29.336
18.695
51.819
35.966
45.134
20.855
113.84
53.612
88.368
16.506
1277.4
53.189
698.70
324.65
0.8
31.993
24.994
30.207
18.045
51.819
34.387
47.304
19.557
113.84
49.498
96.464
14.336
1277.4
44.476
872.06
444.57
0.9
31.993
24.307
31.092
17.411
51.819
32.926
49.533
18.316
113.84
45.910
104.96
12.562
1277.4
37.839
1065.0
584.04
1.0
31.993
23.654
31.993
16.791
51.819
31.570
51.819
17.134
113.84
42.758
113.84
11.182
1277.4
32.655
1277.4
743.07
0.7
0.8
0.9
0.99
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
1
0.1
0.1336
0.1325
0.1249
0.1232
0.1806
0.1785
0.1646
0.1614
0.3253
0.3172
0.2685
0.2577
2.8206
2.1349
0.7453
0.5194
0.2
0.1336
0.1314
0.1258
0.1224
0.1806
0.1765
0.1662
0.1599
0.3253
0.3097
0.2741
0.2526
2.8206
1.6977
0.8903
0.4386
0.3
0.1336
0.1304
0.1267
0.1216
0.1806
0.1746
0.1679
0.1584
0.3253
0.3025
0.2799
0.2477
2.8206
1.3996
1.0568
0.3791
0.4
0.1336
0.1294
0.1277
0.1208
0.1806
0.1727
0.1696
0.1570
0.3253
0.2958
0.2859
0.2429
2.8206
1.1861
1.2446
0.3410
0.5
0.1336
0.1285
0.1286
0.1200
0.1806
0.1709
0.1713
0.1556
0.3253
0.2895
0.2920
0.2383
2.8206
1.0274
1.4538
0.3243
0.6
0.1336
0.1275
0.1296
0.1192
0.1806
0.1692
0.1731
0.1542
0.3253
0.2836
0.2983
0.2339
2.8206
0.9058
1.6844
0.3289
0.7
0.1336
0.1266
0.1305
0.1185
0.1806
0.1676
0.1749
0.1529
0.3253
0.2780
0.3048
0.2297
2.8206
0.8105
1.9363
0.3550
0.8
0.1336
0.1257
0.1315
0.1177
0.1806
0.1660
0.1767
0.1516
0.3253
0.2727
0.3115
0.2256
2.8206
0.7344
2.2097
0.4025
0.9
0.1336
0.1249
0.1325
0.1170
0.1806
0.1644
0.1786
0.1504
0.3253
0.2677
0.3183
0.2217
2.8206
0.6726
2.5044
0.4713
1
0.1336
0.1241
0.1336
0.1163
0.1806
0.1630
0.1806
0.1491
0.3253
0.2630
0.3253
0.2180
2.8206
0.6217
2.8206
0.5615
5
0.1
3.0492
3.0174
2.7831
2.7259
4.1707
4.1067
3.6537
3.5443
7.5964
7.3662
5.9057
5.5633
69.766
52.391
15.944
9.5391
0.2
3.0492
2.9860
2.8119
2.6976
4.1707
4.0443
3.7092
3.4904
7.5964
7.1477
6.0815
5.3966
69.766
41.181
19.904
7.0945
0.3
3.0492
2.9553
2.8410
2.6695
4.1707
3.9836
3.7651
3.4370
7.5964
6.9399
6.2603
5.2329
69.766
33.446
24.369
5.1550
0.4
3.0492
2.9251
2.8702
2.6415
4.1707
3.9244
3.8216
3.3840
7.5964
6.7421
6.4421
5.0723
69.766
27.840
29.339
3.7205
0.5
3.0492
2.8954
2.8996
2.6138
4.1707
3.8667
3.8785
3.3315
7.5964
6.5538
6.6269
4.9146
69.766
23.620
34.814
2.7912
0.6
3.0492
2.8662
2.9291
2.5862
4.1707
3.8104
3.9360
3.2796
7.5964
6.3742
6.8148
4.7601
69.766
20.350
40.794
2.3669
0.7
3.0492
2.8375
2.9589
2.5588
4.1707
3.7555
3.9939
3.2281
7.5964
6.2028
7.0056
4.6085
69.766
17.755
47.280
2.4478
0.8
3.0492
2.8094
2.9888
2.5315
4.1707
3.7020
4.0523
3.1771
7.5964
6.0391
7.1995
4.4599
69.766
15.655
54.270
3.0337
0.9
3.0492
2.7817
3.0189
2.5045
4.1707
3.6498
4.1113
3.1267
7.5964
5.8826
7.3965
4.3144
69.766
13.929
61.766
4.1247
1
3.0492
2.7544
3.0492
2.4776
4.1707
3.5988
4.1707
3.0767
7.5964
5.7330
7.5964
4.1719
69.766
12.489
69.767
5.7209
10
0.1
12.177
12.049
11.108
10.877
16.661
16.404
14.582
14.141
30.351
29.427
23.561
22.182
279.07
209.56
63.558
37.824
0.2
12.177
11.923
11.224
10.763
16.661
16.154
14.805
13.923
30.351
28.550
24.267
21.511
279.07
164.70
79.447
27.978
0.3
12.177
11.800
11.340
10.650
16.661
15.909
15.030
13.707
30.351
27.716
24.986
20.852
279.07
133.72
97.350
20.146
0.4
12.177
11.678
11.458
10.537
16.661
15.671
15.257
13.494
30.351
26.922
25.717
20.205
279.07
111.26
117.27
14.328
0.5
12.177
11.559
11.576
10.425
16.661
15.439
15.487
13.282
30.351
26.166
26.459
19.569
279.07
94.352
139.20
10.525
0.6
12.177
11.442
11.695
10.314
16.661
15.213
15.718
13.072
30.351
25.444
27.214
18.945
279.07
81.240
163.14
8.7356
0.7
12.177
11.327
11.814
10.204
16.661
14.992
15.951
12.864
30.351
24.755
27.980
18.334
279.07
70.829
189.10
8.9606
0.8
12.177
11.213
11.934
10.094
16.661
14.776
16.186
12.658
30.351
24.097
28.758
17.734
279.07
62.401
217.08
11.200
0.9
12.177
11.102
12.055
9.9840
16.661
14.566
16.423
12.454
30.351
23.468
29.548
17.146
279.07
55.468
247.06
15.453
1
12.177
10.992
12.177
9.8760
16.661
14.360
16.661
12.252
30.351
22.866
30.351
16.570
279.07
49.684
279.07
21.721
0.7
0.8
0.9
0.99
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
1
0.1
0.1124
0.1121
0.1105
0.1103
0.1492
0.1478
0.1396
0.1380
0.2867
0.2781
0.2304
0.2209
3.0721
2.1409
0.6883
0.4808
0.2
0.1124
0.1118
0.1107
0.1102
0.1492
0.1465
0.1404
0.1372
0.2867
0.2702
0.2356
0.2165
3.0721
1.6206
0.8360
0.4210
0.3
0.1124
0.1116
0.1108
0.1101
0.1492
0.1453
0.1414
0.1365
0.2867
0.2629
0.2410
0.2124
3.0721
1.2979
1.0130
0.3905
0.4
0.1124
0.1114
0.1110
0.1100
0.1492
0.1442
0.1423
0.1358
0.2867
0.2562
0.2468
0.2086
3.0721
1.0825
1.2192
0.3893
0.5
0.1124
0.1112
0.1112
0.1100
0.1492
0.1432
0.1434
0.1352
0.2867
0.2501
0.2527
0.2050
3.0721
0.9305
1.4548
0.4174
0.6
0.1124
0.111
0.1114
0.1100
0.1492
0.1422
0.1444
0.1346
0.2867
0.2444
0.2590
0.2017
3.0721
0.8186
1.7197
0.4748
0.7
0.1124
0.1108
0.1116
0.1099
0.1492
0.1412
0.1455
0.1341
0.2867
0.2391
0.2655
0.1987
3.0721
0.7333
2.0138
0.5615
0.8
0.1124
0.1106
0.1119
0.1099
0.1492
0.1403
0.1467
0.1336
0.2867
0.2342
0.2723
0.1959
3.0721
0.6667
2.3373
0.6775
0.9
0.1124
0.1105
0.1121
0.1099
0.1492
0.1395
0.1479
0.1332
0.2867
0.2297
0.2793
0.1934
3.0721
0.6134
2.6900
0.8228
1.0
0.1124
0.1104
0.1124
0.1099
0.1492
0.1387
0.1492
0.1328
0.2867
0.2255
0.2867
0.1912
3.0721
0.5699
3.0721
0.9973
5
0.1
2.0631
2.0452
1.9126
1.8802
3.2440
3.1954
2.8523
2.7697
6.9576
6.7191
5.2564
4.9172
76.772
53.314
14.747
8.2979
0.2
2.0631
2.0276
1.9289
1.8642
3.2440
3.1480
2.8942
2.7290
6.9576
6.4945
5.4313
4.7529
76.772
39.906
18.971
6.0738
0.3
2.0631
2.0102
1.9454
1.8483
3.2440
3.1019
2.9365
2.6887
6.9576
6.2827
5.6097
4.5921
76.772
31.412
23.862
4.5166
0.4
2.0631
1.9932
1.9619
1.8325
3.2440
3.0570
2.9793
2.6487
6.9576
6.0826
5.7917
4.4349
76.772
25.626
29.420
3.6261
0.5
2.0631
1.9764
1.9785
1.8167
3.2440
3.0133
3.0224
2.6092
6.9576
5.8934
5.9772
4.2811
76.772
21.466
35.645
3.4025
0.6
2.0631
1.9599
1.9952
1.8011
3.2440
2.9707
3.0659
2.5701
6.9576
5.7143
6.1662
4.1310
76.772
18.350
42.537
3.8457
0.7
2.0631
1.9436
2.0121
1.7856
3.2440
2.9291
3.1098
2.5314
6.9576
5.5445
6.3587
3.9843
76.772
15.939
50.096
4.9557
0.8
2.0631
1.9276
2.029
1.7702
3.2440
2.8887
3.1542
2.4931
6.9576
5.3834
6.5548
3.8412
76.772
14.024
58.321
6.7325
0.9
2.0631
1.9119
2.0460
1.7548
3.2440
2.8492
3.1989
2.4553
6.9576
5.2305
6.7544
3.7016
76.772
12.471
67.213
9.1762
1.0
2.0631
1.8964
2.0631
1.7396
3.2440
2.8108
3.2440
2.4178
6.9576
5.0850
6.9576
3.5656
76.772
11.189
76.772
12.287
10
0.1
8.1632
8.0901
7.5481
7.4154
12.920
12.723
11.334
10.999
27.809
26.853
20.970
19.601
307.09
213.26
58.717
32.755
0.2
8.1632
8.0182
7.615
7.3496
12.920
12.532
11.505
10.833
27.809
25.951
21.675
18.937
307.09
159.58
75.684
23.760
0.3
8.1632
7.9474
7.6822
7.2842
12.920
12.346
11.676
10.669
27.809
25.100
22.394
18.287
307.09
125.56
95.308
17.423
0.4
8.1632
7.8777
7.7498
7.2191
12.920
12.164
11.849
10.507
27.809
24.296
23.126
17.650
307.09
102.37
117.59
13.743
0.5
8.1632
7.8091
7.8178
7.1544
12.920
11.987
12.024
10.346
27.809
23.535
23.872
17.028
307.09
85.681
142.53
12.721
0.6
8.1632
7.7415
7.8862
7.0901
12.920
11.814
12.200
10.186
27.809
22.815
24.632
16.419
307.09
73.175
170.13
14.356
0.7
8.1632
7.675
7.9549
7.0261
12.920
11.646
12.378
10.028
27.809
22.131
25.406
15.823
307.09
63.493
200.38
18.648
0.8
8.1632
7.6096
8.0240
6.9625
12.920
11.4821
12.557
9.8717
27.809
21.482
26.193
15.242
307.09
55.802
233.29
25.598
0.9
8.1632
7.5451
8.0934
6.8992
12.920
11.322
12.738
9.7168
27.809
20.865
26.994
14.674
307.09
49.561
268.86
35.206
1.0
8.1632
7.4816
8.1632
6.8364
12.920
11.166
12.920
9.5634
27.809
20.279
27.809
14.120
307.09
44.407
307.09
47.470
0.7
0.8
0.9
0.99
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
1
0.1
0.8375
0.8019
0.6073
0.5645
1.3038
1.2137
0.8030
0.7166
2.7626
2.3689
1.1868
0.9622
30.134
10.720
2.2263
0.9170
0.2
0.8375
0.7694
0.6299
0.5442
1.3038
1.1356
0.8496
0.6767
2.7626
2.0758
1.3162
0.8670
30.134
6.4830
3.5481
0.9295
0.3
0.8375
0.7395
0.6532
0.5246
1.3038
1.0673
0.8985
0.6392
2.7626
1.8492
1..4571
0.7833
30.134
4.5630
5.3147
1.3867
0.4
0.8375
0.7121
0.6773
0.5058
1.3038
1.0072
0.9496
0.6039
2.7626
1.6691
1.6093
0.7109
30.134
3.4720
7.5260
2.2886
0.5
0.8375
0.6868
0.7021
0.4877
1.3038
0.9539
1.0030
0.5708
2.7626
1.5224
1.7730
0.6500
30.134
2.7762
10.182
3.6354
0.6
0.8375
0.6635
0.7277
0.4705
1.3038
0.9063
1.0586
0.5400
2.7626
1.4006
1.9481
0.6005
30.134
2.2996
13.283
5.4268
0.7
0.8375
0.6418
0.7540
0.4539
1.3038
0.8636
1.1166
0.5115
2.7626
1.2980
2.1346
0.5624
30.134
1.9565
16.828
7.6631
0.8
0.8375
0.6217
0.7811
0.4381
1.3038
0.8252
1.1767
0.4853
2.7626
1.2104
2.3325
0.5357
30.134
1.7004
20.819
10.344
0.9
0.8375
0.6030
0.8089
0.4231
1.3038
0.7903
1.2391
0.4613
2.7626
1.1348
2.5418
0.5204
30.134
1.5038
25.254
13.470
1.0
0.8375
0.5855
0.8375
0.4088
1.3038
0.7587
1.3038
0.4395
2.7626
1.0688
2.7626
0.5166
30.134
1.3493
30.134
17.040
5
0.1
20.936
20.040
15.024
13.885
32.596
30.329
19.803
17.525
69.065
59.182
29.022
23.128
753.34
266.98
50.691
15.905
0.2
20.936
19.216
15.618
13.339
32.596
28.357
21.019
16.463
69.065
51.794
32.378
20.591
753.34
160.14
84.633
15.061
0.3
20.936
18.457
16.227
12.809
32.596
26.624
22.286
15.452
69.065
46.060
36.008
18.328
753.34
111.36
129.61
25.249
0.4
20.936
17.755
16.852
12.295
32.596
25.091
23.605
14.493
69.065
41.478
39.911
16.337
753.34
83.415
185.61
46.470
0.5
20..936
17.104
17.493
11.797
32.596
23.724
24.975
13.586
69.065
37.727
44.087
14.620
753.34
65.445
252.65
78.724
0.6
20.936
16.498
18.150
11.315
32.596
22.498
26.396
12.729
69.065
34.597
48.536
14.176
753.34
53.028
330.73
122.01
0.7
20.936
15.933
18.823
10.848
32.596
21.392
27.869
11.924
69.065
31.943
53.258
12.005
753.34
44.012
419.83
176.33
0.8
20.936
15.405
19.512
10.398
32.596
20.389
29.393
11.171
69.065
29.662
58.254
11.107
753.34
37.220
519.97
241.68
0.9
20.936
14.911
20.216
9.9630
32.596
19.474
30.969
10.469
69.065
27.679
63.523
10.482
753.34
31.956
631.14
318.06
1.0
20.936
14.447
20.936
9.5441
32.596
18.638
32.596
9.8179
69.065
25.938
69.065
10.131
753.34
27.782
753.34
405.48
10
0.1
83.746
80.157
60.064
55.497
130.38
121.31
79.156
70.028
276.26
236.71
115.97
92.358
3013.4
1067.6
201.97
62.631
0.2
83.746
76.859
62.443
53.307
130.38
113.42
84.028
65.771
276.26
207.15
129.41
82.190
3013.4
640.15
337.83
59.156
0.3
83.746
73.819
64.885
51.181
130.38
106.48
89.105
61.719
276.26
184.20
143.95
73.113
3013.4
444.94
517.82
99.809
0.4
83.746
71.006
67.390
49.118
130.38
100.34
94.387
57.872
276.26
165.85
159.57
65.129
3013.4
333.07
741.94
184.59
0.5
83.746
68.397
69.958
47.119
130.38
94.868
99.874
54.230
276.26
150.84
176.29
58.236
3013.4
261.12
1010.2
313.50
0.6
83.746
65.970
72.589
45.182
130.38
89.957
105.57
50.793
276.26
138.30
194.10
52.434
3013.4
211.39
1322.6
486.54
0.7
83.746
63.707
75.284
43.309
130.38
85.526
111.46
47.561
276.26
127.67
213.00
47.725
3013.4
175.28
1679.1
703.71
0.8
83.746
61.592
78.041
41.498
130.38
81.506
117.56
44.534
276.26
118.54
233.00
44.107
3013.4
148.06
2079.7
965.01
0.9
83.746
59.610
80.862
39.751
130.38
77.842
123.87
41.712
276.26
110.59
254.08
41.581
3013.4
126.97
2524.5
1270.4
1.0
83.746
57.749
83.746
38.067
130.38
74.490
130.38
39.095
276.26
103.62
276.26
40.147
3013.4
110.24
3013.4
1620.0
0.7
0.8
0.9
0.99
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
1
0.1
0.4876
0.4746
0.3942
0.3756
0.7704
0.7350
0.5473
0.5057
1.6417
1.4749
0.8662
0.7433
17.673
7.9846
1.9910
1.0357
0.2
0.4876
0.4624
0.4038
0.3666
0.7704
0.7029
0.5693
0.4859
1.6417
1.3411
0.9343
0.6886
17.673
5.3441
2.8135
0.9031
0.3
0.4876
0.4508
0.4136
0.3578
0.7704
0.6738
0.5919
0.4669
1.6417
1.2314
1.0070
0.6384
17.673
4.0252
3.8661
1.0004
0.4
0.4876
0.4399
0.4236
0.3492
0.7704
0.6471
0.6152
0.4485
1.6417
1.1399
1.0842
0.5927
17.673
3.2147
5.1486
1.3276
0.5
0.4876
0.4295
0.4338
0.3408
0.7704
0.6227
0.6393
0.4309
1.6417
1.0623
1.1659
0.5515
17.673
2.6636
6.6610
1.8849
0.6
0.4876
0.4196
0.4441
0.3326
0.7704
0.6003
0.6641
0.4140
1.6417
0.9957
1.2520
0.5148
17.673
2.2653
8.4035
2.6720
0.7
0.4876
0.4103
0.4547
0.3246
0.7704
0.5796
0.6896
0.3979
1.6417
0.9379
1.3427
0.4826
17.673
1.9654
10.376
3.6892
0.8
0.4876
0.4014
0.4655
0.3167
0.7704
0.5604
0.7158
0.3824
1.6417
0.8873
1.4379
0.4549
17.673
1.7327
12.578
4.9363
0.9
0.4876
0.3929
0.4764
0.3091
0.7704
0.5427
0.7428
0.3677
1.6417
0.8424
1.5375
0.4317
17.673
1.5479
15.011
6.4134
1.0
0.4876
0.3849
0.4876
0.3016
0.7704
0.5261
0.7704
0.3537
1.6417
0.8025
1.6417
0.4130
17.673
1.3984
17.673
8.1205
5
0.1
12.190
11.866
9.8393
9.3648
19.261
18.376
13.650
12.589
41.042
36.874
21.539
18.403
441.82
199.51
47.225
22.001
0.2
12.190
11.560
10.083
9.1341
19.261
17.573
14.206
12.084
41.042
33.525
23.270
16.999
441.82
133.20
68.355
17.906
0.3
12.190
11.270
10.331
8.9077
19.261
16.842
14.779
11.595
41.042
30.775
25.110
15.704
441.82
99.857
95.164
19.490
0.4
12.190
10.994
10.584
8.6856
19.261
16.174
15.369
11.124
41.042
28.475
27.059
14.517
441.82
79.213
127.65
26.753
0.5
12.190
10.733
10.841
8.4680
19.261
15.560
15.975
10.669
41.042
26.522
29.117
13.440
441.82
65.056
165.82
39.694
0.6
12.190
10.733
10.841
8.4679
19.261
14.994
16.598
10.232
41.042
24.840
31.284
12.471
441.82
54.734
209.66
58.313
0.7
12.190
10.733
10.841
8.4680
19.261
14.470
17.239
9.8108
41.042
23.375
33.560
11.612
441.82
46.888
259.18
82.611
0.8
12.190
10.484
11.102
8.2545
19.261
13.985
17.896
9.4068
41.042
22.086
35.945
10.861
441.82
40.740
314.38
112.59
0.9
12.190
10.248
11.367
8.0455
19.261
13.533
18.570
9.0197
41.042
20.941
38.439
10.220
441.82
35.807
375.26
148.24
1.0
12.190
9.5999
12.190
7.4446
19.261
13.111
19.261
8.6495
41.042
19.917
41.042
9.6868
441.82
31.774
441.82
189.58
10
0.1
48.759
47.464
39.357
37.458
77.044
73.503
54.600
50.354
164.17
147.50
86.148
73.599
1767.3
798.06
188.64
87.571
0.2
48.759
46.239
40.333
36.535
77.044
70.294
56.824
48.331
164.17
134.10
93.076
67.978
1767.3
532.77
273.23
71.092
0.3
48.759
45.079
41.325
35.629
77.044
67.370
59.116
46.377
164.17
123.10
100.44
62.791
1767.3
399.37
380.53
77.317
0.4
48.759
43.978
42.335
34.740
77.044
64.696
61.475
44.489
164.17
113.90
108.24
58.041
1767.3
316.75
510.52
106.245
0.5
48.759
42.933
43.363
33.868
77.044
62.240
63.901
42.669
164.17
106.09
116.47
53.725
1767.3
260.08
663.22
157.88
0.6
48.759
41.938
44.407
33.014
77.044
59.976
66.395
40.917
164.17
99.359
125.14
49.845
1767.3
218.75
838.63
232.21
0.7
48.759
40.991
45.469
32.177
77.044
57.882
68.956
39.231
164.17
93.497
134.24
46.400
1767.3
187.32
1036.7
329.25
0.8
48.759
40.087
46.548
31.357
77.044
55.939
71.584
37.613
164.17
88.338
143.78
43.390
1767.3
162.69
1257.6
449.00
0.9
48.759
39.224
47.645
30.555
77.044
54.131
74.280
36.063
164.17
83.757
153.76
40.816
1767.3
142.92
1501.1
591.44
1.0
48.759
38.399
48.759
29.770
77.044
52.443
77.044
34.580
164.17
79.656
164.17
38.677
1767.3
126.75
1767.3
756.59
0.7
0.8
0.9
0.99
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
1
0.1
0.2225
0.2195
0.1989
0.1942
0.3446
0.3366
0.2865
0.2751
0.7153
0.6783
0.4938
0.4543
7.4001
4.6647
1.6335
1.1431
0.2
0.2225
0.2165
0.2014
0.1919
0.3446
0.3290
0.2924
0.2697
0.7153
0.6453
0.5149
0.4358
7.4001
3.4961
1.9866
1.0057
0.3
0.2225
0.2137
0.2038
0.1896
0.3446
0.3218
0.2985
0.2643
0.7153
0.6158
0.5369
0.4182
7.4001
2.8469
2.4116
0.9403
0.4
0.2225
0.2110
0.2064
0.1874
0.3446
0.3150
0.3046
0.2591
0.7153
0.5892
0.5597
0.4015
7.4001
2.4268
2.9085
0.9468
0.5
0.2225
0.2084
0.2090
0.1852
0.3446
0.3085
0.3110
0.2540
0.7153
0.5652
0.5835
0.3857
7.4001
2.1278
3.4773
1.0252
0.6
0.2225
0.2058
0.2116
0.1831
0.3446
0.3024
0.3174
0.2491
0.7153
0.5434
0.6081
0.3707
7.4001
1.9014
4.118
1.1755
0.7
0.2225
0.2034
0.2142
0.1810
0.3446
0.2966
0.3240
0.2443
0.7153
0.5236
0.6336
0.3566
7.4001
1.7224
4.8307
1.3977
0.8
0.2225
0.2010
0.2169
0.1789
0.3446
0.2910
0.3308
0.2397
0.7153
0.5055
0.6599
0.3434
7.4001
1.5767
5.6152
1.6918
0.9
0.2225
0.1987
0.2197
0.1769
0.3446
0.2858
0.3376
0.2352
0.7153
0.4889
0.6872
0.3311
7.4001
1.4552
6.4717
2.0578
1.0
0.2225
0.1965
0.2225
0.1750
0.3446
0.2808
0.3446
0.2308
0.7153
0.4736
0.7153
0.3197
7.4001
1.3523
7.4001
2.4958
5
0.1
5.5620
5.4869
4.9618
4.8354
8.6161
8.4163
7.1380
6.8372
17.882
16.960
12.260
11.220
185.00
116.57
39.235
26.109
0.2
5.5620
5.4139
5.0260
4.7731
8.6161
8.2263
7.2921
6.6906
17.882
16.134
12.809
10.728
185.00
87.159
48.425
22.173
0.3
5.5620
5.3430
5.0908
4.7115
8.6161
8.0453
7.4488
6.5466
17.882
15.390
13.377
10.256
185.00
70.707
59.367
19.988
0.4
5.5620
5.2741
5.1562
4.6505
8.6161
7.8728
7.6980
6.4050
17.882
14.717
13.963
9.8020
185.00
59.977
72.060
19.556
0.5
5.5620
5.2071
5.2223
4.5901
8.6161
7.7081
7.7697
6.2660
17.882
14.105
15.687
9.3674
185.00
52.275
86.505
20.874
0.6
5.5620
5.1419
5.2890
4.5304
8.6161
7.5508
7.9339
6.1294
17.882
13.546
15.193
8.9517
185.00
46.388
102.70
23.944
0.7
5.5620
5.0785
5.3563
4.4712
8.6161
7.4004
8.1007
5.9954
17.882
13.035
16.000
8.5551
185.00
41.691
120.65
28.766
0.8
5.5620
5.0169
5.4242
4.4128
8.6161
7.2565
8.2700
5.8640
17.882
12.565
16.500
8.1774
185.00
37.827
140.35
35.339
0.9
5.5620
4.9568
5.4928
4.3549
8.6161
7.1186
8.4418
5.7350
17.882
12.131
17.181
7.8188
185.00
34.577
161.80
43.664
1.0
5.5620
4.8983
5.5620
4.2977
8.6161
6.9863
8.6161
5.6086
17.882
11.731
17.882
7.4791
185.00
31.796
185.00
53.740
10
0.1
22.248
21.948
19.849
19.343
34.464
33.666
28.553
27.349
71.528
67.843
49.040
44.873
740.01
466.29
156.77
104.14
0.2
22.248
21.656
20.106
19.093
34.464
32.907
29.170
26.762
71.528
64.540
51.236
42.903
740.01
348.64
193.58
88.328
0.3
22.248
21.373
20.365
18.846
34.464
32.183
29.797
26.185
71.528
61.564
53.508
41.009
740.01
282.80
237.39
79.512
0.4
22.248
21.098
20.626
18.602
34.464
31493
30.434
25.617
71.528
58.870
55.856
39.190
740.01
239.86
288.20
77.693
0.5
22.248
20.830
20.891
18.360
34.464
30.835
31.081
25.060
71.528
56.422
58.279
37.447
740.01
209.02
346.00
82.874
0.6
22.248
20.569
21.157
18.120
34.464
30.206
31.738
24.513
71.528
54.187
60.777
35.779
740.01
185.44
410.81
95.052
0.7
22.248
20.316
21.426
17.883
34.464
29.604
32.405
23.975
71.528
52.141
63.352
34.187
740.01
166.62
482.61
114.23
0.8
22.248
20.069
21.698
17.649
34.464
29.028
33.081
23.448
71.528
50.259
66.001
32.671
740.01
151.14
561.41
140.41
0.9
22.248
19.829
21.972
17.416
34.464
28.476
33.768
22.930
71.528
48.524
68.727
31.230
740.01
138.11
647.21
173.58
1.0
22.248
19.594
22.248
17.187
34.464
27.946
34.464
22.422
71.528
46.919
71.528
29.864
740.01
126.96
740.01
213.75
0.7
0.8
0.9
0.99
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
1
0.1
0.0260
0.0259
0.0253
0.0252
0.0355
0.0354
0.0343
0.0341
0.0650
0.0645
0.0612
0.0604
0.5874
0.5532
0.3747
0.3374
0.2
0.0260
0.0258
0.0254
0.0251
0.0355
0.0352
0.0344
0.0339
0.0650
0.0641
0.0616
0.0600
0.5874
0.5221
0.3947
0.3201
0.3
0.0260
0.0257
0.0254
0.0250
0.0355
0.0351
0.0346
0.0338
0.0650
0.0636
0.0620
0.0597
0.5874
0.4940
0.4156
0.3038
0.4
0.0260
0.0257
0.0255
0.0250
0.0355
0.0350
0.0347
0.0337
0.0650
0.0632
0.0624
0.0593
0.5874
0.4685
0.4374
0.2883
0.5
0.0260
0.0256
0.0256
0.0249
0.0355
0.0348
0.0348
0.0336
0.0650
0.0628
0.0628
0.0589
0.5874
0.4452
0.4602
0.2737
0.6
0.0260
0.0255
0.0257
0.0248
0.0355
0.0347
0.0350
0.0335
0.0650
0.0624
0.0632
0.0586
0.5874
0.4240
0.4838
0.2601
0.7
0.0260
0.0254
0.0257
0.0248
0.0355
0.0346
0.0351
0.0334
0.0650
0.0620
0.0636
0.0582
0.5874
0.4046
0.5084
0.2473
0.8
0.0260
0.0254
0.0258
0.0247
0.0355
0.0344
0.0352
0.0333
0.0650
0.0616
0.0641
0.0579
0.5874
0.3868
0.5338
0.2355
0.9
0.0260
0.0253
0.0259
0.0247
0.0355
0.0343
0.0354
0.0331
0.0650
0.0612
0.0645
0.0576
0.5874
0.3705
0.5602
0.2246
1.0
0.0260
0.0252
0.0260
0.0246
0.0355
0.0342
0.0355
0.0330
0.0650
0.0608
0.0650
0.0572
0.5874
0.3556
0.5874
0.2145
5
0.1
0.6208
0.6195
0.6089
0.6063
0.8560
0.8533
0.8322
0.8270
1.5796
1.5699
1.4957
1.4774
14.6151
13.7762
9.1902
8.1593
0.2
0.6208
0.6181
0.6102
0.6050
0.8560
0.8506
0.8348
0.8244
1.5796
1.5603
1.5049
1.4684
14.6151
13.0082
9.7294
7.6677
0.3
0.6208
0.6168
0.6115
0.6037
0.8560
0.8479
0.8374
0.8218
1.5796
1.5507
1.5141
1.4593
14.6151
12.3033
10.2845
7.1920
0.4
0.6208
0.6155
0.6128
0.6024
0.8560
0.8453
0.8401
0.8192
1.5796
1.5413
1.5234
1.4503
14.6151
11.6549
10.8556
6.7321
0.5
0.6208
0.6141
0.6142
0.6011
0.8560
0.8426
0.8427
0.8166
1.5796
1.5319
1.5327
1.4413
14.6151
11.0570
11.4424
6.2882
0.6
0.6208
0.6128
0.6155
0.5998
0.8560
0.8400
0.8454
0.8140
1.5796
1.5227
1.5420
1.4324
14.6151
10.5045
12.0452
5.8601
0.7
0.6208
0.6115
0.6168
0.5985
0.8560
0.8373
0.8480
0.8114
1.5796
1.5135
1.5513
1.4234
14.6151
9.9929
12.6639
5.4479
0.8
0.6208
0.6102
0.6181
0.5972
0.8560
0.8347
0.8507
0.8088
1.5796
1.5045
1.5607
1.4146
14.6151
9.5184
13.2984
5.0515
0.9
0.6208
0.6089
0.6195
0.5959
0.8560
0.8321
0.8533
0.8063
1.5796
1.4955
1.5702
1.4057
14.6151
9.0774
13.9488
4.6711
1.0
0.6208
0.6076
0.6208
0.5946
0.8560
0.8296
0.8560
0.8037
1.5796
1.4866
1.5796
1.3969
14.6151
8.6668
14.6151
4.3065
10
0.1
2.4784
2.4731
2.4315
2.4211
3.4193
3.4086
3.3251
3.3044
6.3127
6.2741
5.9791
5.9063
58.4499
55.0988
36.7499
32.6162
0.2
2.4784
2.4678
2.4367
2.4159
3.4193
3.3980
3.3355
3.2940
6.3127
6.2359
6.0157
5.8701
58.4499
52.0297
38.9107
30.6432
0.3
2.4784
2.4626
2.4418
2.4108
3.4193
3.3874
3.3459
3.2837
6.3127
6.1980
6.0524
5.8340
58.4499
49.2118
41.1340
28.7328
0.4
2.4784
2.4573
2.4470
2.4056
3.4193
3.3768
3.3563
3.2734
6.3127
6.1605
6.0893
5.7980
58.4499
46.6183
43.4199
26.8850
0.5
2.4784
2.4521
2.4523
2.4005
3.4193
3.3664
3.3668
3.2631
6.3127
6.1233
6.1262
5.7621
58.4499
44.2260
45.7685
25.0998
0.6
2.4784
2.4469
2.4575
2.3953
3.4193
3.3559
3.3773
3.2529
6.3127
6.0865
6.1633
5.7263
58.4499
42.0146
48.1796
23.3772
0.7
2.4784
2.4417
2.4627
2.3902
3.4193
3.3456
3.3877
3.2426
6.3127
6.0500
6.2005
5.6907
58.4499
39.9662
50.6533
21.7172
0.8
2.4784
2.4366
2.4679
2.3851
3.4193
3.3352
3.3982
3.2324
6.3127
6.0139
6.2378
5.6552
58.4499
38.0652
53.1896
20.1197
0.9
2.4784
2.4314
2.4732
2.3800
3.4193
3.3250
3.4088
3.2222
6.3127
5.9781
6.2752
5.6198
58.4499
36.2978
55.7884
18.5849
1.0
2.4784
2.4263
2.4784
2.3748
3.4193
3.3147
3.4193
3.2120
6.3127
5.9427
6.3127
5.5845
58.4499
34.6518
58.4499
17.1126
3.2. Simulation Result Discussion
From Tables 1–8 and Figures 1–4, we observed that the modified Liu estimator consistently performs better than the Liu estimator and other existing estimators in this study.
0.7
0.8
0.9
0.99
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
OLS
Ridge
Liu
MLiu
1
0.1
0.0608
0.0607
0.0598
0.0595
0.0856
0.0854
0.0835
0.0830
0.1620
0.1611
0.1543
0.1527
1.5604
1.4782
1.0328
0.9358
0.2
0.0608
0.0606
0.0599
0.0594
0.0856
0.0851
0.0837
0.0828
0.1620
0.1602
0.1552
0.1519
1.5604
1.4031
1.0840
0.8901
0.3
0.0608
0.0605
0.0600
0.0593
0.0856
0.0849
0.0839
0.0825
0.1620
0.1593
0.1560
0.1511
1.5604
1.3343
1.1371
0.8463
0.4
0.0608
0.0603
0.0601
0.0592
0.0856
0.0846
0.0842
0.0823
0.1620
0.1585
0.1568
0.1504
1.5604
1.2712
1.1921
0.8042
0.5
0.0608
0.0602
0.0602
0.0591
0.0856
0.0844
0.0844
0.0821
0.1620
0.1576
0.1577
0.1496
1.5604
1.2132
1.2488
0.7641
0.6
0.0608
0.0601
0.0603
0.0590
0.0856
0.0841
0.0846
0.0819
0.1620
0.1568
0.1585
0.1488
1.5604
1.1598
1.3075
0.7257
0.7
0.0608
0.0600
0.0605
0.0589
0.0856
0.0839
0.0849
0.0817
0.1620
0.1559
0.1594
0.1481
1.5604
1.1105
1.3679
0.6892
0.8
0.0608
0.0599
0.0606
0.0588
0.0856
0.0837
0.0851
0.0815
0.1620
0.1551
0.1602
0.1473
1.5604
1.0649
1.4302
0.6546
0.9
0.0608
0.0598
0.0607
0.0587
0.0856
0.0834
0.0854
0.0812
0.1620
0.1543
0.1611
0.1466
1.5604
1.0226
1.4944
0.6218
1.0
0.0608
0.0596
0.0608
0.0586
0.0856
0.0832
0.0856
0.0810
0.1620
0.1535
0.1620
0.1459
1.5604
0.9834
1.5604
0.5908
5
0.1
1.5209
1.5177
1.4931
1.4869
2.1402
2.1338
2.0844
2.0722
4.0504
4.0274
3.8518
3.8085
39.0100
36.9446
25.4493
22.8472
0.2
1.5209
1.5146
1.4961
1.4839
2.1402
2.1275
2.0906
2.0660
4.0504
4.0047
3.8736
3.7869
39.0100
35.0465
26.8065
21.6023
0.3
1.5209
1.5115
1.4992
1.4808
2.1402
2.1213
2.0967
2.0599
4.0504
3.9822
3.8954
3.7654
39.0100
33.2972
28.2010
20.3948
0.4
1.5209
1.5084
1.5023
1.4778
2.1402
2.1150
2.1029
2.0539
4.0504
3.9598
3.9174
3.7440
39.0100
31.6813
29.6329
19.2247
0.5
1.5209
1.5053
1.5054
1.4747
2.1402
2.1088
2.1091
2.0478
4.0504
3.9377
3.9393
3.7226
39.0100
30.1850
31.1023
18.0919
0.6
1.5209
1.5022
1.5085
1.4717
2.1402
2.1027
2.1153
2.0417
4.0504
3.9158
3.9614
3.7013
39.0100
28.7964
32.6090
16.9966
0.7
1.5209
1.4991
1.5116
1.4686
2.1402
2.0965
2.1215
2.0357
4.0504
3.8941
3.9835
3.6801
39.0100
27.5052
34.1531
15.9387
0.8
1.5209
1.4961
1.5147
1.4656
2.1402
2.0904
2.1277
2.0296
4.0504
3.8726
4.0057
3.6590
39.0100
26.3022
35.7347
14.9181
0.9
1.5209
1.4930
1.5178
1.4626
2.1402
2.0843
2.1339
2.0236
4.0504
3.8512
4.0280
3.6379
39.0100
25.1794
37.3536
13.9350
1.0
1.5209
1.4900
1.5209
1.4595
2.1402
2.0783
2.1402
2.0175
4.0504
3.8301
4.0504
3.6169
39.0100
24.1296
39.0100
12.9893
10
0.1
6.0835
6.0709
5.9721
5.9475
8.5606
8.5353
8.3373
8.2882
16.2015
16.1096
15.4061
15.2323
156.0399
147.7764
101.7462
91.3151
0.2
6.0835
6.0584
5.9844
5.9352
8.5606
8.5101
8.3620
8.2636
16.2015
16.0186
15.4934
15.1457
156.0399
140.1802
107.1845
86.3224
0.3
6.0835
6.0459
5.9967
5.9230
8.5606
8.4850
8.3867
8.2392
16.2015
15.9284
15.5810
15.0595
156.0399
133.1789
112.7715
81.4783
0.4
6.0835
6.0335
6.0091
5.9107
8.5606
8.4600
8.4114
8.2147
16.2015
15.8390
15.6688
14.9735
156.0399
126.7096
118.5070
76.7827
0.5
6.0835
6.0211
6.0214
5.8985
8.5606
8.4352
8.4362
8.1903
16.2015
15.7504
15.7569
14.8878
156.0399
120.7180
124.3910
72.2358
0.6
6.0835
6.0088
6.0338
5.8863
8.5606
8.4105
8.4610
8.1659
16.2015
15.6626
15.8453
14.8024
156.0399
115.1568
130.4237
67.8373
0.7
6.0835
5.9965
6.0462
5.8741
8.5606
8.3859
8.4858
8.1416
16.2015
15.5756
15.9339
14.7172
156.0399
109.9844
136.6049
63.5875
0.8
6.0835
5.9842
6.0586
5.8619
8.5606
8.3614
8.5107
8.1173
16.2015
15.4893
16.0229
14.6323
156.0399
105.1643
142.9346
59.4862
0.9
6.0835
5.9720
6.0710
5.8497
8.5606
8.3370
8.5356
8.0931
16.2015
15.4038
16.1120
14.5476
156.0399
100.6645
149.4130
55.5334
1.0
6.0835
5.9598
6.0835
5.8375
8.5606
8.3127
8.5606
8.0689
16.2015
15.3190
16.2015
14.4633
156.0399
96.4563
156.0399
51.7293
Results from Tables 1–8 show that increasing the sample size results in a decrease in the MSE values for each of the estimator. It is evident that MSE values of the estimators increase as the degree of correlation and the number of explanatory variables increase. The simulation results show that the proposed estimator performed best at most levels of the multicollinearity, sample size (), and number of explanatory variables with few exceptions. The only exceptions to its performance are when, and they are defined as follows:
(i),,, and for , 5, and 10(ii),, and for and 10(iii),, and and 0.9 when (iv),,, and for , 5, and 10(v),,, and for , 5, and 10(vi) and for at , 5, and 10 including and 0.9 at
The instances mentioned above are the only times that ridge regression dominates the proposed estimator. The new estimator consistently dominates the OLS and the Liu estimator. Also, from Tables 1–8, we observed that the values of OLS and Liu are the same when .
Consistently when , 0.8, and 0.9, the proposed estimator performs better than other estimators at the different sample sizes irrespective of the values of the biasing parameter and . The fact that the ridge estimator dominates the proposed estimator in some of the exceptions mentioned earlier does not show that it performs better. It only shows that at those intervals, the performance of the new estimator drops. Thus, this necessitates the use of real-life data in the next session because the values of and will be estimated rather than choosing it arbitrarily.
4. Applications
We adopt two datasets to illustrate the theoretical findings of the paper. These include the Portland cement data and the French economy data.
4.1. Portland Dataset
The first user of this dataset was Woods et al. [16] and later adopted by Kaciranlar et al. [17] and Li and Yang [18]. It consists of one predictand, , which is the heat evolved after 180 days of curing measured in calories per gram of cement, and four predictors: X_{1} is the tricalcium aluminate, X_{2} is the tricalcium silicate, X_{3} is the tetracalcium aluminoferrite, and X_{4} is the β-dicalcium silicate. Variance inflation factors (VIFs) and a condition number are adopted to diagnose multicollinearity in the model [19]. The VIFs are 38.50, 254.42, 46.87, and 282.51 while the condition number is approximately 424. Both tests are evidence that the model possesses severe multicollinearity. The regression output is available in Table 9. According to Kejian [3], the optimum biasing parameter is expressed as
Coef.
2.1930
2.1478
2.1695
2.1930
2.1425
1.1533
1.1638
1.1587
1.1533
1.1650
0.7585
0.7261
0.7416
0.7583
0.7221
0.4863
0.4931
0.4899
0.4864
0.4939
MSE
0.0638
0.0636
0.0629
0.0617
0.0608
MSPE
2.8966
2.2665
2.8805
2.1342
1.2789
—
3.3636
1.7256
0.9782
2.6817
Following Özkale and Kaçiranlar [4], we replaced with if .
The ridge biasing parameters are computed by
We also adopt the leave-one-out crossvalidation to validate the performance of the estimators (see [20]). The performance of the estimator is assessed through the mean squared prediction error (MSPE). The result is presented in Table 9.
The theoretical results are computed as follows:
We observed that the theoretical comparisons stated in Sections 2.1, 2.2, and 2.3 are valid since each of the estimates are less than 1. From Table 9, the regression coefficients and MSE of the OLS and Liu estimators are approximately the same because is close to unity. Recall that the Liu estimator becomes OLS when . The proposed estimator possesses the smallest mean square error and average MSE of the validation error. Also, the performances of the estimators largely depend on their biasing parameters.
4.2. French Economy Dataset
The detail about this dataset is initially described in Chatterjee and Price [21] and is later available in the following references Malinvard [22] and Kejian [3]. It comprises one predictand, imports, and three predictor variables (domestic production, stock information, and domestic consumption) with eighteen observations. The variance inflation factors are ,, and and the condition number 32612. Both results indicate the presence of severe multicollinearity. We analyzed the data using the biasing parameters for each of the estimators and present the results in Table 10. The proposed estimator performed the best in the sense of smaller MSE and MSPE. As mentioned earlier, the estimators’ performance is a function of the biasing parameter. The proposed estimator with the biasing parameter performs best. Also, the theoretical results agree with Section 2 and support the simulation and real-life findings.
Coef.
-19.7127
-16.7613
-18.8782
-18.8410
-19.2613
0.0327
0.1419
0.0636
0.0648
0.0493
0.4059
0.3576
0.3922
0.3914
0.3984
0.2421
0.0709
0.1937
0.1918
0.2161
MSE
17.3326
21.30519
16.6017
16.6029
4.9448
MSPE
0.8267
0.4371
0.1411
0.1302
0.0871
—
0.0527
0.0132
0.9424
0.9701
The theoretical results are computed as follows:
5. Some Concluding Remarks
Both ridge regression and Liu estimators are widely accepted in the linear regression model as an alternative to the OLS estimator to circumvent the problem of multicollinearity. In this study, we proposed a modified Liu estimator, which possesses a single parameter which places it in the class of the ridge and Liu estimators. Theoretical comparisons, simulation study, and real-life applications evidently show that the proposed estimator consistently dominates the existing Liu estimator and ridge regression estimator under some conditions. We recommend the use of this estimator for the linear regression model with multicollinearity problem. We note that the proposed estimator can be extended to other regression models, for example, logistic regression, Poisson, ZIP, gamma, and related models, and these possibilities are under current investigation.
Data Availability
Data will be made available on request.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
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