An F Approximation to the Distribution of a Linear Combination of Chi-squared Variables.

A three-parameter F approximation to the distribution of a positive linear combination of central chi-squared variables is described. It is about as easy to implement as the Satterthwaite-Welsh and Hall-Buckley-Eagleson approximations. Some reassuring properties of the F approximation are derived, and numerical results are presented. The numerical results indicate that the new approximation is superior to the Satterthwaite approximation and, for some purposes, better than the Hall-Buckley-Eagleson approximation. It is not quite as good as the Gamma-Weibull approximation due to Solomon and Stephens, but is easier to implement because iterative methods are not required.     

Approximate estimation in nonlinear panel data models

This paper is concerned with the estimation of a general class of nonlinear panel data models in which the conditional distribution of the dependent variable and the distribution of the heterogeneity factors are arbitrary. In general, exact analytical results for this problem do not exist. Here, Laplace and small-sigma appriximations for the marginal likelihood are presented. The computation of the MLE from both approximations is straightforward. It is shown that the accuracy of the Laplace approximation depends on both the sample size and the variance of the individual effects, whereas the accuracy of the small-sigma approximation is 0(1) with respect to the sample size. The results are applied to count, duration and probit panel data models. The accuracy of the approximations is evaluated through a Monte Carlo simulation experiment. The approximations are also applied in an analysis of youth unemployment in Australia.     

The power functions of the likelihood ratio tests for a simply ordered trend in normal means

Likelihood ratio tests for the homogeneity of k normal means with the alternative restricted by an increasing trend are considered as well as the likelihood ratio tests of the null hypothesis that the means satisfy the trend. While the work is primarily a survey of results concerning the power functions of these tests, the extensions of some results to the case of not necessarily equal sample sizes are presented. For the case of known or unknown population variances, exact expressions are given for the power functions for k=3,4, and approximations are discussed for larger k. The topics of consistency, bias and monotonicity of the power functions are included. Also, Bartholomew's conjectures concerning minimal and maximal powers are investigated, with results of a new numerical study given.     

An approximation for the hotelling-lawley trace in the noncentral case

A procedure for estimating power in conjunction with the Hotelling-Lawley trace is developed. By approximating a non-central Wishart distribution with a central Wishart, and using McKeon's (1974) F-type approximation, a relatively simple procedure for obtaining power estimates is obtained. The accuracy of the approximation is investigated by comparing the approximate results with those for a wide range of conditions given in Olson's (1973) extensive Monte Carlo study. Siotani's (1971) asymptotic expansion is used to provide further comparative assessments. It is demonstrated that the approximation is of sufficient accuracy to be used in practical applications.     

Sequential confidence intervals for the number of cells in a multinomial population

A two-sided sequential confidence interval is suggested for the number of equally probable cells in a given multinomial population with prescribed width and confidence coefficient. We establish large-sample properties of the fixed-width confidence interval procedure using a normal approximation, and some comparisons are made. In addition, a simulation study is carried out in order to investigate the finite sample behaviour of the suggested sequential interval estimation procedure.     

三向剖分域上的二元二次样条插值与逼近

本文主要用B－网方法研究了平行六边形区域在三向剖分△_6~3下的二元二次样条插值的存在性、唯一性及逼近度问题。在本文末尾我们举例说明本方法是可行的。     

On Approximability of Boolean Formula Minimization

For a Boolean function given by a Boolean formula (or a binary circuit) S we discuss the problem of building a Boolean formula (binary circuit) of minimal size, which computes the function g equivalent to , or -equivalent to , i.e., . In this paper we prove that if P NP then this problem can not be approximated with a good approximation ratio by a polynomial time algorithm.     

On Combinatorial Approximation of Covering 0-1 Integer Programs and Partial Set Cover

The problems dealt with in this paper are generalizations of the set cover problem, min{cx | Ax b, x {0,1}ⁿ}, where c Q₊ⁿ, A {0,1}^{m × n}, b 1. The covering 0-1 integer program is the one, in this formulation, with arbitrary nonnegative entries of A and b, while the partial set cover problem requires only m–K constrains (or more) in Ax b to be satisfied when integer K is additionall specified. While many approximation algorithms have been recently developed for these problems and their special cases, using computationally rather expensive (albeit polynomial) LP-rounding (or SDP-rounding), we present a more efficient purely combinatorial algorithm and investigate its approximation capability for them. It will be shown that, when compared with the best performance known today and obtained by rounding methods, although its performance comes short in some special cases, it is at least equally good in general, extends for partial vertex cover, and improves for weighted multicover, partial set cover, and further generalizations.     

Batch-Processing Scheduling with Setup Times

The problem is to minimize the total weighted completion time on a single batch-processing machine with setup times. The machine can process a batch of at most B jobs at one time, and the processing time of a batch is given by the longest processing time among the jobs in the batch. The setup time of a batch is given by the largest setup time among the jobs in the batch. This batch-processing problem reduces to the ordinary uni-processor scheduling problem when B = 1. In this paper we focus on the extreme case of B = +, i.e. a batch can contain any number of jobs. We present in this paper a polynomial-time approximation algorithm for the problem with a performance guarantee of 2. We further show that a special case of the problem can be solved in polynomial time.     

Modified Wald statistics for generalized linear models

Summary: Wald statistics in generalized linear models are asymptotically ² distributed. The asymptotic chi–squared law of the corresponding quadratic form shows disadvantages with respect to the approximation of the finite–sample distribution. It is shown by means of a comprehensive simulation study that improvements can be achieved by applying simple finite–sample size approximations to the distribution of the quadratic form in generalized linear models. These approximations are based on a ² distribution with an estimated degree of freedom that generalizes an approach by Patnaik and Pearson. Simulation studies confirm that nominal level is maintained with higher accuracy compared to the Wald statistics.