Confidence interval

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]} .

Definition [edit]

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

.

Boundary points of the confidence interval and 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 . ^{[link 2]}

Another interpretation of the concept of a confidence interval: it can be considered as an interval of parameter values 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 Itself is considered a random variable with a certain a priori distribution (in the simplest case - uniform), and the sample fixed (in classical statistics, exactly the opposite). Bayesian - confidence interval is an interval covering a parameter value saposterory probability :

.

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 .