arXiv subjects

Hirai Mukasa

Publications and source records attributed to Hirai Mukasa.

4 recordsLinked to original sources

arXiv2026

Third-order Halley-type iterative method with positive and bounded correction function

In this paper, we propose a new one-point iterative method for finding simple roots of nonlinear equations by modifying Halley's method. Its correction function is positive and bounded on $\mathbb{R}$ and matches the local expansion of Halley's correction function near a simple root. The method avoids the singularity and sign reversal of Halley's correction function while retaining at least third-order local convergence. We also establish sufficient conditions for global convergence. Numerical experiments show that the proposed method achieves high convergence success rates and favorable iteration counts over a wide range of initial guesses. Further experiments near the singularity of Halley's correction function demonstrate robust convergence behavior of the proposed method. An application to the van der Waals equation illustrates the effectiveness of the proposed method.

math.NA↗

arXiv2026

Bayes linear estimator in the general linear model

The Bayes linear estimator is obtained by minimizing the Bayes risk matrix under squared loss among all linear estimators. In this paper, we study the statistical properties and equivalence problems of Bayes linear estimators in the general linear model. First, we examine linear sufficiency and linear completeness of Bayes linear estimators. Second, we derive necessary and sufficient conditions under which two Bayes linear estimators coincide. These conditions clarify when a Bayes linear estimator based on a simpler covariance structure retains Bayes-risk optimality under the original covariance structure. Moreover, several examples, including Rao's mixed-effects model and the spatial error model, show that our results can simplify the estimation procedure. Finally, we establish equivalent conditions for the equality of residual sums of squares when Bayes linear estimators are considered.

math.ST↗

arXiv2026

Parameter-free conditions for equality of general ridge estimators under spatial error models

This paper investigates when two general ridge estimators coincide under spatial error models. First, in the general linear model, we derive a necessary and sufficient condition based on the commutativity of an extended dispersion matrix and an orthogonal projector, thereby extending the classical result of Zyskind. Next, we establish parameter-free conditions for first-order spatial autoregressive and spatial moving average processes. Also, we obtain a necessary and sufficient condition valid for all values of the spatial correlation coefficient in the specified parameter range and characterize all penalty matrices for which the two general ridge estimators coincide. A numerical experiment illustrates the theoretical results and shows that the proposed conditions can simplify the two-step estimation procedure by eliminating the need to estimate the spatial correlation coefficient.

math.ST↗

arXiv2025

Equality between two general ridge estimators and equivalence of their residual sums of squares

General ridge estimators are typical linear estimators in a general linear model. The class of them includes some shrinkage estimators in addition to classical linear unbiased estimators such as the ordinary least squares estimator and the weighted least squares estimator. We derive necessary and sufficient conditions under which two general ridge estimators coincide. In particular, two noteworthy conditions are added to those from previous studies. The first condition is given as a seemingly column space relationship to the covariance matrix of the error term, and the second one is based on the biases of general ridge estimators. Another problem studied in this paper is to derive an equivalence condition such that equality between two residual sums of squares holds when general ridge estimators are considered.

math.ST↗