Accomplishment Based Renewal For Research in Specification Testing, Nonparametric Estimation and Neural Networks

人工神经网络(ANN)是认知科学家最近发展起来的一类丰富而有趣的非线性模型。尽管这些模型的灵感来自于大脑神经结构的某些特征，但由于它们的简单性和极大的灵活性，它们在计量经济学和经济学中具有相当大的潜力。具体地说，ANN模型作为非参数估计和规格测试的基础是很有前途的，因为它们能够将一般函数空间的任意元素逼近到任意精度。在以前的NSF拨款下，这个项目专注于建立在非参数估计和规格测试中使用ANN模型所必需的数学和统计基础。在这次基于成果的更新中，提出的研究是继续发展这些基础，建立基于ANN的非参数估计的性质，并将所得到的理论应用于构造模型规格检验。数学和统计理论的研究领域有两个。第一个领域是建立Sobolev空间中ANN模型的逼近度结果。第二个领域是得到Hilbert空间中被构造为相依过程(具体地说是混合过程)的部分和的随机元的中心极限定理和不变性原理。应用领域直接利用所提出的理论结果来建模经济主体的学习。

Artificial neural networks (ANN's) are a rich and interesting class of nonlinear models developed recently by cognitive scientists. Although inspired by certain features of the neural architecture of the brain, these models have considerable potential for use in econometrics and economics because of their simplicity and great flexibility. Specifically, ANN models show promise as the basis for nonparmetric estimation and for specification testing because of their ability to approximate arbitrary elements of general function spaces to arbitrary accuracy. Under previous NSF grants, this project focused on establishing the mathematical and statistical foundations necessary for using ANN models in nonparametric estimation and specification testing. The research proposed in this accomplishment based renewal is to continue development of these foundations, to establish properties of ANN- based nonparametric estimators, and to apply the resulting theory to construct model specification tests. The areas in which mathematical and statistical theory will be pursued are two. The first area is the establishment of degree of approximation results for ANN models in Sobolev spaces. The second area is to obtain central limit theorems and invariance principles for random elements constructed as partial sums of a dependent process (specifically, a mixingale process) in a Hilbert space. The application areas make direct use of the proposed theoretical results to model learning by economic agents.