Mathematical Statistics

1Basic Ideas of Mathematical Statistics

Elementary statistical computations have been carried out for thousands of years. For example, the arithmetic mean from a number of measures or observation data has been known for a very long time.

First descriptive statistics arose starting with the collection of data, for example, at the national census or in registers of medical cards, and followed by compression of these data in the form of statistics or graphical representations (figures). Mathematical statistics developed on the fundament of probability theory from the end of 19th century on. At the beginning of the 20th century, Karl Pearson and Sir Ronald Aylmer Fisher were notable pioneers of this new discipline. Fisher’s book (1925) was a milestone providing experimenters such basic concepts as his well‐known maximum likelihood method and analysis of variance as well as notions of sufficiency and efficiency. An important information measure is still called the Fisher information (see Section 1.4).

Concerning historical development we do not want to go into detail. We refer interested readers to Stigler (1986, 1990). Instead we will describe the actual state of the theory. Nevertheless many stimuli come from real applications. Hence, from time to time we will include real examples.

Although the probability calculus is the fundament of mathematical statistics, many practical problems containing statements about random variables cannot be solved with this calculus alone. For example, ...