Statistics: classical or Bayesian. Two ways in comparison. (Statistik: Klassisch oder Bayes. Zwei Wege im Vergleich.) (German) Zbl 1341.62028

Tschirk, Wolfgang

Springer-Lehrbuch. Heidelberg: Springer Spektrum (ISBN 978-3-642-54384-5/pbk; 978-3-642-54385-2/ebook). xi, 166 p. (2014).

This monograph represents an introduction to both classical and Bayesian theory of estimation and testing. The motivation arises from three historical examples which will be discussed at the end of this exposition.
The start is an introduction to probability theory together with a careful discussion of probability concepts due to Popper and Jaynes. Point estimation and confidence intervals are mainly investigated for the normal and the binomial case. For these cases, also the classical test procedures are presented. In the Bayesian approach it is assumed that the unknown parameters are as well random variables with a specific a priori distribution. This allows for the computation of a posterior distribution. The posterior distribution opens the way for Bayesian statistical inference. Point estimation consists of computation of posterior mean, posterior median, etc. The concept of a confidence interval or a credibility interval is now much simpler. Moreover, it is possible to compute the probability of a hypothesis. Priors are diffuse or conjugate, i.e. normal/normal, binomial/beta.
Three historical examples are considered in some detail. The first example is the Semmelweis-conjecture. Semmelweis argued that by more careful hygienic behaviour the mortality of women on a birth-station could seriously be decreased. Despite an overwhelming statistical evidence it was very difficult to turn around the heads of the medical authorities. The second example concerns the Millikan experiment of measuring the elementary electrical charge. The third example is about a psychological problem due to Milgram, entitled “Obedience to authority”. A fourth example could be a Bayesian confidence interval for the HbA1c.
This monograph is well written and suited for a seminar in statistics.

Reviewer: Hilmar Drygas (Kassel)

MSC:

62-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62C10	Bayesian problems; characterization of Bayes procedures
62F03	Parametric hypothesis testing
62F10	Point estimation
62A01	Foundations and philosophical topics in statistics
62C12	Empirical decision procedures; empirical Bayes procedures

Keywords:

probability; point estimation; testing; posterior distribution