View source for Confidence estimation

{{TEX|want}}

A method in mathematical statistics for the construction of a set of approximate values of the unknown parameters of probability distributions.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246201.png" /> be a random vector assuming values in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246202.png" /> in a Euclidean space and let the probability distribution of this vector belong to the parametric family of distributions defined by the densities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246205.png" />, with respect to some measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246206.png" />. It is assumed that the true value of the parametric point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246207.png" /> corresponding to the result of observations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246208.png" /> is unknown. Confidence estimation consists in constructing a certain set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c0246209.png" />, depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462010.png" />, containing the value of a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462011.png" /> corresponding to the unknown true value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462012.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462013.png" /> be the range of values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462015.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462017.png" />, be some family of sets belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462019.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462020.png" />; moreover, it is assumed that for an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462021.png" /> and any value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462022.png" /> the probability of the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462023.png" /> is defined. This probability is given by the integral

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462024.png" /></td> </tr></table>

and is said to be the covering probability of the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462025.png" /> by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462026.png" /> for a given value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462027.png" />.

If the true value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462028.png" /> is unknown, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462029.png" /> (from the family of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462031.png" />) which corresponds to the results of observations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462032.png" /> is said to be a confidence set or an [[Interval estimator|interval estimator]] for the unknown true value of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462033.png" />. The confidence probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462034.png" />, which can be expressed, in terms of the covering probability, by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462035.png" /></td> </tr></table>

is employed as the probability characteristic of the interval estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462036.png" /> constructed according to the above rule. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462037.png" /> is the probability of covering by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462038.png" /> of the value of a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462039.png" /> corresponding to the unknown true parametric point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462040.png" />.

If the confidence probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462041.png" /> is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462042.png" />, the interval estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462043.png" /> is said to be similar to the sampling space. This name is due to the analogy between the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462044.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462045.png" /></td> </tr></table>

In a more general situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462046.png" /> depends on the unknown <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462047.png" />, and for this reason the quality of the interval estimator is usually characterized in practical work by the confidence level

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462048.png" /></td> </tr></table>

where the lower bound is calculated over the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462049.png" />. (The confidence level is sometimes called the confidence coefficient.)

Optimization of confidence estimation is defined by the requirements to be met by interval estimators. For instance, if the objective is to construct confidence sets similar to sampling spaces, with a given confidence level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462050.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462051.png" />), the first requirement is expressed by the identity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462052.png" /></td> </tr></table>

It is natural to look for an interval estimator that covers the true value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462053.png" /> with a probability at least that of covering an arbitrary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462054.png" />. In other words, the second requirement, known as the requirement of unbiasedness, is expressed by the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462055.png" /></td> </tr></table>

Under these conditions, the  "best"  interval estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462056.png" /> may reasonably be taken as one which covers any value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462057.png" /> other than the true value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462058.png" /> with a smaller probability. Hence the third requirement of  "highest accuracy" : For any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462059.png" /> other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462060.png" /> and meeting the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462061.png" /></td> </tr></table>

the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462062.png" /></td> </tr></table>

must be valid.

The task of finding interval estimators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462063.png" /> satisfying all three requirements is equivalent to the task of constructing unbiased most-powerful statistical tests similar to the sampling space and having significance level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462064.png" />. The problem of the existence of a solution to this problem and its constructive description form the base of the general theory of statistical hypothesis testing.

Confidence estimation is most often used when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462065.png" /> is a scalar function. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462067.png" />, be independent random variables subject to the same [[Normal distribution|normal distribution]], with the unknown parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462069.png" />. The problem is to construct an interval estimator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462070.png" />. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462071.png" /></td> </tr></table>

Since the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462072.png" /> is subject to the [[Student distribution|Student distribution]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462073.png" /> degrees of freedom, and since this distribution does not depend on the unknown parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462075.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462077.png" />), it follows that, for any positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462078.png" />, the probability of occurrence of the event

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462079.png" /></td> </tr></table>

depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462080.png" />. If this interval is taken as the interval estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462081.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462082.png" />, it will correspond to the confidence probability

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462083.png" /></td> </tr></table>

which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024620/c02462084.png" />. Such an interval estimator is known as a confidence interval, while its end points are known as confidence bounds; in this case the confidence interval is a confidence estimator similar to the sampling space. In this example the interval estimator is most accurate unbiased.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Wilks,   "Mathematical statistics" , Wiley  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Schmetterer,   "Introduction to mathematical statistics" , Springer  (1974)  (Translated from German)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>