# Introduction

• Norbert Schmitz
Part of the Lecture Notes in Statistics book series (LNS, volume 79)

## 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.

## Keywords

Decision Procedure Sampling Plan Statistical Decision Sequential Procedure Defective Item
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

1. 1.
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.Google Scholar
2. 2.
[y] denotes the smallest integer ≥ y and u α/2 is the α/2-quantile of the N(0, l)-distribution.Google Scholar
3. 3.
Dodge, H.F./ Romig, H.G.: A method of sampling inspection. Bull. Syst. Techn. J. 8(1929), 613–631Google Scholar
4. 4.
Bartky, W.: Multiple sampling with constant probability. Ann. Math. Statist. 14(1943), 363–377
5. 5.
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
6. 6.
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.Google Scholar
7. 7.
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).
8. 8.
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).Google Scholar
9. 9.
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.Google Scholar
10. 10.
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).Google Scholar
11. 11.
Wald, A./ Wolfowitz, J.: Optimum character of the sequential probability ratio test. Ann. Math. Statist. 19(1948), 326–339.
12. 12.
Arrow, K.I./ Blackwell, D./ Girshick, M.A.: Bayes and minimax solutions of sequential decision problems. Econometrica 17(1949), 213–244.
13. 13.
This situation will be called ”iid case”.Google Scholar
14. 14.
Exceptions arise in ”open ended tests” in which infinite samples are accepted for one hypothesis (e.g. for the superior drug).Google Scholar
15. 15.
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.Google Scholar
16. 16.
All these values have been computed by Dr. Marion Harenbrock (Institut für Mathematische Statistik der Univ. Münster)Google Scholar
17. 17.
The index M will be omitted when there is no possibility of misunderstanding.Google Scholar
18. 18.
To avoid further terminological complications we assume the same set M for all stages; (external) restrictions will be regarded when defining sampling plans.Google Scholar
19. 19.
[x] describes the largest integer ≤ x.Google Scholar
20. 20.
”0” corresponds to the decision ”no further observation”.Google Scholar
21. 21.