Principal component ridge type estimator for the inverse Gaussian regression model

Inverse Gaussian regression models are useful for regression data where both variables are nonnegative and the variance of the dependent variable depends on the independent variable, Zero intercept inverse Gaussian regression models are presented with non-constant variance, constant ratio of variance to the mean and constant coefficient of variation, For purposes of calibration, the prediction band is used to give point and interval estimators for the independent variable, The results are illustrated with a real data set.

ABSTRACTThe Inverse Gaussian Regression Model (IGRM) is used when the response variable is positively skewed and follows the inverse Gaussian distribution. In this article, we propose a Liu-type estimator to combat multicollinearity in the IGRM. The variance of the Maximum Likelihood Estimator (MLE) is overstated due to the presence of severe multicollinearity. Moreover, some estimation methods are suggested to estimate the optimal value of the shrinkage parameter. The performance of the proposed estimator is compared with the MLE and some other existing estimators in the sense of mean squared error through Monte Carlo simulation and different real-life applications. Under certain conditions, it is concluded that the proposed estimator is superior to the MLE, ridge, and Liu estimator.

Modelling follow up time at a single-lane roundabout

In this article, a biased estimator is proposed to combat multi-collinearity in the logistic regression model. The proposed estimator is a general estimator which includes other biased estimators, such as the ridge estimator and the Liu estimator as special cases. Necessary and sufficient conditions for the superiority of the new biased estimator over the maximum likelihood estimator, the ridge estimator are obtained and some properties in the mean squared error sense are discussed. Furthermore, a Monte Carlo simulation study is given to illustrate some of the theoretical results.

The binary logistic regression model (LRM) is practical in situations when the response variable (RV) is dichotomous. The maximum likelihood estimator (MLE) is generally considered to estimate the LRM parameters. However, in the presence of multicollinearity (MC), the MLE is not the correct choice due to its inflated standard deviation (SD) and standard errors (SE) of the estimates. To combat MC, commonly used biased estimators, i.e., the Ridge estimators (RE) and Liu estimators (LEs), are preferred. However, most of the time, the traditional LE attains a negative value for its Liu parameter (LP), which is considered to be a major drawback. Therefore, to overcome this issue, we proposed a new adjusted LE for the binary LRM. Owing to numerical evaluation purposes, Monte Carlo simulation (MCS) study is performed under different conditions where bias and mean squared error are the performance criteria. Findings showed the superiority of our proposed estimator in comparison with the other estimation methods due to the existence of high but imperfect multicollinearity, which clearly means that it is consistent when the regressors are multicollinear. Furthermore, the findings demonstrated that whenever there is MC, the MLE is not the best choice. Finally, a real application is being considered to be evidence for the advantage of the intended estimator. The MCS and the application findings pointed out that the considered adjusted LE for the binary logistic regression model is a more efficient estimation method whenever the regressors are highly multicollinear.

Mixed Poisson regression models (MPRMs) are widely used for analyzing overdispersed count data. However, the presence of multicollinearity among explanatory variables poses challenges when estimating regression coefficients using the maximum likelihood estimator (MLE), leading to inflated variances. The Poisson Modification of the Quasi-Lindley regression model (PMQLRM), a recently introduced alternative within MPRMs, faces similar issues. To address this, we propose a Liu-type estimator for the PMQLRM as an effective remedy for multicollinearity. Several existing methods are utilized to estimate the Liu-type parameter, and the theoretical superiority conditions of the proposed estimator over the MLE, ridge regression estimator, and Liu estimator are established using the scalar mean squared error (MSE) criterion. A Monte Carlo simulation study is conducted to compare the performance of different estimators based on the MSE. Additionally, a real-world dataset is analyzed to demonstrate the practical advantages of the proposed method. The findings indicate that the Poisson-modification of the Quasi-Lindley Liu-type estimator outperforms the MLE and other biased estimators when multicollinearity is present, offering a more stable and reliable alternative for parameter estimation in mixed Poisson regression models.

We define a new four-parameter model called the odd log-logistic generalized inverse Gaussian distribution which extends the generalized inverse Gaussian and inverse Gaussian distributions. We obtain some structural properties of the new distribution. We construct an extended regression model based on this distribution with two systematic structures, which can provide more realistic fits to real data than other special regression models. We adopt the method of maximum likelihood to estimate the model parameters. In addition, various simulations are performed for different parameter settings and sample sizes to check the accuracy of the maximum likelihood estimators. We provide a diagnostics analysis based on case-deletion and quantile residuals. Finally, the potentiality of the new regression model to predict price of urban property is illustrated by means of real data.

The Beta Regression Model (BRM) is commonly used while analyzing data where the dependent variable is restricted to the interval $[0,1]$ for example proportion or probability. The Maximum Likelihood Estimator (MLE) is used to estimate the regression coefficients of BRMs. But in the presence of multicollinearity, MLE is very sensitive to high correlation among the explanatory variables. For this reason, we introduce a new biased estimator called the Beta Liu-Type Estimator (BLTE) to overcome the multicollinearity problem in the case that dependent variable follows a Beta distribution. The proposed estimator is a general estimator which includes other biased estimators, such as the Ridge Estimator, Liu Estimator, and the estimators with two biasing parameters as special cases in BRM. The performance of the proposed new estimator is compared to the MLE and other biased estimators in terms of the Estimated Mean Squared Error (EMSE) criterion by conducting a simulation study. Finally, a numerical example is given to show the benefit of the proposed estimator over existing estimators.

The beta regression is a widely known statistical model when the response (or the dependent) variable has the form of fractions or percentages. In most of the situations in beta regression, the explanatory variables are related to each other which is commonly known as the multicollinearity problem. It is well-known that the multicollinearity problem affects severely the variance of maximum likelihood (ML) estimates. In this article, we developed a new biased estimator (called a two-parameter estimator) for the beta regression model to handle this problem and decrease the variance of the estimation. The properties of the proposed estimator are derived. Furthermore, the performance of the proposed estimator is compared with the ML estimator and other common biased (ridge, Liu, and Liu-type) estimators depending on the mean squared error criterion by making a Monte Carlo simulation study and through two real data applications. The results of the simulation and applications indicated that the proposed estimator outperformed ML, ridge, Liu, and Liu-type estimators.

<p>Supplementary Table S1. Analyses of MSI, DNA methylation, KRAS, BRAF, and PIK3CA mutations, and neoantigen load. Supplementary Table S2. List of antibodies and fluorophores used in the immunofluorescence procedure. Supplementary Table S3. Statistical methods. Supplementary Table S4. Macrophage density, M1-like macrophage density, M2-like macrophage density, and M1:M2 macrophage density ratio in overall tissue regions and patient survival with inverse probability weighting (IPW). Supplementary Table S5. Densities of macrophage populations defined by positivity for single polarization marker in tumor intraepithelial and stromal regions and patient survival with inverse probability weighting (IPW). Supplementary Table S6. Clinical, pathological, and molecular characteristics of colorectal cancer cases according to year of diagnosis. Supplementary Table S7. Macrophage densities and M1:M2 macrophage density ratio in tumor intraepithelial and stromal regions and patient survival in strata of MSI status with inverse probability weighting (IPW). Supplementary Table S8. Macrophage densities and M1:M2 macrophage density ratio in tumor intraepithelial and stromal regions and patient survival in strata of year of diagnosis with inverse probability weighting (IPW). Figure S1. Comparison of staining patterns between multiplex immunofluorescence and standard immunohistochemistry. Figure S2. Flowchart of the cyclic immunofluorescence procedure. Figure S3. Analysis of immunofluorescence slides to quantify tumor intraepithelial and stromal macrophage densities. Figure S4. Macrophage densities across 10 TMAs. Figure S5. Fluorescence signal intensities across 10 TMAs. Figure S6. t-SNE analysis of a random sample of 0.5% of all cells based on fluorophore signal intensities shows no clear clustering according to the TMAs. Figure S7. Core-to-core correlation of macrophage densities in two randomly chosen cores of tumors with two or more cores. Figure S8. Inverse probability weighting-adjusted Kaplan-Meier survival curves of colorectal cancer survival according to the ordinal quartile categories (Q1-Q4) of intraepithelial (A) and stromal (B) macrophage densities. Figure S9. Forest plots of inverse probability weighting-adjusted Cox regression models of colorectal cancer specific survival according to the densities of intraepithelial and stromal M1-like polarized macrophages, with M1-like macrophages defined using different cut-offs (10-50%) of the M1-end of the macrophage polarization (M1:M2) index distribution. Figure S10. Forest plots of inverse probability weighting-adjusted Cox regression models of colorectal cancer specific survival according to the densities of intraepithelial and stromal M2-like polarized macrophages, with M2-like macrophages defined using different cut-offs (10-50%) of the M2-end of the macrophage polarization (M1:M2) index distribution. Figure S11. Forest plots of inverse probability weighting-adjusted Cox regression models of colorectal cancer specific survival according to the intraepithelial and stromal M1:M2-density ratio with M1-like and M2-like macrophages defined using different cut-offs (10-50%) of both tails of the macrophage polarization (M1:M2) index distribution. Figure S12. Densities of M1-like and M2-like macrophages in relation to histologic lymphocytic reaction patterns.</p>

S ome P oints of D iscussion • Why hasn't the PIG model been widely used before this? • What types of data are best modeled using a PIG regression? • How do we model data with a very high initial peak and long right skew? • How do we know whether a PIG is a better-fitted model than negative binomial or Poisson models? P oisson I nverse G aussian M odel A ssumptions The Poisson inverse Gaussian (PIG) model is similar to the negative binomial model in that both are mixture models. The negative binomial model is a mixture of Poisson and gamma distributions , whereas the inverse Gaussian model is a mixture of Poisson and inverse Gaussian distributions . Those of you who are familiar with generalized linear models will notice that there are three GLM continuous distributions: normal, gamma, and inverse Gaussian. The normal distribution is typically parameterized to a lognormal distribution when associated with count models, presumably because the log link forces the distribution to have only nonnegative values. The Poisson and negative binomial (both NB2 and NB1) models have log links. Recall that the negative binomial is a mixture of the Poisson and gamma distributions, with variances of μ and μ 2 / v , respectively. We inverted v so that there is a direct relationship between the mean, dispersion, and variance function. Likewise, the inverse Gaussian is a mixture of Poisson and inverse Gaussian distributions, with an inverse Gaussian variance of μ 3 Φ.

The ill-posed least squares problems often arise in many engineering applications such as machine learning, intelligent navigation algorithms, surveying and mapping adjustment model, and linear regression model. A new biased estimation (BE) method based on Neumann series is proposed in this article to solve the ill-posed problems more effectively. Using Neumann series expansion, the unbiased estimate can be expressed as the sum of infinite items. When all the high-order items are omitted, the proposed method degenerates into the ridge estimation or generalized ridge estimation method, whereas a series of new biased estimates can be acquired by including some high-order items. Using the comparative analysis, the optimal biased estimate can be found out with less computation. The developed theory establishes the essential relationship between BE and unbiased estimation and can unify the existing unbiased and biased estimate formulas. Moreover, the proposed algorithm suits for not only ill-conditioned equations but also rank-defect equations. Numerical results show that the proposed BE method has improved accuracy over the existing robust estimation methods to a certain extent.

Background: Dengue Hemorrhagic Fever (DHF) is a major health problem in East Java Province due to population density, sanitation, availability of medical personnel, and altitude. DHF count data often experiences overdispersion, challenging the Poisson regression assumption in the Generalized Linear Model (GLM). Objective: To compare the performance of Poisson, Negative Binomial (NBR), and Poisson Inverse Gaussian (PIG) ​​models in modeling DHF cases. Method: A cross-sectional quantitative study analyzed 38 districts/cities in East Java (saturation sampling) using secondary data processed with . Analysis: Descriptive statistics, multicollinearity test (VIF<10), overdispersion test, and model comparison via AIC/deviance. Results: NBR achieved the lowest AIC (412.10) compared to Poisson (1466.60) and PIG (412.41), confirming significant overdispersion (deviance/df ≈37.16). Population density and altitude had a significant effect (p<0.05). Conclusion: NBR is optimal for DHF modeling, supporting targeted environmental and health interventions in East Java.

This paper introduces a new class of efficient and debiased two-step shrinkage estimators for a linear regression model in the presence of multicollinearity. We derive the proposed estimators’ mean square error and define the necessary and sufficient conditions for superiority over the existing estimators. In addition, we develop an algorithm for selecting the shrinkage parameters for the proposed estimators. The comparison of the new estimators versus the traditional ordinary least squares, ridge regression, Liu, and the two-parameter estimators is done by a matrix mean square error criterion. The Monte Carlo simulation results show the superiority of the proposed estimators under certain conditions. In the presence of high but imperfect multicollinearity, the two-step shrinkage estimators’ performance is relatively better. Finally, two real-world chemical data are analyzed to demonstrate the advantages and the empirical relevance of our newly proposed estimators. It is shown that the standard errors and the estimated mean square error decrease substantially for the proposed estimator. Hence, the precision of the estimated parameters is increased, which of course is one of the main objectives of the practitioners.

As bias estimation methods are developed, it becomes necessary to obtain the bound on bias estimation for more complex bias and sensor models. Three-dimensional (3-D) sensors, such as radars commonly used in applications, contain both scale and additive biases in sine space which result in a nonlinear estimation problem that may have poor observability and accuracy depending on the geometry of the sensors. By converting the sine space and range measurements to Cartesian using an unbiased conversion, it is possible, via creation of pseudomeasurements, to eliminate the need to estimate the target's state thereby reducing the sensor bias estimator complexity. The present paper evaluates the Cramer–Rao lower bound (CRLB) for estimating scale and additive biases in sine space for 3-D sensors and compares it with a maximum likelihood formulation implemented via iterated least squares, which is thereby shown to be statistically efficient. Additionally, the importance of measurement diversity is investigated with respect to the CRLB.