The generalized Gaussian error calculus considers unknown systematic errors to spawn biased estimators. Beyond, random errors are asked to conform to the idea of what the author calls well-defined measuring conditions.
The approach features the properties of a building kit: any overall uncertainty turns out to be the sum of a contribution due to random errors, to be taken from a confidence interval as put down by Student, and a contribution due to unknown systematic errors, as expressed by an appropriate worst case estimation.
Generalized Gaussian Error Calculus
Michael Grabe
Generalized Gaussian Error Calculus
With 47 Figures
123
Dr. rer. nat. Michael Grabe Am Hasselteich 5 38104 Braunschweig, Germany
[email protected]
ISBN 978-3-642-03304-9 e-ISBN 978-3-642-03305-6 DOI 10.1007/978-3-642-03305-6 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009940174 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar Steinen Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Lucy, Niklas, Finley, and Rafael
Preface
The book of nature is written in the language of mathematics Galileo Galilei, 1623
Metrology strives to supervise the flow of the measurand’s true values through consecutive, arbitrarily interlocking series of measurements. To highlight this feature the term traceability has been coined. Traceability is said to be achieved, given the true values of each of the physical quantities entering and leaving the measurement are localized by specified measurement uncertainties. The classical Gaussian error calculus is known to be confined to the treatment of random errors. Hence, there is no distinction between the true value of a measurand on the one side and the expectation of the respective estimator on the other. This became apparent not until metrologists considered the effect of so-called unknown systematic errors. Unknown systematic errors are time-constant quantities unknown with respect to magnitude and sign. While random errors are treated via distribution densities, unknown systematic errors can only be assessed