# 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 $X$ be a random variable taking values in a sample space $( \mathfrak X , \mathfrak B , {\mathsf P} _ \theta )$, $\theta \in \Theta$, $\Theta$ an interval on the real axis, where the true value of $\theta$ is unknown. An interval $( a _ {1} ( X) , a _ {2} ( X) ) \subseteq \Theta$ with boundaries that are functions of the random variable $X$ being observed is called an interval estimator, or confidence interval, for $\theta$; the number

$$p = \inf _ {\theta \in \Theta } \ {\mathsf P} \{ a _ {1} ( X) < \theta < a _ {2} ( X) \mid \theta \}$$

is called the confidence coefficient of this confidence interval, and $a _ {1} ( X)$ and $a _ {2} ( X)$ 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 $\theta$.

Suppose that on a set $T \subset \mathbf R ^ {1}$ a family of functions

$$u ( \theta , \cdot ) = \ ( u _ {1} ( \theta , \cdot ) \dots u _ {k} ( \theta , \cdot ) ) : \ T \rightarrow \mathfrak U \subset \mathbf R ^ {k} ,$$

$$\theta = ( \theta _ {1} \dots \theta _ {m} ) \in \Theta \subset \mathbf R ^ {m} ,$$

has been given, and suppose that it is required to estimate the function $u ( \theta , \cdot )$ corresponding to the unknown true value of $\theta$ using the realization of a random vector $X = ( X _ {1} \dots X _ {n} )$ taking values in the sample space $( \mathfrak X , \mathfrak B , {\mathsf P} _ \theta )$, $\mathfrak X \subset \mathbf R ^ {m}$, $\theta \in \Theta$. To each $t \in T$ corresponds a set $B ( t)$, which is the image of $\Theta$ under $u ( \cdot , t ) : \Theta \rightarrow B ( t) \subset \mathfrak U$. By definition, a set $C ( X , t ) \subset B ( t)$ is called a confidence set for $u ( \theta , t )$ if $u ( \theta , \cdot )$ at $t \in T$ has confidence probability

$${\mathsf P} \{ u ( \theta , t ) \in C ( X , t ) \ \mid \theta \} = p ( \theta , t )$$

and confidence coefficient

$$p ( t) = \ \inf _ {\theta \in \Theta } \ p ( \theta , t ) .$$

The totality of all confidence sets $C ( X , t )$ forms in $\mathfrak U$ the confidence region $C ( X)$ for $u ( \Theta , \cdot ) : T \rightarrow \mathfrak U$ with confidence probability

$${\mathsf P} \{ C ( X) \ni u ( \theta , \cdot ) : T \rightarrow \mathfrak U \mid \theta \} = \widetilde{p} ( \theta )$$

and confidence coefficient

$$p = \inf _ {\theta \in \Theta } \widetilde{p} ( \theta ) .$$

Sets of the type $C ( X , t )$ or $C ( X)$ are called interval estimators for one value $u ( \theta , t )$ of a function $u ( \theta , \cdot )$ at a point and for the function $u ( \theta , \cdot )$, 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