Abstract
In ”classical” mathematical statistics one assumes that a statistical decision (e.g. point estimator for a parameter, test of hypotheses, curve estimator, confidence interval for a parameter, etc.) has to be made on the basis of a fixed number n of observed values x 1,…, x n — large parts of the corresponding theory may be found in the books by Lehmann [Le], Bickel/Doksum [B/D] or Witting [Wi1], [Wi2]. For many practical applications this is just the description of the situation the statistician is confronted with: After the observations have been made the investigator shows up with his data and asks for advice on how to analyze them.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Notes
Proceeding in this way the statistician at the same time reaches a new ”position” within the statistical investigation: While in classical statistics he often has to try to make the best from data which have been collected without consulting him, he now has influence on the pattern (the sampling rules and the final sample size) of the experiment.
[y] denotes the smallest integer ≥ y and u α/2 is the α/2-quantile of the N(0, l)-distribution.
Dodge, H.F./ Romig, H.G.: A method of sampling inspection. Bull. Syst. Techn. J. 8(1929), 613–631
Bartky, W.: Multiple sampling with constant probability. Ann. Math. Statist. 14(1943), 363–377
For an extended description of the work of that group we refer to the article ”The Statistical Research Group, 1942–1945” by W.A. Wallis; J. Amer. Statist. Assoc. 75 (1980), 320–335
Abraham Wald, born Oct. 31. 1902 in Cluj (Romania), studied in Vienna, emigrated in 1938 to USA, died Dec. 13. 1950 by an aeroplane crash in India. He made important contributions to geometry, mathematical economics and to statistical decision theory.
For the anecdote that Wald had as enemy alien no access to his own research results comp, the article by W.A. Wallis (footnote 5).
Statistical Research Group, Columbia University (1945): Final Report (submitted to the Applied Mathematics Panel, National Defense Research Committee in completion of research under Contract OEMsr-618 between the trustees of Columbia University and the Office of Scientific Research and Development).
We refer to the report ”Sequential tests in industrial statistics”, J. Royal Statist. Soc., Supplement 8 (1946), 1–26 (with discussion) by G.A. Barnard.
Barnard, G.A.: Economy in sampling with special reference to engineering experimentation. Brit. Min. Supply Adv. Serv., Stat. Math. and Qual. Control. Techn. Report QC/R/7,I (1944).
Wald, A./ Wolfowitz, J.: Optimum character of the sequential probability ratio test. Ann. Math. Statist. 19(1948), 326–339.
Arrow, K.I./ Blackwell, D./ Girshick, M.A.: Bayes and minimax solutions of sequential decision problems. Econometrica 17(1949), 213–244.
This situation will be called ”iid case”.
Exceptions arise in ”open ended tests” in which infinite samples are accepted for one hypothesis (e.g. for the superior drug).
Dr. K.-H. Eger (TH Chemnitz, Germany) has computed these values by using the algorithm contained in his book [Eg]; for computational reasons he used pi = 0,90016837105.
All these values have been computed by Dr. Marion Harenbrock (Institut für Mathematische Statistik der Univ. Münster)
The index M will be omitted when there is no possibility of misunderstanding.
To avoid further terminological complications we assume the same set M for all stages; (external) restrictions will be regarded when defining sampling plans.
[x] describes the largest integer ≤ x.
”0” corresponds to the decision ”no further observation”.
a1,…,ao):=().
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer-Verlag New York, Inc.
About this chapter
Cite this chapter
Schmitz, N. (1993). Introduction. In: Schmitz, N. (eds) Optimal Sequentially Planned Decision Procedures. Lecture Notes in Statistics, vol 79. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2736-6_1
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2736-6_1
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97908-3
Online ISBN: 978-1-4612-2736-6
eBook Packages: Springer Book Archive