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Central limit theorem for directional statistics – Central limit theorem applied to the case of directional statistics; Delta method – to compute the limit distribution of a function of a random variable. Erdős–Kac theorem – connects the number of prime factors of an integer with the normal probability distribution
Directional statistics is the subdiscipline of statistics that deals with directions (unit vectors in R n), axes (lines through the origin in R n) or rotations in R n. The means and variances of directional quantities are all finite, so that the central limit theorem may be applied to the particular case of directional statistics. [2]
The central limit theorem implies that those statistical parameters will have asymptotically normal distributions. The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
The Generalized Central Limit Theorem (GCLT) was an effort of multiple mathematicians (Berstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937. [14] The first published complete proof (in French) of the GCLT was in 1937 by Paul Lévy. [15]
This section illustrates the central limit theorem via an example for which the computation can be done quickly by hand on paper, unlike the more computing-intensive example of the previous section. Sum of all permutations of length 1 selected from the set of integers 1, 2, 3
Local asymptotic normality is a generalization of the central limit theorem. It is a property of a sequence of statistical models, which allows this sequence to be asymptotically approximated by a normal location model, after a rescaling of the parameter.
This theorem can be used to disprove the central limit theorem holds for by using proof by contradiction. This procedure involves proving that Lindeberg's condition fails for X k {\displaystyle X_{k}} .
Pages in category "Central limit theorem" The following 11 pages are in this category, out of 11 total. ... Central limit theorem for directional statistics;