Propagation of uncertainty: Expressions of second and third order uncertainty with third and fourth moments

We present two new forms of the remainder in Taylor’s formula involving a generalization of the Taylor-Langrange formula. An asymptotic formula of the Taylor’s remainder for real analytic functions is given as application.

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New forms of the Taylor's remainder

We introduce a formula which generalizes Taylor's theorem from powers of linear terms z-x to functional terms \phi(z)-\phi(x), leading to a formula which reduces in a special case to Cauchy's generalized mean value theorem. In other words, regarding Cauchy's mean value theorem as an extension of the simple mean value theorem, we provide the analogous extension of Taylor's theorem. The filling of this gap is easy and requires only mathematics on an undergraduate level, so that the mentioned analogy might be a useful tool for illustration at schools and universities.

A simple unifying formula for Taylor's Theorem and Cauchy's Mean Value Theorem.

We propose a modification of local polynomial time series fitting which improves the efficiency of the conventional method when the observation error is strongly mixing. This generalizes the work of Xiao et. al. in 2003, who considered an error process with an invertible linear representation. Here, we do not suppose a certain functional structure on the random observation error. Furthermore, we allow for heteroscedasticity. The procedure is based on a pre-whitening transformation of the data. The dependent variable as well as the unknown variance function are estimated via preliminary local polynomial regression. Establishing its asymptotic distribution, we show that the proposed estimator is more efficient than the conventional one. In a simulation study, the performance of our estimator on finite samples is investigated.

L. M. Blumenthal

Papers

Discussions: Concerning the Remainder Term in Taylor's Formula