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Confidence interval

Lecture



Confidence interval is a term used in mathematical statistics with an interval (as opposed to a point) estimate of statistical parameters, which is preferable for a small sample size. A confidence interval is an interval that covers an unknown parameter with a given reliability.

The method of confidence intervals was developed by the American statistician Jerzy Neumann, based on the ideas of the English statistics by Ronald Fisher [ref 1] .

Content

  • 1 Definition
    • 1.1 Examples
  • 2 Bayesian confidence interval
  • 3 Notes

Definition [edit]

The confidence interval of the parameter   Confidence interval random distribution   Confidence interval with a confidence level of 100% - p [note 1] , generated by the sample   Confidence interval called interval with borders   Confidence interval and   Confidence interval which are realizations of random variables   Confidence interval and   Confidence interval such that

  Confidence interval .

Boundary points of the confidence interval   Confidence interval and   Confidence interval are called confidential limits .

Interpretation of the confidence interval based on intuition will be as follows: if p is large (say, 0.95 or 0.99), then the confidence interval will almost certainly contain the true value   Confidence interval . [link 2]

Another interpretation of the concept of a confidence interval: it can be considered as an interval of parameter values   Confidence interval compatible with the experimental data and do not contradict them.

Examples [edit]

  • Confidence interval for the expectation of a normal sample;
  • Confidence interval for normal sample variance.

Bayesian Confidence Interval [edit]

In Bayesian statistics, there is a similar, but different in some key details, definition of a confidence interval. Here is the estimated parameter   Confidence interval Itself is considered a random variable with a certain a priori distribution (in the simplest case - uniform), and the sample   Confidence interval fixed (in classical statistics, exactly the opposite). Bayesian   Confidence interval - confidence interval is an interval   Confidence interval covering a parameter value   Confidence interval saposterory probability   Confidence interval :

  Confidence interval .

As a rule, classical and Bayesian confidence intervals differ. In English literature, the Bayesian confidence interval is commonly referred to as the term credible interval , and the classic is called the confidence interval .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis