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Showing papers on "Asymptotology published in 1969"


Journal ArticleDOI
TL;DR: In this paper, the authors studied the asymptotic theory of Bayes solutions in estimating and testing when hypothesis and alternative are separated at least by an indifference region, under the assumption that the observations are independent and indentically distributed.
Abstract: This paper deals with the asymptotic theory of Bayes solutions in (i) Estimation (ii) Testing when hypothesis and alternative are separated at least by an indifference region, under the assumption that the observations are independent and indentically distributed. The estimation results which are partial generalizations of results of LeCam begin with a proof of the convergence of the normalized posterior density to the appropriate normal density in a strong sense. From this result we derive the asymptotic efficiency of Bayes estimates obtained from smooth loss functions and in particular of the posterior mean. The last two theorems of this section deal with asymptotic expansions for the posterior risk in such estimation problems. The section on testing contains a limit theorem for the n-th root of the posterior risk under weak conditions on the prior and the loss function. Finally we discuss generalizations and some open problems.

95 citations


Journal ArticleDOI
TL;DR: In this paper, a general asymptotic method based on the work of Krylov-Bogoliubov is developed to obtain the response of nonlinear over damped systems.
Abstract: A general asymptotic method based on the work of Krylov-Bogoliubov is developed to obtain the response of nonlinear over damped systems. A second-order system with both roots real is treated first and the method is then extended to higher-order systems. Two illustrative examples show good agreement with results obtained by numerical integration.

53 citations



Journal ArticleDOI
TL;DR: In this article, an order-of-magnitude analysis applied to the governing equations and boundary conditions quantifies the error resulting from the neglect of various factors such as density difference, initial superheat and variable properties.
Abstract: The paper considers one-dimensional freezing and thawing of ice–water systems for the conditions first examined by Stefan. An order-of-magnitude analysis applied to the governing equations and boundary conditions quantifies the error resulting from the neglect of various factors. Principal among these are density difference, initial superheat and variable properties. Asymptotic solutions for the temperature distribution and interface history are developed for a wide range of boundary conditions: prescribed temperature or heat flux, prescribed convection and prescribed radiation. Comparison with known results reveals the general adequacy of the asymptotic solutions and an estimate of the error incurred.

16 citations


Journal ArticleDOI
TL;DR: In this article, the authors consider systems of the form (1) on the semiaxis, where is a column vector with components, is an matrix, and is a parameter.
Abstract: In this paper we consider systems of the form (1)on the semiaxis , where is a column vector with components, is an matrix, and is a parameter. We pose the problem of finding the asymptotic behavior of the solutions of equation (1) as and .

10 citations



Journal ArticleDOI
TL;DR: In this article, it was shown that the conclusion of Theorem 1 will follow from Lemma 1 when applied to equation (15) if we assume, instead of (6), The hypothesis given in Trench's theorem is sufficient to imply ( * ) but not (6).
Abstract: Remarks 1, 3 and 5 are incorrect as stated. They should be supplemented by the following observations: (i) In case the perturbing term is linear in y , i.e. f(t, y) = B(t)y , the conclusion of Theorem 1 will follow from Lemma 1 when applied to equation (15) if we assume, instead of (6), The hypothesis given in Trench's theorem is sufficient to imply ( * ) but not (6). A similar comment applies to Remark 5.

2 citations