Performance of ridge estimator in inverse Gaussian regression model

The presence of multicollinearity among the explanatory variables has undesirable effects on the maximum likelihood estimator (MLE). Ridge estimator (RE) is a widely used estimator in overcoming this issue. The RE enjoys the advantage that its mean squared error (MSE) is less than that of MLE. The inverse Gaussian regression (IGR) model is a well-known model in the application when the response variable positively skewed. The purpose of this paper is to derive the RE of the IGR under multicollinearity problem. In addition, the performance of this estimator is investigated under numerous methods for estimating the ridge parameter. Monte Carlo simulation results indicate that the suggested estimator performs better than the MLE estimator in terms of MSE. Furthermore, a real chemometrics dataset application is utilized and the results demonstrate the excellent performance of the suggested estimator when the multicollinearity is present in IGR model.     

Trace Class Markov Chains for Bayesian Inference with Generalized Double Pareto Shrinkage Priors

Bayesian shrinkage methods have generated a lot of interest in recent years, especially in the context of high‐dimensional linear regression. In recent work, a Bayesian shrinkage approach using generalized double Pareto priors has been proposed. Several useful properties of this approach, including the derivation of a tractable three‐block Gibbs sampler to sample from the resulting posterior density, have been established. We show that the Markov operator corresponding to this three‐block Gibbs sampler is not Hilbert–Schmidt. We propose a simpler two‐block Gibbs sampler and show that the corresponding Markov operator is trace class (and hence Hilbert–Schmidt). Establishing the trace class property for the proposed two‐block Gibbs sampler has several useful consequences. Firstly, it implies that the corresponding Markov chain is geometrically ergodic, thereby implying the existence of a Markov chain central limit theorem, which in turn enables computation of asymptotic standard errors for Markov chain‐based estimates of posterior quantities. Secondly, because the proposed Gibbs sampler uses two blocks, standard recipes in the literature can be used to construct a sandwich Markov chain (by inserting an appropriate extra step) to gain further efficiency and to achieve faster convergence. The trace class property for the two‐block sampler implies that the corresponding sandwich Markov chain is also trace class and thereby geometrically ergodic. Finally, it also guarantees that all eigenvalues of the sandwich chain are dominated by the corresponding eigenvalues of the Gibbs sampling chain (with at least one strict domination). Our results demonstrate that a minor change in the structure of a Markov chain can lead to fundamental changes in its theoretical properties. We illustrate the improvement in efficiency resulting from our proposed Markov chains using simulated and real examples.     

Restricted ridge estimator in the logistic regression model

It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, Schaefer et al. presented a ridge estimator in the logistic regression model. Making use of the ridge estimator, when some linear restrictions are also present, we introduce a restricted ridge estimator in the logistic regression model. Statistical properties of this newly defined estimator will be studied and comparisons are done in the simulation study in the sense of mean squared error criterion. A real-data example and a simulation study are introduced to discuss the performance of this estimator.     

投入产出系统计算方法的探讨

投入产出系统计算工作量主要集中在计算完全需要系数矩阵(I-A)~(-1)。本文给出两种计算(I-A)~(-1)的方法。这两种方法都只用到矩阵乘法,原理简单,易于掌握,便于推广。     

四元数矩阵之迹的几个定理

定义了四元数矩阵的范数,并利用范数的性质,建立了几个四元数矩阵之迹的不等式     

Some finite sample properties of generalized ridge regression estimators

In this paper we study the mean square error properties of the generalized ridge estimator. We obtain the exact and the approximate bias and the mean square error of the operational generalized ridge estimator in terms of G( ) functions. We show, among other things, that the operational generalized ridge estimator does not dominate the ordinary least squares estimator up to a certain order of approximation. Finally, we note that the iterative procedures to obtain coverging ridge estimators should be used with caution.     

Comparison of regression estimators using Pitman measures of nearness

Received: August 5, 1999; revised version: June 14, 2000     

Two kinds of restricted modified estimators in linear regression model

In this paper, we introduce two kinds of new restricted estimators called restricted modified Liu estimator and restricted modified ridge estimator based on prior information for the vector of parameters in a linear regression model with linear restrictions. Furthermore, the performance of the proposed estimators in mean squares error matrix sense is derived and compared. Finally, a numerical example and a Monte Carlo simulation are given to illustrate some of the theoretical results.     

On the ridge regression estimator with sub-space restriction

In the linear regression model with elliptical errors, a shrinkage ridge estimator is proposed. In this regard, the restricted ridge regression estimator under sub-space restriction is improved by incorporating a general function which satisfies Taylor’s series expansion. Approximate quadratic risk function of the proposed shrinkage ridge estimator is evaluated in the elliptical regression model. A Monte Carlo simulation study and analysis based on a real data example are considered for performance analysis. It is evident from the numerical results that the shrinkage ridge estimator performs better than both unrestricted and restricted estimators in the multivariate t-regression model, for some specific cases.     

Performance analysis of the preliminary test estimator with series of stochastic restrictions

In this paper, the problem of estimation of the regression coefficients in a multiple regression model is considered under the multicollinearity situation when there are series of stochastic linear restrictions available on the regression parameter vector. We have considered the preliminary test ridge regression estimators (PTRREs) based on the Wald, likelihood ratio, and lagrangian multiplier tests. Tables for the maximum and minimum guaranteed efficiency of the PTRREs are obtained, which allow us to determine the optimum choice of the level of significance corresponding to the optimum estimator. Some numerical results support the findings.