PARAMETER ESTIMATION FOR THE FIXED BLOCK EFFECTS MODEL USING A USUAL DESIGN

ABSTRACT

This paper is devoted to the fixed block effects model analysed with most of the classical designs. First, we find regularities conditions for such designs. Then, we obtain explicitly all the least squares estimators of the model. A particular attention is given to orthogonal blocked designs and their optimal properties.     

C0(X)中的Chebyshev中心的存在性

设X是一Banach空间，Co(X)表示X中所有按范数拓扑收敛于零的序列构成的空间(赋上确界范数)．证明了Co(X)中的每个紧子集均有中心充要条件是X中每个紧子集均有中心，而且，若x满足条件(Q)，则Co(X)中的每个有界集有中心充要条件是X是拟一致凸的．据此构造了一Banach空间X满足：X的每个紧子集有中心、X满足条件(Q)和X不是拟一致凸的，这样Banach空间Co(X)中的每个紧子集有中心，但并不是每个有界集均有中心．     

局部完全对称的亚半正定矩阵的行列式的不等式

本文得到一个局部完全对称的亚半正定矩阵的行列式的不等式，拓广了[2，3]的结果．     

Joint Inventory and Pricing Coordination with Incomplete Demand Information

In retailing operations, retailers face the challenge of incomplete demand information. We develop a new concept named K‐approximate convexity, which is shown to be a generalization of K‐convexity, to address this challenge. This idea is applied to obtain a base‐stock list‐price policy for the joint inventory and pricing control problem with incomplete demand information and even non‐concave revenue function. A worst‐case performance bound of the policy is established. In a numerical study where demand is driven from real sales data, we find that the average gap between the profits of our proposed policy and the optimal policy is 0.27%, and the maximum gap is 4.6%.     

关于(k)=∑sin~kA-∑cos~kA(k=3,5)的上下界

通过局部地采用优超方法给出了三角函数（k）=∑sinkA－∑coskA（k=3，5）的上下确界     

Smoothing Splines and Shape Restrictions

Constrained smoothing splines are discussed under order restrictions on the shape of the function m . We consider shape constraints of the type m ^{( r )}≥ 0, i.e. positivity, monotonicity, convexity, .... (Here for an integer r ≥ 0, m ^{( r )} denotes the r th derivative of m .) The paper contains three results: (1) constrained smoothing splines achieve optimal rates in shape restricted Sobolev classes; (2) they are equivalent to two step procedures of the following type: (a) in a first step the unconstrained smoothing spline is calculated; (b) in a second step the unconstrained smoothing spline is "projected" onto the constrained set. The projection is calculated with respect to a Sobolev-type norm; this result can be used for two purposes, it may motivate new algorithmic approaches and it helps to understand the form of the estimator and its asymptotic properties; (3) the infinite number of constraints can be replaced by a finite number with only a small loss of accuracy, this is discussed for estimation of a convex function.     

Optimal allocation of stratified samples with several variance constraints and equal workloads over time by geometric programming

We apply geometric programming, developed by Duffin, Peterson and Zener (1967), to the optimal allocation of stratified samples with several variance constraints arising from several estimates of deficiency rates in the quality control of administrative decisions. We develop also a method for imposing constraints on sample sizes to equalize workloads over time, as required by the practicalities of clerical work for quality control.

We allocate samples by an extension of the work of Neyman (1934), following the exposition of Cochran (1977). Davis and Schwartz (1987) developed methods for multiconstraint Neyman allocation by geometric programming for integrated sampling. They also applied geometric programming to Neyman allocation of a sample for estimating college enrollments by Cornell (1947) and Cochran (1977). This paper continues the application of geometric programming to Neyman allocation with multiple constraints on variances and workloads and minimpal sampling costs.     

BAYESIAN HYPER‐LASSOS WITH NON‐CONVEX PENALIZATION

The Lasso has sparked interest in the use of penalization of the log‐likelihood for variable selection, as well as for shrinkage. We are particularly interested in the more‐variables‐than‐observations case of characteristic importance for modern data. The Bayesian interpretation of the Lasso as the maximum a posteriori estimate of the regression coefficients, which have been given independent, double exponential prior distributions, is adopted. Generalizing this prior provides a family of hyper‐Lasso penalty functions, which includes the quasi‐Cauchy distribution of Johnstone and Silverman as a special case. The properties of this approach, including the oracle property, are explored, and an EM algorithm for inference in regression problems is described. The posterior is multi‐modal, and we suggest a strategy of using a set of perfectly fitting random starting values to explore modes in different regions of the parameter space. Simulations show that our procedure provides significant improvements on a range of established procedures, and we provide an example from chemometrics.     

Estimation of systems of equations subject to curvature constraints

Two computationally manageable estimation procedures for maintaining curvature constraints are compared. Using an efficient nonlinear optimizing algorithm, the well-known Cholesky factorization is compared with the eigenvalue decomposition method for estimating a system of input demand and output supply equations. Statistical tests are provided. The Cholesky factorization is superior when efficient optimizers are used. The eigenvalue decomposition may become a better choice when the user is limited to less efficient algorithms.     

空间的凸性和度量投影的连续性

用最佳逼近论中有关概念给出了空间严格凸、CLUR和LUR一些结论,并用它们证得了保证度量投影连续性的一个定理。