^{1}

^{2}

In this article, the restricted almost unbiased ridge logistic estimator (RAURLE) is proposed to estimate the parameter in a logistic regression model with exact linear re-strictions when there exists multicollinearity among explanatory variables. The performance of the proposed estimator over the maximum likelihood estimator (MLE), ridge logistic estimator (RLE), almost unbiased ridge logistic estimator (AURLE), and restricted maximum likelihood estimator (RMLE) with respect to different ridge parameters is investigated through a simulation study in terms of scalar mean square error.

Multicollinearity inflates the variance of the maximum likelihood estimator (MLE) in the logistic regression. As a result, one may not obtain an efficient estimate for the parameter

Wu and Asar (2016) [

Consider the following logistic regression model

which follows Bernoulli distribution with parameter

where ^{th} row of X, which is an

where^{th} element equals

In the presence of multicollinearity, Schaefer et al. (1984) [

where

The asymptotic properties of LRE:

However the LRE is a biased estimator which produces inconsistent estimates for the parameter (Wu and Asar, 2016 [

where

And the asymptotic properties of AURLE:

As another remedial action for multicollinearity, one may use the exact linear restrictions in addition to the sample logistic regression model (1). The resulting esti- mator is called as Restricted estimator.

Suppose that the following exact restriction is given in addition to the general logistic regression model (1).

where H is a

In the presence of the above restriction (11) in addition to the logistic regression model (1), Duffy and Santner (1989) [

The asymptotic mean and variance of

and

Consequently the bias of

To improve the performance of the estimators further, in this section, by combining AURLE and RMLE, we propose a new estimator which is called as the Restricted Almost Unbiased Ridge Logistic Estimator (RAURLE) and defined as

where

The asymptotic properties of

and

Consequently, the mean square error can be obtained as,

Now we consider the existing methods to obtain an estimated value for the ridge parameter k, since RAURLE depends on k. Many researchers suggested various methods of estimating the ridge parameter in the ridge regression approach and recently this estimation method is added to the logistic regression. In this research, we have considered the following existing ridge parameter estimation methods to compare the performance of the proposed estimator with some existing estimators in logistic regression.

1) Hoerl and Kennard (1970) [

where

2) Hoerl et al. (1975) [

where p is the number of predictor variables in the model (1).

3) Lawless and Wang (1976) [

4) Lindley and Smith (1972) [

5) Schaefer et al. (1984) [

It is difficult to compare the mean square error of the estimators theoretically, since none of the estimators MLE, RLE, AURLE, RMLE and RAURLE are not always superior. So, we use Monte Carlo simulation to examine the performance of the proposed estimator over the existing estimators under different levels of multicolli- nearity. Following McDonald and Galarneau (1975) [

where

where ^{th} simulation. The simulation results are given in Tables 1-3. It can be noticed from the Tables 1-3 that the scalar mean square error of the proposed estimator RAURLE is smaller compared to MLE, RLE, AURLE and RMLE, with respect to all the selected values of n, r, and k, considered in this research. Further, the new estimator RAURLE has better performance when

In this paper, we proposed a restricted almost unbiased ridge logistic estimator (RAURLE) in logistic regression with exact linear restrictions when the explanatory variables are highly correlated. Through a Monte Carlo simulation study, we examined

Estimator | ||||||
---|---|---|---|---|---|---|

0.80 | MLE | 2.7913 | 2.7913 | 2.7913 | 2.7913 | 2.7913 |

RLE | 2.1156 | 1.7907 | 2.5850 | 2.3325 | 2.5182 | |

AURLE | 2.6754 | 2.5265 | 2.7811 | 2.7393 | 2.7733 | |

RMLE | 0.7946 | 0.7946 | 0.7946 | 0.7946 | 0.7946 | |

RAURLE | 0.7727 | 0.7420 | 0.7919 | 0.7850 | 0.7911 | |

0.90 | MLE | 5.3804 | 5.3804 | 5.3804 | 5.3804 | 5.3804 |

RLE | 3.2110 | 1.1707 | 3.2801 | 3.7876 | 2.8573 | |

AURLE | 4.7335 | 2.5165 | 4.7767 | 5.0440 | 4.4847 | |

RMLE | 1.3230 | 1.3230 | 1.3230 | 1.3230 | 1.3230 | |

RAURLE | 1.2127 | 0.7413 | 1.2202 | 1.2680 | 1.1662 | |

0.95 | MLE | 10.5921 | 10.5921 | 10.5921 | 10.5921 | 10.5921 |

RLE | 3.3890 | 1.0535 | 3.1522 | 5.6636 | 2.4171 | |

AURLE | 6.6049 | 2.6589 | 6.2946 | 8.8763 | 5.2217 | |

RMLE | 2.0985 | 2.0985 | 2.0985 | 2.0985 | 2.0985 | |

RAURLE | 1.4868 | 0.7045 | 1.4316 | 1.8598 | 1.2326 | |

0.99 | MLE | 52.3691 | 52.3691 | 52.3691 | 52.3691 | 52.3691 |

RLE | 3.2128 | 0.5469 | 11.0650 | 12.1737 | 8.0410 | |

AURLE | 9.1283 | 1.5910 | 24.7708 | 26.5211 | 19.5062 | |

RMLE | 4.1985 | 4.1985 | 4.1985 | 4.1985 | 4.1985 | |

RAURLE | 1.0587 | 0.2943 | 2.3991 | 2.5345 | 1.9761 |

Estimator | ||||||
---|---|---|---|---|---|---|

0.80 | MLE | 1.0027 | 1.0027 | 1.0027 | 1.0027 | 1.0027 |

RLE | 0.9559 | 0.7576 | 0.8381 | 0.9900 | 0.7688 | |

AURLE | 1.0014 | 0.9640 | 0.9860 | 1.0026 | 0.9677 | |

RMLE | 0.3818 | 0.3818 | 0.3818 | 0.3818 | 0.3818 | |

RAURLE | 0.3814 | 0.3698 | 0.3767 | 0.3818 | 0.3710 | |

0.90 | MLE | 1.9144 | 1.9144 | 1.9144 | 1.9144 | 1.9144 |

RLE | 1.5371 | 1.1484 | 1.3535 | 1.8586 | 1.1580 | |

AURLE | 1.8685 | 1.7054 | 1.8081 | 1.9134 | 1.7111 | |

RMLE | 0.6081 | 0.6081 | 0.6081 | 0.6081 | 0.6081 | |

RAURLE | 0.5961 | 0.5518 | 0.5799 | 0.6079 | 0.5534 | |

0.95 | MLE | 3.7477 | 3.7477 | 3.7477 | 3.7477 | 3.7477 |

RLE | 3.1236 | 1.9762 | 2.1164 | 3.4727 | 1.6703 | |

AURLE | 3.6853 | 3.1647 | 3.2627 | 3.7361 | 2.9106 | |

RMLE | 0.9656 | 0.9656 | 0.9656 | 0.9656 | 0.9656 | |

RAURLE | 0.9522 | 0.8360 | 0.8585 | 0.9631 | 0.7775 | |

0.99 | MLE | 18.4345 | 18.4345 | 18.4345 | 18.4345 | 18.4345 |

RLE | 8.3450 | 2.8305 | 5.5908 | 12.9410 | 3.7151 | |

AURLE | 14.4989 | 7.0897 | 11.4944 | 17.4029 | 8.6919 | |

RMLE | 2.0647 | 2.0647 | 2.0647 | 2.0647 | 2.0647 | |

RAURLE | 1.6809 | 0.8901 | 1.3698 | 1.9668 | 1.0678 |

Estimator | ||||||
---|---|---|---|---|---|---|

0.80 | MLE | 0.5813 | 0.5813 | 0.5813 | 0.5813 | 0.5813 |

RLE | 0.5721 | 0.5497 | 0.5668 | 0.5784 | 0.5230 | |

AURLE | 0.5812 | 0.5803 | 0.5811 | 0.5813 | 0.5779 | |

RMLE | 0.2734 | 0.2734 | 0.2734 | 0.2734 | 0.2734 | |

RAURLE | 0.2732 | 0.2730 | 0.2731 | 0.2733 | 0.2720 | |

0.90 | MLE | 1.1084 | 1.1084 | 1.1084 | 1.1084 | 1.1084 |

RLE | 0.9929 | 0.6402 | 0.9460 | 1.0985 | 0.9585 | |

AURLE | 1.1015 | 0.9746 | 1.0945 | 1.1084 | 1.0966 | |

RMLE | 0.4193 | 0.4193 | 0.4193 | 0.4193 | 0.4193 | |

RAURLE | 0.4170 | 0.3744 | 0.4147 | 0.4193 | 0.4154 | |

0.95 | MLE | 2.1685 | 2.1685 | 2.1685 | 2.1685 | 2.1685 |

RLE | 1.8938 | 1.1041 | 1.6649 | 2.1269 | 1.8481 | |

AURLE | 2.1486 | 1.8086 | 2.0982 | 2.1681 | 2.1412 | |

RMLE | 0.6627 | 0.6627 | 0.6627 | 0.6627 | 0.6627 | |

RAURLE | 0.6574 | 0.5631 | 0.6437 | 0.6626 | 0.6553 | |

0.99 | MLE | 10.6602 | 10.6602 | 10.6602 | 10.6602 | 10.6602 |

RLE | 5.4949 | 0.9743 | 4.8691 | 9.6580 | 5.3288 | |

AURLE | 8.9707 | 2.7172 | 8.4656 | 10.6080 | 8.8449 | |

RMLE | 1.4734 | 1.4734 | 1.4734 | 1.4734 | 1.4734 | |

RAURLE | 1.2608 | 0.4198 | 1.1957 | 1.4670 | 1.2446 |

the performance of the proposed estimator over some existing estimators MLE, RLE, AURLE and RMLE in terms of scalar mean square error. Also, five different choices of existing ridge parameter estimates were used to compare the estimators. The results show that the newly proposed estimator outperforms all the other estimators considered in this study under the selected values of n, r, and k by means of SMSE.

We thank the Editor and the referee for their comments and suggestions, and the Postgraduate Institute of Science, University of Peradeniya, Sri Lanka for providing necessary facilities to complete this research.

Varathan, N. and Wijekoon, P. (2016) On the Restricted Almost Unbiased Ridge Estimator in Logistic Regression. Open Journal of Statistics, 6, 1076-1084. http://dx.doi.org/10.4236/ojs.2016.66087