1 Introduction
The maximum likelihood estimation technique is commonly used method to estimate the parameters of the logistic regression model. The multicollinearity severely affects the varinace of the estimates of parameters in the logistic regression model. As a result model produces inefficient estimates. To overcome this issue, several alternative estimators have been proposed in the literature. These estimators were introduced mainly based on two types. The first type of estimators are based only on the sample information and the second type of estimators are based on the sample and priori available information which may be in the form of exact or stochastic linear restrictions. Logistic Ridge Estimator (LRE) by Schaefer et al. (1984), the Principal Component Logistic Estimator (PCLE) by Aguilera et al. (2006), the Modified Logistic Ridge Estimator (MLRE) by Nja et al. (2013), the Logistic Liu Estimator (LLE) by Mansson et al. (2012), the LiuType Logistic Estimator (LTLE) by Inan and Erdogan (2013), the Almost Unbiased Ridge Logistic Estimator (AURLE) by Wu and Asar (2016), the Almost Unbiased Liu Logistic Estimator (AULLE) by Xinfeng (2015), and the Optimal Generalized Logistic Estimator (OGLE) by Varathan and Wijekoon (2017) are some of the first type of estimators proposed in the literature. Under the second type of estimators, the Restricted Maximum Likelihood Estimator (RMLE) by Duffy and Santner (1989), the Restricted Logistic Liu Estimator (RLLE) by S iray et al. (2015), the Modified Restricted Liu Estimator by Wu (2015), the Restricted Logistic Ridge Estimator (RLRE) by Asar et al. (2016a), the Restricted LiuType Logistic Estimator (RLTLE) by Asar et al. (2016b), and the Restricted Almost Unbiased Ridge logistic Estimator (RAURLE) by Varathan and Wijekoon (2016a) were introduced to improve the performance of the logistic model when the exact linear restrictions are available in addition to sample model. When the restrictions on the parameters are stochastic, the Stochastic Restricted Maximum Likelihood Estimator (SRMLE) (Nagarajah and Wijekoon, 2015), the Stochastic Restricted Ridge Maximum Likelihood Estimator (SRRMLE) (Varathan and Wijekoon, 2016b), and the Stochastic Restricted Liu Maximum Likelihood Estimator (SRLMLE) (Varathan and Wijekoon, 2016c) were proposed in the literature. In this research, following Xinfeng (2015), a new estimator namely, Stochastic restricted almost unbiased logistic Liu Estimator (SRAULLE) is proposed for the logistic regression model with the presence of stochastic linear restrictions as prior information. The organization of the paper is as follows. The model specification and estimation are given in Section 2. Proposed estimators and their asymptotic properties are discussed in Section 3. In Section 4, the conditions for superiority of SRAULLE over some existing estimators are derived with respect to mean square error (MSE) criterion. A Monte Carlo simulation study is conducted to investigate the performance of the proposed estimator in the scalar mean squared error (SMSE) sense in Section 5. A numerical example is discussed in Section 6. Finally, some conclusive remarks are given in Section 7.
2 Model Specification and estimation
Consider the logistic regression model
(2.1) 
which follows Binary distribution with parameter as
(2.2) 
where is the row of , which is an data matrix with explanatory variables and is a vector of coefficients,
are independent with mean zero and variance
of the response . The Maximum likelihood estimate (MLE) of can be obtained as follows:(2.3) 
where ; is the column vector with element equals and . Note that
is an unbiased estimate of
and its covariance matrix is(2.4) 
The MSE and SMSE of are
and
When the multicollinearity presents in the logistic regression model (2.1), many alternative estimators have been proposed in the literature. Among those, in this research we consider the Logistic Liu estimator (LLE) by Mansson et al., (2012) and the Almost Unbiased Logistic Liu Estimator (AULLE) by Xinfeng (2015) under the first type of estimators.
(2.7)  
(2.8) 
The asymptotic properties of LLE:
Consequently, the bias vector and the mean square error matrix of LLE are obtained as
and
respectively.
The asymptotic properties of AULLE:
Then, the bias vector and the mean square error matrix of AULLE are obtained as
and
respectively.
As an alternative technique to stabilize the variance of the estimator due to multicollinearity, one can use prior information, if available, in addition to the sample model (2.1) either as exact linear restrictions or stochastic linear restrictions.
Suppose that the following stochastic linear prior information is
given in addition to the general logistic regression model (2.1).
(2.17) 
where is an stochastic known vector, is a of full rank known elements and is an random vector of disturbances with mean 0 and dispersion matrix , which is assumed to be known positive definite matrix. Further, it is assumed that is stochastically independent of , i.e) .
In the presence of stochastic linear restrictions (2.17) in addition to the logistic regression model (2.1), Nagarajah and Wijekoon (2015) introduced the Stochastic Restricted Maximum Likelihood Estimator (SRMLE).
(2.18) 
The asymptotic properties of SRMLE:
(2.19) 
and
(2.21)  
The MSE of SRMLE is
3 The Proposed Estimator
In this section, by replacing by in (2.8), we propose a new estimator which is called as the Stochastic restricted almost unbiased logistic Liu Estimator (SRAULLE) and defined as
(3.1) 
where .
The asymptotic properties of are
and
Consequently, the mean square error can be obtained as,
4 Mean square error comparisons
When different estimators are available for the same parameter vector in the regression model one must solve the problem of their comparison. Usually, as a general measure, the mean square error matrix is used, and is defined by
where is the dispersion matrix, and denotes the bias vector.
The Scalar Mean Squared Error (SMSE) of the estimator can be defined as
(4.2) 
For two given estimators and , the estimator is said to be superior to under the MSE criterion if and only if
(4.3) 
4.1 Comparison of SRAULLE with MLE
To compare the estimators and , we consider their MSE differences as below:
where and . One can obviously say that and are positive definite matrices and is nonnegative definite matrix. Further by Lemma 1 (see Appendix A), it is clear that is positive definite matrix. By Lemma 2 (see Appendix A), if , then is a positive definite matrix, where is the largest eigen value of . Based on the above arguments, it can be concluded that the estimator SRAULLE is superior to MLE if and only if .
4.2 Comparison of SRAULLE with LLE
Consider the MSE differences of and
where and . One can easily say that and are positive definite matrices and and are nonnegative definite matrices. Further by Lemma 1, it is clear that and are positive definite matrices. By Lemma 2, if , then is a positive definite matrix, where is the largest eigen value of . Based on the above results, it can be said that the estimator SRAULLE is superior to LLE if and only if .
4.3 Comparison of SRAULLE with AULLE
Consider the MSE differences of and
Since the above mean square error difference is positive definite, it can be concluded that SRAULLE is always superior than AULLE.
4.4 Comparison of SRAULLE with SRMLE
Consider the MSE differences of and
where and . It can be easily seen that
and
are positive definite matrices and is nonnegative definite matrix. Further by Lemma 1, it is clear that
is positive definite matrix. By Lemma 2, if
, then is a
positive definite
matrix, where is the largest eigen value of . Based on the above results, it can be
said
that the estimator SRAULLE is superior to SRMLE if and only if .
According to the results obtained from above mean square error
comparisons it can be concluded that the proposed estimator SRAULLE
is always superior than AULLE. However, under certain conditions
SRAULLE performs well over MLE, LLE, and SRMLE with respect to the
mean square error sense.
5 A Simulation study
To examine the performance of the proposed estimator; SRAULLE with the existing estimators: MLE, LLE, AULLE and SRMLE in this section, we conduct the Monte Carlo simulation study. The simulations are based on different levels of multicollinearity; , 0.8, 0.9 and 0.99 and different sample sizes; , 50, 75 and 100. The Scalar Mean Square Error (SMSE) is considered for the comparison. Following McDonald and Galarneau (1975) and Kibria (2003), the explanatory variables are generated as follows:
(5.1) 
where are independent standard normal pseudo random numbers and is specified so that the theoretical correlation between any two explanatory variables is given by . Four explanatory variables are generated using (5.1).
The dependent variable in (2.1) is obtained from the Bernoulli() distribution where . The parameter values of are chosen so that and .
Following Asar et al. (2016b), Wu and Asar (2015) and Mansson et al. (2012), the optimum value of the biasing parameter can be obtained by minimizing SMSE value with respect to . However, for simplicity in this paper we consider some selected values of in the range .
Moreover, we consider the following restrictions.
(5.2) 
The simulation is repeated 1000 times by generating new pseudo random numbers and the simulated SMSE values of the estimators are obtained using the following equation.
(5.3) 
where is any estimator considered in the simulation. The simulation results are displayed in Tables 5.1  5.3 (Appendix). As we observed from the theoretical results, the proposed estimator SRAULLE is superior to AULLE in the mean square error sense with respect to all the sample sizes , 50, 75 , and 100 and all the , 0.8, 0.9, and 0.99. From the Tables 5.1 5.3, it is further noted that if the multicollinearity is very high (for example ) the proposed estimator SRAULLE is a very good alternative to MLE, LLE, AULLE and SRMLE regardless of the values of and . However, the performance of LLE is considerably good for very small values and moderate values. Moreover, as we expected, MLE has the worst performance in all of the cases (having the largest SMSE values).
6 Numerical example
In order to check the performance of the new estimator SRAULLE, in this section, we used a real data set, which is taken from the Statistics Sweden website (http://www.scb.se/). The data consists the information about 100 municipalities of Sweden. The explanatory variables considered in this study are Population (), Number unemployed people (), Number of newly constructed buildings (), and Number of bankrupt firms (). The variable Net population change (
) is considered as response variable, which is defined as
The correlation matrix of the explanatory variables , , , and is displayed in Table 6.1. It can be noticed from the Table 6.1 that, all the pair wise correlations are very high (greater than 0.95). Hence a clear high multicollinearity exists in this data set. Further, the condition number being a measure of multicollinearity is obtained as 188 showing that there exists severe multicollinearity with this data set. Moreover, we use the same restrictions as in (5.2) for the prior information.
The SMSE values of MLE, LLE, AULLE, SRMLE, and SRAULLE for some selected values of biasing parameter in the range are given in the Table 6.2.
It can be clearly noticed from the Table 6.2 that the proposed estimator SRAULLE outperforms the estimators MLE, LLE, AULLE, and SRMLE in the SMSE sense, with respect to all the selected values of biasing parameter in the range except . Further, SRAULLE is having better performance compared to AULLE for all the values of .
Table 6.1: The correlation matrix of the explanatory variables
1.000  0.998  0.971  0.970  
0.998  1.000  0.960  0.958  
0.971  0.960  1.000  0.987  
0.970  0.958  0.987  1.000 
7 Concluding Remarks
In this research, we proposed the Stochastic restricted almost
unbiased logistic Liu estimator (SRAULLE) for logistic regression
model in the presence of linear stochastic restriction when the
multicollinearity problem exists. The conditions for superiority of
the proposed estimator over some existing estimators were derived
with respect to MSE criterion. Further, a numerical example and a
Monte Carlo simulation study were done to illustrate the theoretical
findings. Results reveal that the proposed estimator is always
superior to AULLE in the mean square error sense and it can be a
better alternative to the other existing estimators under certain
conditions.
References

Aguilera, A. M., Escabias, M., Valderrama, M. J., (2006). Using principal components for estimating logistic regression with highdimensional multicollinear data. Computational Statistics & Data Analysis 50: 19051924.

Asar, Y., Arashi, M., Wu, J.,(2016a). Restricted ridge estimator in the logistic regression model. Commun. Statist. Simmu. Comp.. Online. DOI: 10.1080/03610918.2016.1206932

Asar, Y., Erişoǧlu, M., Arashi, M., (2016b). Developing a restricted twoparameter Liutype estimator: A comparison of restricted estimators in the binary logistic regression model. Commun. Statist. Theor. Meth. Online. DOI: 10.1080/03610926.2015.1137597

Duffy, D. E., Santner, T. J., (1989). On the small sample prosperities of normrestricted maximum likelihood estimators for logistic regression models. Commun. Statist. Theor. Meth. 18: 959980.

Inan, D., Erdogan, B. E., (2013). LiuType logistic estimator. Communications in Statistics Simulation and Computation. 42: 15781586.

Kibria, B. M. G., (2003). Performance of some new ridge regression estimators.
Commun. Statist. Theor. Meth. 32: 419435. 
Mansson, G., Kibria, B. M. G., Shukur, G., (2012). On Liu estimators for the logit regression model.
The Royal Institute of Techonology, Centre of Excellence for Science and Innovation Studies (CESIS), Sweden, Paper No. 259. 
McDonald, G. C., and Galarneau, D. I., (1975). A Monte Carlo evaluation of some ridge type estimators. Journal of the American Statistical Association 70: 407416.

Nja, M. E., Ogoke, U. P., Nduka, E. C., (2013). The logistic regression model with a modified weight function. Journal of Statistical and Econometric Method Vol.2, No. 4: 161171.

Rao, C. R., Toutenburg, H., Shalabh and Heumann, C., (2008). Linear Models and Generalizations.Springer. Berlin.

Rao, C. R.,and Toutenburg, H.,(1995). Linear Models :Least Squares and Alternatives, Second Edition. SpringerVerlag New York, Inc.

Schaefer, R. L., Roi, L. D., Wolfe, R. A., (1984). A ridge logistic estimator. Commun. Statist. Theor. Meth. 13: 99113.

Şiray, G. U., Toker, S., Kaçiranlar, S., (2015).On the restricted Liu estimator in logistic regression model. Communicationsin Statistics Simulation and Computation 44: 217232.

Nagarajah, V., Wijekoon, P., (2015). Stochastic Restricted Maximum Likelihood Estimator in Logistic Regression Model. Open Journal of Statistics. 5, 837851. DOI: 10.4236/ojs.2015.57082

Varathan, N., Wijekoon, P., (2016a). On the restricted almost unbiased ridge estimator in logistic regression. Open Journal of Statistics. 6, 10761084. DOI: 10.4236/ojs.2016.66087

Varathan, N., Wijekoon, P., (2016b). Ridge Estimator in Logistic Regression under stochastic linear restriction. British Journal of Mathematics & Computer Science. 15 (3), 1. DOI: 10.9734/BJMCS/2016/24585

Varathan, N., Wijekoon, P., (2016c). Logistic Liu Estimator under stochastic linear restrictions. Statistical Papers. Online. DOI: 10.1007/s0036201608566

Varathan, N., Wijekoon, P., (2017). Optimal Generalized Logistic Estimator, Communications in StatisticsTheory and Methods , DOI: 10.1080/03610926.2017.1307406

Wu, J., (2015). Modified restricted Liu estimator in logistic regression model. Computational Statistics. Online. DOI: 10.1007/s0018001506093

Wu, J., Asar, Y., (2016). On almost unbiased ridge logistic estimator for the logistic regression model. Hacettepe Journal of Mathematics and Statistics. 45(3), 989998.
DOI: 10.15672/HJMS.20156911030 
Wu, J., Asar, Y., (2015). More on the restricted Liu Estimator in the logistic regression model. Communications in Statistics Simulation and Computation. Online.
DOI: 10.1080/03610918.2015.1100735 
Xinfeng, C., (2015). On the almost unbiased ridge and Liu estimator in the logistic regression model.International Conference on Social Science, Education Management and Sports Education. Atlantis Press: 16631665.
Appendix
Lemma 1: Let : and : such that and . Then . (Rao and Toutenburg, 1995)
Lemma 2: Let the two matrices ,, then if and only if .
(Rao et al., 2008)
Table 5.1: The estimated MSE values for different when
d = 0.01
d = 0.1
d = 0.2
d = 0.3
d = 0.4
d = 0.5
d = 0.6
d = 0.7
d = 0.8
d = 0.9
d = 0.99
MLE
1.4798
1.4798
1.4798
1.4798
1.4798
1.4798
1.4798
1.4798
1.4798
1.4798
1.4798
LLE
0.8052
0.8600
0.9233
0.9893
1.0579
1.1291
1.2030
1.2794
1.3585
1.4402
1.5160
AULLE
1.2383
1.2780
1.3186
1.3552
1.3875
1.4153
1.4383
1.4564
1.4694
1.4772
1.4798
SRMLE
1.0138
1.0138
1.0138
1.0138
1.0138
1.0138
1.0138
1.0138
1.0138
1.0138
1.0138
SRAULLE
0.8536
0.8798
0.9067
0.9309
0.9524
0.9708
0.9861
0.9981
1.0068
1.0120
1.0137
MLE
1.9817
1.9817
1.9817
1.9817
1.9817
1.9817
1.9817
1.9817
1.9817
1.9817
1.9817
LLE
0.8781
0.9602
1.0566
1.1584
1.2658
1.3786
1.4968
1.6205
1.7497
1.8843
2.0101
AULLE
1.4912
1.5699
1.6513
1.7253
1.7912
1.8481
1.8955
1.9329
1.9599
1.9763
1.9817
SRMLE
1.1793
1.1793
1.1793
1.1793
1.1793
1.1793
1.1793
1.1793
1.1793
1.1793
1.1793
SRAULLE
0.9039
0.9479
0.9935
1.0350
1.0720
1.1041
1.1308
1.1518
1.1670
1.1762
1.1793
MLE
3.5707
3.5707
3.5707
3.5707
3.5707
3.5707
3.5707
3.5707
3.5707
3.5707
3.5707
LLE
0.9334
1.0954
1.2945
1.5137
1.7530
2.0124
2.2919
2.5915
2.9112
3.2510
3.5740
AULLE
1.9075
2.1522
2.4151
2.6625
2.8885
3.0881
3.2573
3.3924
3.4908
3.5506
3.5705
SRMLE
1.5271
1.5271
1.5271
1.5271
1.5271
1.5271
1.5271
1.5271
1.5271
1.5271
1.5271
SRAULLE
0.8744
0.9702
1.0733
1.1703
1.2590
1.3375
1.4039
1.4570
1.4957
1.5192
1.5270
MLE
33.1595
33.1595
33.1595
33.1595
33.1595
33.1595
33.1595
33.1595
33.1595
33.1595
33.1595
LLE
0.4893
1.2132
2.5451
4.4324
6.8751
9.8731
13.4266
17.5353
22.1995
27.4190
32.5914
AULLE
1.2984
3.5907
7.2401
11.5879
16.2114
20.7438
24.8751
28.3515
30.9757
32.6065
33.1540
SRMLE
2.4804
2.4804
2.4804
2.4804
2.4804
2.4804
2.4804
2.4804
2.4804
2.4804
2.4804
SRAULLE
0.2878
0.4482
0.7008
1.0004
1.3183
1.6294
1.9128
2.1510
2.3308
2.4425
2.4800
Table 5.2: The estimated MSE values for different when
d = 0.01
d = 0.1
d = 0.2
d = 0.3
d = 0.4
d = 0.5
d = 0.6
d = 0.7
d = 0.8
d = 0.9
d = 0.99
MLE
1.0662
1.0662
1.0662
1.0662
1.0662
1.0662
1.0662
1.0662
1.0662
1.0662
1.0662
LLE
0.6249
0.6600
0.7004
0.7422
0.7854
0.8300
0.8761
0.9236
0.9726
1.0230
1.0696
AULLE
0.9318
0.9543
0.9770
0.9975
1.0154
1.0307
1.0434
1.0533
1.0604
1.0647
1.0662
SRMLE
0.7173
0.7173
0.7173
0.7173
0.7173
0.7173
0.7173
0.7173
0.7173
0.7173
0.7173
SRAULLE
0.6334
0.6474
0.6617
0.6744
0.6856
0.6952
0.7031
0.7092
0.7137
0.7164
0.7173
MLE
1.5025
1.5025
1.5025
1.5025
1.5025
1.5025
1.5025
1.5025
1.5025
1.5025
1.5025
LLE
0.7156
0.7739
0.8419
0.9134
0.9883
1.0667
1.1485
1.2337
1.3224
1.4145
1.5003
AULLE
1.1894
1.2403
1.2926
1.3400
1.3819
1.4181
1.4481
1.4717
1.4887
1.4990
1.5024
SRMLE
0.8807
0.8807
0.8807
0.8807
0.8807
0.8807
0.8807
0.8807
0.8807
0.8807
0.8807
SRAULLE
0.7111
0.7388
0.7671
0.7928
0.8155
0.8351
0.8513
0.8641
0.8733
0.8789
0.8807
MLE
2.8448
2.8448
2.8448
2.8448
2.8448
2.8448
2.8448
2.8448
2.8448
2.8448
2.8448
LLE
0.8150
0.9431
1.0989
1.2687
1.4526
1.6505
1.8626
2.0887
2.3288
2.5830
2.8239
AULLE
1.6588
1.8370
2.0267
2.2037
2.3644
2.5056
2.6248
2.7198
2.7888
2.8307
2.8446
SRMLE
1.2238
1.2238
1.2238
1.2238
1.2238
1.2238
1.2238
1.2238
1.2238
1.2238
1.2238
SRAULLE
0.7500
0.8216
0.8976
0.9684
1.0325
1.0888
1.1363
1.1741
1.2015
1.2182
1.2237
MLE
26.9632
26.9632
26.9632
26.9632
26.9632
26.9632
26.9632
26.9632
26.9632
26.9632
26.9632
LLE
0.4165
1.0607
2.1916
3.7594
5.7641
8.2058
11.0845
14.4001
18.1527
22.3422
26.4863
AULLE
1.2822
3.2802
6.3089
9.8379
13.5461
17.1550
20.4289
23.1753
25.2440
26.5281
26.9589
SRMLE
2.2638
2.2638
2.2638
2.2638
2.2638
2.2638
2.2638
2.2638
2.2638
2.2638
2.2638
SRAULLE
0.1830
0.3484
0.5958
0.8823
1.1824
1.4737
1.7377
1.9589
2.1255
2.2288
2.2635
Table 5.3: The estimated MSE values for different when
d = 0.01
d = 0.1
d = 0.2
d = 0.3
d = 0.4
d = 0.5
d = 0.6
d = 0.7
d = 0.8
d = 0.9
d = 0.99
MLE
0.4775
0.4775
0.4775
0.4775
0.4775
0.4775
0.4775
0.4775
0.4775
0.4775
0.4775
LLE
0.3791
0.3887
0.3977
0.4076
0.4177
0.4280
0.4384
0.4489
0.4595
0.4703
0.4801
AULLE
0.4647
0.4669
0.4691
0.4710
0.4728
0.4742
0.4754
0.4763
0.4770
0.4774
0.4775
SRMLE
0.3940
0.3940
0.3940
0.3940
0.3940
0.3940
0.3940
0.3940
0.3940
0.3940
0.3940
SRAULLE
0.3837
0.3854
0.3872
0.3888
0.3902
0.3913
0.3923
0.3930
0.3936
0.3939
0.3940
MLE
0.6770
0.6770
0.6770
0.6770
0.6770
0.6770
0.6770
0.6770
0.6770
0.6770
0.6770
LLE
0.4802
0.4968
0.5156
0.5349
0.5545
0.5745
0.5948
0.6156
0.6367
0.6582
0.6778
AULLE
0.6404
0.6467
0.6530
0.6586
0.6634
0.6675
0.6709
0.6736
0.6755
0.6766
0.6770
SRMLE
0.5120
0.5120
0.5120
0.5120
0.5120
0.5120
0.5120
0.5120
0.5120
0.5120
0.5120
SRAULLE
0.4850
0.4896
0.4943
0.4984
0.5020
0.5050
0.5075
0.5095
0.5109
0.5117
0.5120
MLE
1.3107
1.3107
1.3107
1.3107
1.3107
1.3107
1.3107
1.3107
1.3107
1.3107
1.3107
LLE
0.6774
0.7261
0.7824
0.8409
0.9016
0.9646
1.0299
1.0974
1.1671
1.2391
1.3058
AULLE
1.1005
1.1356
1.1712
1.2032
1.2312
1.2552
1.2750
1.2906
1.3017
1.3085
1.3107
SRMLE
0.8007
0.8007
0.8007
0.8007
0.8007
0.8007
0.8007
0.8007
0.8007
0.8007
0.8007
SRAULLE
0.6742
0.6953
0.7167
0.7359
0.7528
0.7673
0.7792
0.7886
0.7953
0.7993
0.8007
MLE
13.1308
13.1308
13.1308
13.1308
13.1308
13.1308
13.1308
13.1308
13.1308
13.1308
13.1308
LLE
0.5461
0.9882
1.6442
2.4735
3.4762
4.6523
6.0017
7.5245
9.2207
11.0902
12.9211
AULLE
1.6681
2.8645
4.3929
6.0178
7.6355
9.1563
10.5043
11.6171
12.4466
12.9580
13.1291
SRMLE
2.0259
2.0259
2.0259
2.0259
2.0259
2.0259
2.0259
2.0259
2.0259
2.0259
2.0259
SRAULLE
0.2834
0.4630
0.6941
0.9407
1.1869
1.4187
1.6245
1.7945
1.9212
1.9994
2.0256
Table 5.4: The estimated MSE values for different when
d = 0.01
d = 0.1
d = 0.2
d = 0.3
d = 0.4
d = 0.5
d = 0.6
d = 0.7
d = 0.8
d = 0.9
d = 0.99
MLE
0.4172
0.4172
0.4172
0.4172
0.4172
0.4172
0.4172
0.4172
0.4172
0.4172
0.4172
LLE
0.3380
0.3450
0.3528
0.3608
0.3689
0.3771
0.3853
0.3937
0.4021
0.4107
0.4185
AULLE
0.4078
0.4094
0.4111
0.4125
0.4137
0.4148
0.4157
0.4163
0.4168
0.4171
0.4172
SRMLE
0.3423
0.3423
0.3423
0.3423
0.3423
0.3423
0.3423
0.3423
0.3423
0.3423
0.3423
SRAULLE
0.3350
0.3363
0.3375
0.3387
0.3396
0.3404
0.3411
0.3416
0.3420
0.3422
0.3423
MLE
0.5932
0.5932
0.5932
0.5932
0.5932
0.5932
0.5932
0.5932
0.5932
0.5932
0.5932
LLE
0.4341
0.4477
0.4629
0.4785
0.4943
0.5104
0.5268
0.5435
0.5605
0.5777
0.5935
AULLE
0.5662
0.5709
0.5755
0.5796
0.5832
0.5863
0.5888
0.5907
0.5921
0.5930
0.5932
SRMLE
0.4469
0.4469
0.4469
0.4469
0.4469
0.4469
0.4469
0.4469
0.4469
0.4469
0.4469
SRAULLE
0.4278
0.4311
0.4344
0.4373
0.4398
0.4420
0.4437
0.4451
0.4461
0.4467
0.4469
MLE
1.1311
1.1311
1.1311
1.1311
1.1311
1.1311
1.1311
1.1311
1.1311
1.1311
1.1311
LLE
0.6264
0.6659
0.7113
0.7584
0.8070
0.8573
0.9092
0.9627
1.0179
1.0746
1.1271
AULLE
0.9785
1.0041
1.0300
1.0532
1.0736
1.0910
1.1053
1.1166
1.1246
1.1295
1.1311
SRMLE
0.6958
0.6958
0.6958
0.6958
0.6958
0.6958
0.6958
0.6958
0.6958
0.6958
0.6958
SRAULLE
0.6086
0.6232
0.6381
0.6513
0.6630
0.6729
0.6811
0.6875
0.6921
0.6949
0.6958
MLE
10.8045
10.8045
10.8045
10.8045
10.8045
10.8045
10.8045
10.8045
10.8045
10.8045
10.8045
LLE
0.5945
0.9872
1.5492
2.2435
3.0700
4.0288
5.1198
6.3430
7.6985
9.1863
10.6384
AULLE
1.7707
2.7752
4.0155
5.3061
6.5735
7.7539
8.7932
9.6474
10.2821
10.6727
10.8032
SRMLE
1.8820
1.8820
1.8820
1.8820
1.8820
1.8820
1.8820
1.8820
1.8820
1.8820
1.8820
SRAULLE
0.3585
0.5315
0.7427
0.9609
1.1742
1.3722
1.5462
1.6889
1.7948
1.8600
1.8817
Table 6.2: The SMSE values of estimators for the Numerical example
d = 0.01
d = 0.1
d = 0.2
d = 0.3
d = 0.4
d = 0.5
MLE
0.0009457555
0.0009457555
0.0009457555
0.0009457555
0.0009457555
0.0009457555
LLE
0.0009441630
0.0009443098
0.0009444729
0.0009446361
0.0009447993
0.0009449624
AULLE
0.0009457541
0.0009457543
0.0009457546
0.0009457548
0.0009457550
0.0009457551
SRMLE
0.0009445487
0.0009445487
0.0009445487
0.0009445487
0.0009445487
0.0009445487
SRAULLE
0.0009445472
0.0009445475
0.0009445477
0.0009445480
0.0009445481
0.0009445483
d = 0.6
d = 0.7
d = 0.8
d = 0.9
d = 0.99
MLE
0.0009457555
0.0009457555
0.0009457555
0.0009457555
0.0009457555
LLE
0.0009451256
0.0009452888
0.0009454521
0.0009456153
0.0009457622
AULLE
0.0009457553
0.0009457554
0.0009457554
0.0009457555
0.0009457555
SRMLE
0.0009445487
0.0009445487
0.0009445487
0.0009445487
0.0009445487
SRAULLE
0.0009445484
0.0009445485
0.0009445486
0.0009445487
0.0009445487
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