Mathematical statistics

As mathematical statistics is defined as the branch of statistics that analyzes the methods and procedures of statistics by mathematical means or reason with their help only. Together with probability theory forms the mathematical statistics, known as the stochastic part of mathematics. Most essentially synonymous, the terms inductive and inferential statistics are ( inferential statistics ) used, the characterization of the complementary to the descriptive statistics of the statistics.

The mathematical basis of mathematical statistics, the probability theory.

What deals statistic?

Subject of the statistics are populations whose members all possess a particular characteristic. We are looking for statements about how often take this feature within the population of its possible values ​​. Often the statements limited to derived quantities such as the average of the feature values ​​that hold the members of the population.

An example is the age distribution often graphed as age pyramid, which may be in the population, for example, the German population. Because a precise determination of the age distribution of the Germans presupposes an elaborate full survey as a census, one looks for methods that largely reliable statements on the basis of sample surveys are already possible. As in the example of the Political Barometer will be to only members of randomly selected subsets of the population, so-called samples examined for the feature of interest.

Methodology of mathematical statistics

If the age distribution in the population known probabilities for the observable within sample age distributions could be calculated with the formulas of probability theory, which are subject due to the random selection of the sample random fluctuations. In mathematical statistics one uses those calculations to inversely deduce from the sample results to the population can: Here, the concrete observed for a sample characteristic values ​​are those frequency distributions characterized within the population on the basis with which the observation made result can be explained in a plausible manner. The focus of theoretical investigations are not only the conclusions reached, but also estimates how numerically exactly and how safe are those forecasts.

The one user interest frequency distributions are only indirectly subject of the methods of mathematical statistics. Instead, these methods refer to random variables. In this case, such random variables in particular are regarded whose probability distribution corresponds to the relative frequencies of feature values ​​. Especially for the example given in the age distribution is a realized value of the random variable equal to the age of a randomly selected Germans. In this way, the observations of a sample obtained can be interpreted as a so-called stochastic realizations of independent and identically distributed random variables. The knowledge is represented in this case by a family of probability distributions or by a corresponding family of probability measures. We speak of a distributional assumptions. These can both statements about possible feature values ​​, such as in relation to their integrality, as well as on the type of distribution, for example, " the values ​​are normally distributed " involve.

The central field of mathematical statistics, the estimation theory, developed within the appropriate estimation method. Methodologically taken the approach that you compare starting investigated by the distribution assumption certain classes of estimating functions with respect to different quality criteria (such sufficiency or efficiency). In an estimator such it can be both a monovalent approximation of a desired parameter of the population as well as a range estimate in the form of a so-called confidence interval. Specific assumptions about the population can be verified by appropriate statistical tests. Here, a 0-1 decision over the rejection or retention of the hypothesis is based brought about by a hypothesis based on the sampling results.

Statistical models

An utter formalization on the basis of mathematical objects is achieved with the concept of the statistical model, often referred to as a statistical space. Differing from the previously described, more application- oriented scenario can be dispensed to the definition of a population:

The possible sample results are summarized to a lot, the sampling area. The fact observable events are formally characterized by the sample space defined σ - algebra. The distribution assumption, that is the candidate probability distributions correspond to a family of probability measures. A statistical model is thus formally a triple. Is a real parameter vector, ie, it is called a parametric model with parameter space. In the event of a real parameter called one-parameter model.

A measurable function is called in another measurement space sampling function, or statistics. An estimator, or just an estimator of a parameter of the parameter is a sampling function.

Example

A (possibly pronged ) coin is tossed once. The probability that in a litter falls head is unknown. It is observed how often is the coin head. The corresponding statistical model for this is given by

• As the sample space,
• The power set of,
• As the set of possible values ​​of the unknown parameter,
• Is the binomial distribution with parameters and.

An obvious for the parameter estimator in this case is represented by the relative frequency for.