# Interval estimator

*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 [3]–[5]), the Neyman method of confidence intervals ([5], [8], [9]), and the method proposed by L.N. Bol'shev [6].

#### References

[1] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) |

[2] | R.A. Fisher, "Statistical methods and scientific inference" , Hafner (1973) |

[3] | S.N. Bernshtein, "On "fiducial" probabilities of Fisher" Izv. Akad. Nauk SSSR Ser. Mat. , 5 (1941) pp. 85–94 (In Russian) (English abstract) |

[4] | L.N. Bol'shev, "Criticism on "Bernshtein: On fiducial probabilities of Fisher" " , Colected works of S.N. Bernstein , 4 , Moscow (1964) pp. 566–569 (In Russian) |

[5] | J. Neyman, "Silver jubilee of my dispute with Fisher" J. Oper. Res. Soc. Japan , 3 : 4 (1961) pp. 145–154 |

[6] | L.N. Bol'shev, "On the construction of confidence limits" Theor. Probab. Appl. , 10 (1965) pp. 173–177 Teor. Veroyatnost. i Primenen. , 10 : 1 (1965) pp. 187–192 |

[7] | L.N. Bol'shev, E.A. Loginov, "Interval estimates in the presence of nuisance parameters" Theor. Probab. Appl. , 11 (1966) pp. 82–94 Teor. Veryatnost. i Primenen. , 11 : 1 (1966) pp. 94–107 |

[8] | J. Neyman, "Fiducial argument and the theory of confidence intervals" Biometrika , 32 : 2 (1941) pp. 128–150 |

[9] | J. Neyman, "Outline of a theory of statistical estimation based on the classical theory of probability" Philos. Trans. Roy. Soc. London , 236 (1937) pp. 333–380 |

#### Comments

#### References

[a1] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |

**How to Cite This Entry:**

Interval estimator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Interval_estimator&oldid=17171