首项系数为正的Sturm-Liouville问题的特征值等式

对于给定的首项系数函数为正的Sturm-Liouville方程,利用在自伴边界条件空间上自然圈、边界条件的极限和特征值的单调性的几个结果,给出了在耦合边界条件与分离边界条件下的特征值间的一些等式的证明.     

Crisis cycles in the slightly unstable block random model

ABSTRACT

The paper proposes a new approach for studying the time to time appearing breakdowns in economy. Block random model can describe stability of large complicated systems with variable number of participants. Theoretical background of the model is given by a theorem about the eigenvalues of block random matrices [Juhász F. On the characteristic values of non-symmetric block random matrices. J Theoret Probab. 1990;67:199–205; On the structural eigenvalues of block random matrices. Linear Algebra Appl. 1996;246:225–231]. The model takes into account not only effects of participants but of groups formed from them as well. Slight instability means group level stability and participant level instability [Juhász F. On the turbulence of slightly unstable block random systems. In: Taylor C, et al., editors. Numerical methods for laminar and turbulent flow. Atlanta; 1995. p. 113–121]. Lability index of block random systems is introduced for measuring instability. It is showed that lability index of a slightly unstable block random model is growing while number of participants increases. Alteration in the number of participants makes it possible to describe crisis cycles.     

关于矩阵乘积特征值估计的一点注记

由矩阵奇异值的表示定理,给出了一个关于矩阵乘积的特征值的估计,进而说明[1]、[2]的结果都是没有意义的。     

一类带谱参数的奇异Sturm-Liouville算子Ⅰ

研究了一类具有转换条件且边界条件中带谱参数的奇异Storm-Liouville问题．将对上述问题特征值的研究，转化为考虑定义在Hilbert空间H中一个算子A的特征值问题．     

Estimating multivariate random effects without replication

This paper considers the estimation of multivariate random effects that are measured with error, but for which there are no replications. Using structural simplification of the correlation of the data, separate estimates are generated for the covariance of the random effects and the covariance of the error. An estimator of the random effects based on a truncated eigen structure is defined, and matrix mean squared error and its trace (risk) are analyzed, with comparison to the maximum likelihood estimator (m.l.e) and also to the Stein-like estimator of Efron and Morris (1972). It is shown that the estimator has risk which is smaller than the risk of the maximum likelihood estimator and the Efron-Morris estimator in most cases.     

Test of linear trend in eigenvalues of k covariance matrices with applications in common principal components analysis

A test for linear trend among a set of eigenvalues of k covariance matrices is developed. A special case of this test is Flury's (1986) test for the equality of eigenvalues. The linear trend hypothesis appears to be more relevant to data analysis than the equality hypothesis. Examples show how the linear trend hypothesis can be acceptable while the equality hypothesis is rejected.     

Bootstrapping the latent roots of certain random matrices

Eigenvalues and functions of eigenvalues play an important role in the reduction of the dimensionality of data in multivariate analysis. However, even under the usual normal model context, the associated distributional theory is extremely complicated. In this paper, bootstrap algorithms for ap-proximating the distributions of functions of certain eigenvalues are given, with applications to confidence interval construction for population param-eters. Extensive Monte Carlo simulation results demonstrate the small sample performance of the bootstrap simultaneous confidence sets     

A simulation study on SPSS ridge regression and ordinary least squares regression procedures for multicollinearity data

This study compares the SPSS ordinary least squares (OLS) regression and ridge regression procedures in dealing with multicollinearity data. The LS regression method is one of the most frequently applied statistical procedures in application. It is well documented that the LS method is extremely unreliable in parameter estimation while the independent variables are dependent (multicollinearity problem). The Ridge Regression procedure deals with the multicollinearity problem by introducing a small bias in the parameter estimation. The application of Ridge Regression involves the selection of a bias parameter and it is not clear if it works better in applications. This study uses a Monte Carlo method to compare the results of OLS procedure with the Ridge Regression procedure in SPSS.     

The score test for the two‐sample occupancy model

The score test statistic from the observed information is easy to compute numerically. Its large sample distribution under the null hypothesis is well known and is equivalent to that of the score test based on the expected information, the likelihood‐ratio test and the Wald test. However, several authors have noted that under the alternative hypothesis this no longer holds and in particular the score statistic from the observed information can take negative values. We extend the anthology on the score test to a problem of interest in ecology when studying species occurrence. This is the comparison of two zero‐inflated binomial random variables from two independent samples under imperfect detection. An analysis of eigenvalues associated with the score test in this setting assists in understanding why using the observed information matrix in the score test can be problematic. We demonstrate through a combination of simulations and theoretical analysis that the power of the score test calculated under the observed information decreases as the populations being compared become more dissimilar. In particular, the score test based on the observed information is inconsistent. Finally, we propose a modified rule that rejects the null hypothesis when the score statistic is computed using the observed information is negative or is larger than the usual chi‐square cut‐off. In simulations in our setting this has power that is comparable to the Wald and likelihood ratio tests and consistency is largely restored. Our new test is easy to use and inference is possible. Supplementary material for this article is available online as per journal instructions.     

New results on the convergence of random matrices

This paper extends the previous convergence results in Cerqueti and Costantini (2008) to a more general case using larger normed set of functions. In this regard, the weight-based convergence of the random matrices and their generalized eigenvalues is obtained under less restrictive requirements for the weights.