Almost unbiased ridge estimator in the count data regression models

The ridge regression model has been consistently demonstrated to be an attractive shrinkage method to reduce the effects of multicollinearity. The Poisson regression model is a well-known model in application when the response variable is count data. However, it is known that multicollinearity negatively affects the variance of maximum likelihood estimator of the Poisson regression coefficients. To address this problem, a Poisson ridge regression model has been proposed by numerous researchers. In this paper, a new Poisson ridge estimator (NPRRM) is proposed and derived. The idea behind the NPRRM is to get diagonal matrix with small values of diagonal elements that leading to decrease the shrinkage parameter, and therefore, the resultant estimator can be better with small amount of bias. Our Monte Carlo simulation results suggest that the NPRRM estimator can bring significant improvement relative to other existing estimators. In addition, the real application results demonstrate that the NPRRM estimator outperforms both Poisson ridge regression and maximum likelihood estimators in terms of predictive performance.

The ridge regression model has been consistently demonstrated to be an attractive shrinkage method to reduce the effects of multicollinearity. The logistic regression model is a well-known model in application when the response variable is binary data. However, it is known that multicollinearity negatively affects the variance of maximum likelihood estimator of the logistic regression coefficients. To address this problem, a logistic ridge estimator has been proposed by numerous researchers. In this paper, a new Jackknifing logistic ridge estimator (NLRE) is proposed and derived. The idea behind the NLRE is to get diagonal matrix with small values of diagonal elements that leading to decrease the shrinkage parameter and, therefore, the resultant estimator can be better with small amount of bias. Our Monte Carlo simulation results suggest that the NLRE estimator can bring significant improvement relative to other existing estimators. In addition, the real application results demonstrate that the NLRE estimator outperforms both logistic ridge estimator and maximum likelihood estimator in terms of predictive performance.

The ridge estimator has been consistently demonstrated to be an attractive shrinkage method to reduce the effects of multicollinearity. The negative binomial regression model (NBRM) is a well-known model in application when the response variable is a count data with overdispersion. However, it is known that the variance of maximum likelihood estimator (MLE) of the NBRM coefficients can negatively affected in the presence of multicollinearity. In this paper, the generalized ridge estimator is proposed to overcome the limitation of ridge estimator. Several methods for estimating the shrinkage matrix have been adapted. Our Monte Carlo simulation results suggest that the proposed estimator, regardless the type of estimating method of shrinkage matrix is better than the MLE estimator and ridge estimator, in terms of MSE. In addition, some estimating method of shrinkage matrix can bring significant improvement relative to others.

Recently, research papers have shown a strong interest in modeling count data. The over-dispersion or under-dispersion are frequently seen in the count data. The count data modeling with a broad range of dispersion has been successfully accomplished using the Conway-Maxwell-Poisson regression (COMP) model. The multicollinearity issue is known to have a detrimental impact on the maximum likelihood estimator's variance. To solve this issue, biased estimators as a ridge estimator have repeatedly shown to be an appealing way to minimize the effects of multicollinearity. In this paper, we suggested the jackknife ridge estimator (JCOMPRE) and the modified version of the jackknife ridge estimator (MJCOMPRE) for the COMP model. The proposed estimators (JCOMPRE and MJCOMPRE) are reducing the effects of the multicollinearity and the biasedness of the ridge estimator at the same time. According to the findings of the Monte Carlo simulation study and real-life applications, the proposed estimators have a minimum bias and a minimum mean squared error. This means that the proposed estimators are more efficient than the maximal likelihood estimator and the ridge estimator.

Preface Introduction Types of Epidemiologic Study Designs Prospective vs. Retrospective Study Designs. Cross-Sectional vs. Longitudinal Studies. A Note on Terminology. The Measurement of Effect. Design-Based vs. Model-Based Inference. Prospective (Cohort) Studies Measures of Risk: Risk Difference. Risk Ratio. Odds Ratio. A Comparison of Taylor Series and Test-Based Methods. Risk Ratio vs. Odds Ratio. Retrospective (Case-Control) Studies Measures of Risk: Risk Ratio. Odds Ratio. Cross-Sectional Studies Measures of Prevalence. Point and Interval Estimation. Stratified Analysis Confounding and Interaction. Adjustment for Confounding in a 2 x 2 Table. Measures of Risk. Measures of Prevalence. Matching Matched-Pair Analysis. Matched-Set Analysis Involving Multiple Controls Per Case. Matched-Set Analysis Using Varying Number of Controls Per Case. Efficacy of Matching. Matched Follow-Up Studies. Multivariate Models Linear Models. Binary Logistic Regression. Proportional Hazards Regression. Poisson Regression. Prospective (Cohort) Studies with Person-Time Data Measures of Risk: Incidence Rate Difference. Incidence Rate Ratio. Stratified Analysis with Person-Time Data Measures of Risk. Other Summary Effect Measures. Attributable Risk Percent Attributable Risk. Population Attributable Risk. Confidence Intervals. Adjustment for Confounding. Appendices Pearson Chi-Square vs. Mantel-Haenszel Chi-Square. Variance of a Function of a Random Variable. The Odds Ratio. More Confidence Intervals for the Risk Ratio. Maximum Likelihood Estimation. Variance of the Conditional Maximum Likelihood Estimator of the Odds Ratio and the Log Odds Ratio. Variance of the Maximum Likelihood Estimator of the Common Odds Ratio in a Stratified Analysis. Some Important Probability Distributions. Randomized and Mid-p Tests and Mid-p Confidence Intervals. Mantel-Haenszel Chi-Square Statistic for Stratified 2 x 2 Tables. Analysis of a 2 x l Table. Maximum Likelihood Estimation for the Logistic Regression Model. Logistic Regression Model for a Case-Control Design. Odds Ratio as an Approximation to a Risk Ratio. Exercises References Related Bibliography Indexes Author Index. Subject Index. Data Index.

The Poisson regression model (PRM) is the standard statistical method of analyzing count data, and it is estimated by a Poisson maximum likelihood (PML) estimator. Such an estimator is affected by outliers, and some robust Poisson regression estimators have been proposed to solve this problem. PML estimators are also influenced by multicollinearity. Biased Poisson regression estimators have been developed to address this problem, including Poisson ridge regression and Poisson almost unbiased ridge estimators. However, the above mentioned estimators do not deal with outliers and multicollinearity problems together in a PRM. Therefore, we propose two robust ridge estimators to deal with the two problems simultaneously in the PRM, namely, the robust Poisson ridge regression (RPRR) estimator and the robust Poisson almost unbiased ridge (RPAUR) estimator. Theoretical comparisons and Monte‐Carlo simulations are conducted to investigate the performance of the proposed estimators relative to the performance of other approaches to PRM parameter estimation. The simulation results indicate that the RPAUR estimator outperforms the other estimators in all situations where both problems exist. Finally, real data are used to confirm the results of this paper.

The most used approach for survival data is the Cox proportional hazards regression model. Multicollinearity, however, is known to have a detrimental impact on the variance of maximum likelihood estimator of the Cox regression coefficients. It has been repeatedly shown that the ridge estimator is a desirable shrinking technique to lessen the consequences of multicollinearity. In this paper, a linearized ridge estimator is developed to transform the biasing parameter of the ordinary ridge estimator to a linearized version and study its performance in the Cox regression model under multicollinearity. Our Monte Carlo simulation findings and the real data application indicate that the suggested estimator may significantly reduce mean squared error in comparison to other competing estimators.

Bayesian optimal design for non-linear model under non-regularity condition

The beta regression model is a commonly used when the response variable has the form of fractions or percentages. The maximum likelihood (ML) estimator is used to estimate the regression coefficients of this model. However, it is known that multicollinearity problem affects badly the variance of ML estimator. Therefore, this paper introduces the Liu‐type estimator for the beta regression model to handle the multicollinearity problem. The performance of the proposed (Liu‐type) estimator is compared to the ML estimator and other biased (ridge and Liu) estimators depending on the mean squared error (MSE) criterion by conducting a simulation study and through an empirical application. The results indicated that the proposed estimator outperformed the ML, ridge, and Liu estimators.

The Liu estimator is used to get precise estimatesby introducing bootstrap technique to reduce the problem of multicollinearity in Poisson regression model. In the presence of multicollinearity, the variance of maximum likelihood estimator (MLE) becomes overstated and theinference based on MLEdoes not remain trustworthy. In this article, we proposed some new Poisson bootstrap Liu and ridge estimators. The proposed estimators are then compared with the non-bootstrap Liu and ridge estimators. Based on mean-squared error criterion, the simulation study revealed showed that the proposed estimators showed efficient results as compared to other existing estimators. Finally, a real example is used to illustrate the application ofproposed estimators.

In the logistic regression, it is known that multicollinearity affects the variance of Maximum Likelihood Estimator (MLE). To overcome this issue, several researchers proposed alternative estimators when exact linear restrictions are available in addition to sample model. In this paper, we propose a new estimator called Stochastic Restricted Ridge Maximum Likelihood Estimator (SRRMLE) for the logistic regression model when the linear restrictions are stochastic. Moreover, the conditions for superiority of SRRMLE over some existing estimators are derived with respect to Mean Square Error (MSE) criterion. Finally, a Monte Carlo simulation is conducted for comparing the performances of the MLE, Ridge Type Logistic Estimator (LRE) and Stochastic Restricted Maximum Likelihood Estimator (SRMLE) for the logistic regression model by using Scalar Mean Squared Error (SMSE).

Asymptotic variances of maximum likelihood estimator for the correlation coefficient from a BVN distribution with one variable subject to censoring

The number of HIV and AIDS cases is a count data correlated with each other. The common model to regress two response variables in the form of count data is bi-response Poisson regression. One infraction of assumptions in the Poisson regression is over dispersion. Negative binomial regression is to be solution to this over dispersion case. This study aims to estimate bi-response negative binomial nonparametric regression model based on local linear estimators for modeling the number of HIV and AIDS cases in East Java. The results give the deviance value as the goodness of fit criterion of bi-response negative binomial nonparametric regression model by using local linear estimator of 2.263 which is smaller than the deviance value of the parametric regression model of 3.926. It means that the bi-response negative binomial nonparametric regression model by using local linear estimator is better than that by using parametric regression model approach to explain the relationship between HIV and AIDS cases affected by drug users in East Java.

In this paper, a new regression estimator is proposed as an alternative to the ridge estimator in the case of multicollinearity in Bell regression model, called an almost unbiased ridge estimator. Also, we provide the theoretical properties of the new almost unbiased ridge estimator, and some theorems showing under which conditions that the almost unbiased ridge estimator is superior to its competitors. We consider a comprehensive simulation study to demonstrate the superiority of the almost unbiased ridge estimator compared to the usual Bell ridge estimator and the maximum likelihood estimator. The usefulness and superiority of the introduced regression estimator is shown via a real-world data example. According to the results of the simulation study and real-world data example, we conclude that the new almost unbiased ridge regression estimator is superior to its competitors in terms of the mean square error criterion.

Zero-inflated negative binomial regression (ZINB) models are commonly used for count data that show overdispersion and extra zeros. The correlation among variables of the count data leads to the presence of a multicollinearity problem. In this case, the maximum likelihood estimator (MLE) will not be an efficient estimator as the value of the mean squared error (MSE) will be large. Several alternative estimators, such as ridge estimators, have been proposed to solve the multicollinearity problem. In this paper, we propose an estimator called an almost unbiased ridge estimator for the ZINB model (AUZINBRE) to solve the multicollinearity problem in the correlated count data. The performance of the AUZINBRE is investigated using a Monte Carlo simulation study. The MSE is used as a measure to compare the results of the proposed estimators with those of the ridge estimators and the MLE. In addition, the AUZINBRE is applied to a real dataset.