Neyman method of confidence intervals
One of the methods of confidence estimation, which makes it possible to obtain interval estimators (cf. Interval estimator) for unknown parameters of probability laws from results of observations. It was proposed and developed by J. Neyman (see [1], [2]). The essence of the method consists in the following. Let be random variables whose joint distribution function
depends on a parameter
,
. Suppose, next, that as point estimator of the parameter
a statistic
is used with distribution function
,
. Then for any number
in the interval
one can define a system of two equations in
:
![]() | (*) |
Under certain regularity conditions on , which in almost-all cases of practical interest are satisfied, the system (*) has a unique solution
![]() |
such that
![]() |
The set is called the confidence interval (confidence estimator) for the unknown parameter
with confidence probability
. The statistics
and
are called the lower and upper confidence bounds corresponding to the chosen confidence coefficient
. In turn, the number
![]() |
is called the confidence coefficient of the confidence interval . Thus, Neyman's method of confidence intervals leads to interval estimators with confidence coefficient
.
Example 1. Suppose that independent random variables are subject to one and the same normal law
whose mathematical expectation
is not known (cf. Normal distribution). Then the best estimator for
is the sufficient statistic
, which is distributed according to the normal law
. Fixing
in
and solving the equations
![]() |
one finds the lower and upper confidence bounds
![]() |
corresponding to the chosen confidence coefficient . Since
![]() |
the confidence interval for the unknown mathematical expectation of the normal law
has the form
![]() |
and its confidence coefficient is precisely .
Example 2. Let be a random variable subject to the binomial law with parameters
and
(cf. Binomial distribution), that is, for any integer
,
![]() |
![]() |
where
![]() |
is the incomplete beta-function (,
,
). If the "success" parameter
is not known, then to determine the confidence bounds one has to solve, in accordance with Neyman's method of confidence intervals, the equations
![]() |
where . From tables of mathematical statistics the roots
and
of these equations are determined, which are the upper and lower confidence bounds, respectively, with confidence coefficient
. The coefficient of the resulting confidence interval
is precisely
. Obviously, if an experiment gives
, then
, and if
, then
.
Neyman's method of confidence intervals differs substantially from the Bayesian method (cf. Bayesian approach) and the method based on Fisher's fiducial approach (cf. Fiducial distribution). In it the unknown parameter of the distribution function
is treated as a constant quantity, and the confidence interval
is constructed from an experiment in the course of which the value of the statistic
is calculated. Consequently, according to Neyman's method of confidence intervals, the probability for
to hold is the a priori probability for the fact that the confidence interval
"covers" the unknown true value of the parameter
. In fact, Neyman's confidence method remains valid if
is a random variable, because in the method the interval estimator is constructed from carrying out an experiment and consequently does not depend on the a priori distribution of the parameter. Neyman's method differs advantageously from the Bayesian and the fiducial approach by being independent of a priori information about the parameter
and so, in contrast to Fisher's method, is logically sound. In general, Neyman's method leads to a whole system of confidence intervals for the unknown parameter, and in this context arises the problem of constructing an optimal interval estimator having, for example, the properties of being unbiased, accurate or similar, which can be solved within the framework of the theory of statistical hypothesis testing.
References
[1] | J. Neyman, "On the problem of confidence intervals" Ann. Math. Stat. , 6 (1935) pp. 111–116 |
[2] | J. Neyman, "Outline of a theory of statistical estimation based on the classical theory of probability" Philos. Trans. Roy. Soc. London. Ser. A. , 236 (1937) pp. 333–380 |
[3] | L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) |
[4] | 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 |
[5] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |
Neyman method of confidence intervals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neyman_method_of_confidence_intervals&oldid=49488