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Parametric estimation of distributions

Lecture



Parametric estimation of distributions is implemented in cases where the type of distribution is known.   Parametric estimation of distributions and according to the training sample, it is only necessary to estimate the values ​​of the parameters of these distributions. A priori species knowledge   Parametric estimation of distributions in practice it is not often, however, given the convenience of this approach, sometimes it is assumed, for example, that   Parametric estimation of distributions - normal law. Such assumptions are not always convincing, but they are used nonetheless if the learning results lead to acceptable recognition errors.

So, learning is reduced to estimating the values ​​of parameters.   Parametric estimation of distributions with a previously known form of these distributions. A special place among distributions is occupied by the normal law. This is due to the fact that, as is known from mathematical statistics, if a random variable is generated by the influence of a sufficiently large number of random factors with arbitrary distribution laws and there is no clearly dominant among these influences, then the quantity of interest has a normal distribution law. For one-dimensional case

  Parametric estimation of distributions

(for simplicity, we will henceforth consider the one-dimensional case, and interested students can refer to the literature cited at the end of the lecture outline).

The parameters of this distribution are two quantities:   Parametric estimation of distributions - expected value,   Parametric estimation of distributions - dispersion. They also need to be evaluated by the sample. One of the most simple is the method of moments. It is applicable for distributions   Parametric estimation of distributions depending on   Parametric estimation of distributions parameters having   Parametric estimation of distributions finite first moments that can be expressed as explicit functions   Parametric estimation of distributions parameters   Parametric estimation of distributions . Then, calculating the sample   Parametric estimation of distributions   Parametric estimation of distributions her first moments and equating them   Parametric estimation of distributions get the system of equations

  Parametric estimation of distributions ,

from which estimates are determined   Parametric estimation of distributions .

For one-dimensional normal law

  Parametric estimation of distributions   Parametric estimation of distributions .

  Parametric estimation of distributions   Parametric estimation of distributions .


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Pattern recognition

Terms: Pattern recognition