ESTIMATING EQUATIONS AND NON-LINEAR FUNCTIONAL RELATIONSHIPS

A nonlinear functional relationship is defined as an R-dimensional manifold in P-dimensional space. The formulation of the model may be explicitly in terms of R-dimensional vectors of incidental parameters or implicitly by a (P-R)-dimensional vector function of constraints. The objective is to estimate and make inference about a K-vector of parameters θ which defines the manifold. Each observed P-vector has its expectation lying on the manifold, and the error vector has a variance matrix defined in terms of a further vector of parameters The theory of estimating equations in the presence of incidental parameters is extended and applied to the explicit formulation, to give equations suitable for estimating θ given knowledge of only the first two moments. The method has a geometrical interpretation. Estimating equations for are chosen to be those which would be optimal if the normality assumption were true. First order corrections to the biases of these estimates are included. An example where the manifold is a circle centred on the origin is used to illustrate the theory. Further examples incorporate more general features, including the estimation of two variance parameters and estimation in higher dimensions.     

Exclusion restrictions in instrumental variables equations

In this paper we consider two-stage estimators of parameters of a structural equation in a model with recursive exclusion restrictions on the instrumental variables equations. The estimations considered are simple OLS and GLS estimators after substitution of estimates of the systematic part of the IV equations for the endogenous variables. It is known in the literature that neither imposing the restrictions in the first stage nor ignoring them will in general be more efficient than the alternative. We introduce a class of mixed instrumental variables estimators (MIV) with these possibilities as special cases which yields an estimator which is not only more efficient than the two stage estimators considered in the literature but as efficient as an efficient system estimator like 3SLS.     

Partitioned algorithms for maximum likelihood and other non-linear estimation

There are a variety of methods in the literature which seek to make iterative estimation algorithms more manageable by breaking the iterations into a greater number of simpler or faster steps. Those algorithms which deal at each step with a proper subset of the parameters are called in this paper partitioned algorithms. Partitioned algorithms in effect replace the original estimation problem with a series of problems of lower dimension. The purpose of the paper is to characterize some of the circumstances under which this process of dimension reduction leads to significant benefits.Four types of partitioned algorithms are distinguished: reduced objective function methods, nested (partial Gauss-Seidel) iterations, zigzag (full Gauss-Seidel) iterations, and leapfrog (non-simultaneous) iterations. Emphasis is given to Newton-type methods using analytic derivatives, but a nested EM algorithm is also given. Nested Newton methods are shown to be equivalent to applying to same Newton method to the reduced objective function, and are applied to separable regression and generalized linear models. Nesting is shown generally to improve the convergence of Newton-type methods, both by improving the quadratic approximation to the log-likelihood and by improving the accuracy with which the observed information matrix can be approximated. Nesting is recommended whenever a subset of parameters is relatively easily estimated. The zigzag method is shown to produce a stable but generally slow iteration; it is fast and recommended when the parameter subsets have approximately uncorrelated estimates. The leapfrog iteration has less guaranteed properties in general, but is similar to nesting and zigzagging when the parameter subsets are orthogonal.     

The automatic percentile method: Accurate confidence limits in parametric models

The problem of constructing confidence limits for a scalar parameter is considered. Under weak conditions, Efron's accelerated bias-corrected bootstrap confidence limits are correct to second order in parametric familles. In this article, a new method, called the automatic percentile method, for setting approximate confidence limits is proposed as an attempt to alleviate two problems inherent in Efron's method. The accelerated bias-corrected method is not fully automatic, since it requires the calculation of an analytical adjustment; furthermore, it is typically not exact, though for many situations, particularly scalar-parameter familles, exact answers are available. In broader generality, the proposed method is exact when exact answers exist, and it is second-order accurate otherwise. The automatic percentile method is automatic, and for scalar parameter models it can be iterated to achieve higher accuracy, with the number of computations being linear in the number of iterations. However, when nuisance parameters are present, only second-order accuracy seems obtainable.     

The rules of conditional inference: Is there a universal definition of nonformation?

Summary We consider the ideas of sufficiency and ancillarity for parametric models with nuisance parameters, and more generally Barndorff-Nielsen's notion of nonformation. The original four definitions of non-formation, namelyB-,S-,G- andM-nonformation, each cover different types of models. We stress the interpretation of nonformation in terms of the idea of perfect fit. This leads to a new definition of nonformation, calledI-nonformation, which is well suited for inference in exponential families. We also consider Rémon's concept ofL-sufficiency, and a recent extension toL-nonformation, due to Barndorff-Nielsen, which unifies and extendsB-,S- andG- nonformation. We study the relations between these six definitions, and show that they are all special cases ofM-nonformation. All animals are equal, but some animals are more equal than others. From ‘Animal Farm’, by G. Orwell (1945).     

Calculating the Hazard Function and Probability of Tumor for Cancer Risk Assessment When the Parameters Are Time-Dependent

The probability of tumor and hazard function are calculated in a stochastic two-stage model for carcinogenesis when the parameters of the mode are time-dependent. The method used is called the method of characteristics.     

On stochastic processes for quantitative trait locus mapping under selective genotyping

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait) on the interval [0, T] representing a chromosome. The originality of this study is that we are under selective genotyping: only the individuals with extreme phenotypes are genotyped. We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on [0, T] and under local alternatives with a QTL at t^☆ on [0, T]. We show that the LRT process is asymptotically the square of a ‘non-linear interpolated and normalized Gaussian process’. We have an easy formula in order to compute the supremum of the square of this interpolated process. We prove that we have to genotype symmetrically and that the threshold is exactly the same as in the situation where all the individuals are genotyped.     

Adjusted Supremum Score‐Type Statistics for Evaluating Non‐Standard Hypotheses

Supremum score test statistics are often used to evaluate hypotheses with unidentifiable nuisance parameters under the null hypothesis. Although these statistics provide an attractive framework to address non‐identifiability under the null hypothesis, little attention has been paid to their distributional properties in small to moderate sample size settings. In situations where there are identifiable nuisance parameters under the null hypothesis, these statistics may behave erratically in realistic samples as a result of a non‐negligible bias induced by substituting these nuisance parameters by their estimates under the null hypothesis. In this paper, we propose an adjustment to the supremum score statistics by subtracting the expected bias from the score processes and show that this adjustment does not alter the limiting null distribution of the supremum score statistics. Using a simple example from the class of zero‐inflated regression models for count data, we show empirically and theoretically that the adjusted tests are superior in terms of size and power. The practical utility of this methodology is illustrated using count data in HIV research.     

同轴双电机耦合系统功率平衡研究

双电机同轴硬联运行时,2台电机速度被强制同步,由于2台相同电机参数略有差异,导致功率分配不平衡,针 对这一问题提出了基于直接转矩控制的双电机主从模糊控制系统。基于功率平衡要求,通过MATLAB仿真分析双电机 运行负载分配情况,提出双电机功率平衡运行条件。根据直接转矩控制,使用主从控制和传统PID设计的系统基本上解 决了功率不平衡问题,但其动态性能不佳。为了改善动态性能,设计了“模糊参数自整定PID”控制器。仿真结果表明, 该系统最终达到了功率平衡,有效防止双电机同轴运行时1台电机负载过大的问题,而模糊参数自整定PID与传统PID 控制的系统相比,可以提高功率跟踪的动态性能。     

基于蚁群BP神经网路算法的六维力传感器解耦研究

针对传统BP神经网络在六维力传感器解耦训练过程中,由于其初始参数的选取不确定性导致神经网络出现震 荡、收敛速度缓慢和陷入局部极值等问题,提出一种基于蚁群BP神经网络算法并应用于六维力传感器解耦研究。该算 法利用蚁群算法在种群寻优方面的优越性,通过局部和全局信息素更新相结合的方式更新信息素,提高蚁群算法搜索的 效率,为BP神经网络提供一组最优的训练初始参数,网络收敛速度得到很大地提高,同时局部极值和震荡等缺点也有一 定的改善。实验仿真结果表明,在六维力传感器神经网络模型训练过程中,达到同样的目标误差,基于蚁群BP神经网络 算法的迭代次数Ⅳ比传统算法少50%,运行时间r快60%。这说明蚁群BP种经网络算法在六维力传感器解耦研究中 有着很好的应用效果。