高等数学教学中的反问题及反例

反问题及反例推动了数学历史的发展，它代表的是一种创新思维．因此，在教学中恰当的应用反问题及反例可以帮助学生全面、准确地理解数学概念及定理，激发学习兴趣，培养思维的灵活性、缜密性和批判性，进而发展创新思维，获得创新能力．     

An inclusion-consistent solution to the problem of absurd confidence statements: 1. Consistent exact confidence-interval estimation

An example is given of a uniformly most accurate unbiased confidence belt which yields absurd confidence statements with 100% occurrence. In several known examples, as well as in the 100%-occurrence counterexample, an optimal confidence belt provides absurd statements because it is inclusion-inconsistent with either a null or an all-inclusive belt or both. It is concluded that confidence-theory optimality criteria alone are inadequate for practice, and that a consistency criterion is required. An approach based upon inclusion consistency of belts [C(x) C C C(x), for some x, implies γ ≤ γ for confidence coefficients] is suggested for exact interval estimation in continuous parametric models. Belt inclusion consistency, the existence of a proper-pivotal vector [a pivotal vector T(X, θ) such that the effective range of T(x,.) is independent of x], and the existence of a confidence distribution are proven mutually equivalent. This consistent approach being restrictive, it is shown, using Neyman's anomalous 1954 example, how to determine whether any given parametric function can be estimated consistently and exactly or whether a consistent nonexact solution must be attempted.     

On Efron's Parameterization

This paper discusses a curved exponential family of distributions which is defined by a differential equation with respect to the expectation parameters in the two–dimensional exponential family. The differential equation considered here is the same as the one given by Efron (1975) for the trinomial distribution. This equation is extended here to a general exponential family, and called Efron's parameterization in the two–dimensional exponential family. The solution of Efron's parameterization is obtained explicitly in an exponential family, although Kumagai & Inagaki (1996) showed that there exists no proper solution of Efron's equation for the trinomial distribution, in line with the counterexample given by Efron (1975 p. 1206). The paper gives some characterizations of Efron's parameterization with special reference to Fisher's circle model. The implications of these characterizations are the two–dimensional normal distribution and a spiral curve in the plane.     

葛梯尔反例中的“偷天换日＂

通过对“谬误推理”与“二律背反”本质的阐释,尝试性地解决知识论中葛梯尔问题带来的挑战。简要地介绍了“葛梯尔反例”的背景以及一些相应的理论,分析了康德解决“谬误推理”和“二律背反”的原理、方法,通过对“葛梯尔反例”的再分析,得到其问题的实质在于认识论与形而上学的混淆,并提出相应的解决办法。     

On the bootstrap in cube root asymptotics

The authors study the application of the bootstrap to a class of estimators which converge at a nonstandard rate to a nonstandard distribution. They provide a theoretical framework to study its asymptotic behaviour. A simulation study shows that in the case of an estimator such as Chernoff's estimator of the mode, usually the basic bootstrap confidence intervals drastically undercover while the percentile bootstrap intervals overcover. This is a rare instance where basic and percentile confidence intervals, which have exactly the same length, behave in a very different way. In the case of Chernoff's estimator, if the distribution is symmetric, it is possible to bootstrap from a smooth symmetric estimator of the distribution for which the basic bootstrap confidence intervals will have the claimed coverage probability while the percentile bootstrap interval will have an asymptotic coverage of 1!