Walking on Plane and Matrix Square

We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, φ(n) is an even number. But it is not true for all even number, we could find some numbers which would not be the value of any φ(n). The Sum of two odd numbers is one even number. Gold Bach stated “Every even integer greater than 2 can be written as the sum of two primes”. Other than two, all prime numbers are odd numbers. So we can write, every even integer greater than 2 as the sum of two primes. German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio. We could find the series which is generated by one and inverse the golden ratio. Also we can note consecutive golden ratio numbers converge to the golden ratio. Lothar Collatz stated integers converge to one. It is also known as 3k + 1 problem. Tao redefined Collatz conjecture as 3k − 1 problem. We could not prove it directly but one parallel proof will prove this conjecture.

The Lower-bound Estimate of D[X_(2m)] and Fast Construction of Matrix X_(2m)

In this paper, the character matrix x n is studied, fast construction method of matrix X 2m is provided. And it is proved that the lower bound estimate of the number of an Latin squares matrix D[X m] is2m(2m)!+∑mi=2[(2m)!] 2∏ij=1K j!∏rj=1b j!.

The Lower-bound Estimate of D[X2m] and Fast Construction of Matrix X2m

In this paper, the character matrix x n is studied, fast construction method of matrix X 2m is provided. And it is proved that the lower bound estimate of the number of an Latin squares matrix D[X m] is2m(2m)!+∑mi=2[(2m)!] 2∏ij=1K j!∏rj=1b j!.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

THE SUPERIORITY OF EMPIRICAL BAYES ESTIMATION OF PARAMETERS IN PARTITIONED NORMAL LINEAR MODEL

In this article,the empirical Bayes（EB）estimators are constructed for the estimable functions of the parameters in partitioned normal linear model.The superiorities of the EB estimators over ordinary least-squares（LS）estimator are investigated under mean square error matrix（MSEM）criterion.

The Generalized Mixed Least Square Methodsfor the Estimation of Weights in AHP

This paper proposes a class of generalized mixed least square methods(GMLSM) forthe estimation of weights in the analytic hierarchy process and studies their good properties such asinvariance under transpose, invariance under change of scale, and also gives a simple convergent iterativealgorithm and some numerical examples. The well-known eigenvector method(EM) is then compared.Theoretical analysis and the numerical results show that the iterative times of the GMLSM are generallyfewer than that of the MLSM, and the GMLSM are preferable to the EM in several important respects.

THE SUPERIORITIES OF BAYES LINEAR UNBIASED ESTIMATION IN PARTITIONED LINEAR MODEL

In this article, the Bayes linear unbiased estimation （BALUE） of parameters is derived for the partitioned linear model. The superiorities of the BALUE over ordinary least square estimator （LSE） are studied in terms of the Bayes mean square error matrix （BMSEM） criterion and Pitman closeness （PC） criterion.

Modified Almost Unbiased Liu Estimator in Linear Regression Model

In this paper,we propose a new biased estimator namely modified almost unbiased Liu estimator by combining almost unbiased Liu estimator(AULE)andridge estimator(RE)in a linear regression model when multicollinearity presents amongthe independent variables.Necessary and sufficient conditions for the proposed estimator over the ordinary least square estimator,RE,AULE and Liu estimator(LE)in the mean squared error matrix sense are derived,and the optimal biasing parameters are obtained.To illustrate the theoretical findings,a Monte Carlo simulation study is carried out and a numerical example is used.

D-Optimal Designs for Hierarchical Linear Models with Heteroscedastic Errors

This paper investigates the optimal design problem for the prediction of the individual parameters in hierarchical linear models with heteroscedastic errors.An equivalence theorem is established to characterize D-optimality of designs for the prediction based on the mean squared error matrix.The admissibility of designs is also considered and a sufficient condition to simplify the design problem is obtained.The results obtained are illustrated in terms of a simple linear model with random slope and heteroscedastic errors.

Linear Bayes Estimators Applied to the Inverse Gaussian Lifetime Model

For the two-parameter inverse Gaussian distribution denoted by IG（μ,A）, the authors employ a linear Bayes procedure to estimate the parameters μ and A. The superiority of the proposed linear Bayes estimator （LBE） over both the classical UMVUE and the maximum likelihood estimator （MLE） is established in terms of the mean squared error matrix （MSEM） criterion. Compared with the usual Bayes estimator, which is obtained by an MCMC method, the proposed LBE is simple and easy to use. Some numerical results are presented to verify that the LBE performs well.

On the Stochastic Restricted Modified Almost Unbiased Liu Estimator in Linear Regression Model

In this article,we propose a new biased estimator,namely stochastic restricted modified almost unbiased Liu estimator by combining modified almost unbiased Liu estimator(MAULE)and mixed estimator(ME)when the stochastic restrictions are available and the multicollinearity presents.The conditions of supe-riority of the proposed estimator over the ordinary least square estimator,ME,ridge estimator,Liu estimator,almost unbiased Liu estimator,stochastic restricted Liu esti-mator and MAULE in the mean squared error matrix sense are obtained.Finally,a numerical example and a Monte Carlo simulation are given to illustrate the theoretical findings.