for the unknown true value of a scalar parameter of a probability distribution
An interval belonging to the set of admissible values of the parameters, with boundaries that are functions of the results of observations subject to the given distribution. Let be a random variable taking values in a sample space , , an interval on the real axis, where the true value of is unknown. An interval with boundaries that are functions of the random variable being observed is called an interval estimator, or confidence interval, for ; the number
is called the confidence coefficient of this confidence interval, and and are called the lower, respectively, upper, confidence bounds. The concept of an interval estimator has been generalized to the more general case when it is required to estimate some function, or some value of it, depending on a parameter .
Suppose that on a set a family of functions
has been given, and suppose that it is required to estimate the function corresponding to the unknown true value of using the realization of a random vector taking values in the sample space , , . To each corresponds a set , which is the image of under . By definition, a set is called a confidence set for if at has confidence probability
and confidence coefficient
The totality of all confidence sets forms in the confidence region for with confidence probability
and confidence coefficient
Sets of the type or are called interval estimators for one value of a function at a point and for the function , respectively.
There are several approaches to the construction of interval estimators for independent parameters of a distribution. The best known are the Bayesian approach, based on the Bayes theorem, Fisher's method, based on the fiducial distribution (for Fisher's method, see –), the Neyman method of confidence intervals (, , ), and the method proposed by L.N. Bol'shev .
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Interval estimator. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interval_estimator&oldid=17171