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More precisely Markov's theorem can be stated as follows: [2] [3] given two braids represented by elements , ′ in the braid groups ,, their closures are equivalent links if and only if ′ can be obtained from applying to a sequence of the following operations:
The theorem was named after Carl Friedrich Gauss and Andrey Markov, although Gauss' work significantly predates Markov's. [3] But while Gauss derived the result under the assumption of independence and normality, Markov reduced the assumptions to the form stated above. [4] A further generalization to non-spherical errors was given by Alexander ...
If the experimental errors, , are uncorrelated, have a mean of zero and a constant variance, , the Gauss–Markov theorem states that the least-squares estimator, ^, has the minimum variance of all estimators that are linear combinations of the observations. In this sense it is the best, or optimal, estimator of the parameters.
Download as PDF; Printable version ... In mathematics, the Markov spectrum, devised by Andrey Markov, is a ... Markov's theorem and 100 years of the uniqueness ...
In linear regression, the use of the least-squares estimator is justified by the Gauss–Markov theorem, which does not assume that the distribution is normal. From the perspective of generalized linear models, however, it is useful to suppose that the distribution function is the normal distribution with constant variance and the link function ...
Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true.
In probability theory, Markov's inequality gives an upper bound on the probability that a non-negative random variable is greater than or equal to some positive constant. Markov's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality.
If the assumptions of OLS regression hold, the solution = (), with =, is an unbiased estimator, and is the minimum-variance linear unbiased estimator, according to the Gauss–Markov theorem. The term λ n I {\displaystyle \lambda nI} therefore leads to a biased solution; however, it also tends to reduce variance.