Tolerance intervals
Random intervals, constructed for independent identically-distributed random variables with unknown distribution function , containing with given probability
at least a proportion
(
) of the probability measure
.
Let be independent and identically-distributed random variables with unknown distribution function
, and let
,
be statistics such that, for a number
(
) fixed in advance, the event
has a given probability
, that is,
![]() | (1) |
In this case the random interval is called a
-tolerance interval for the distribution function
, its end points
and
are called tolerance bounds, and the probability
is called a confidence coefficient. It follows from (1) that the one-sided tolerance bounds
and
(i.e. with
, respectively
) are the usual one-sided confidence bounds with confidence coefficient
for the quantiles
and
, respectively, that is,
![]() |
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Example. Let be independent random variables having a normal distribution
with unknown parameters
and
. In this case it is natural to take the tolerance bounds
and
to be functions of the sufficient statistic
, where
![]() |
Specifically, one takes and
, where the constant
, called the tolerance multiplier, is obtained as the solution to the equation
![]() |
where is the distribution function of the standard normal law; moreover,
does not depend on the unknown parameters
and
. The tolerance interval constructed in this way satisfies the following property: With confidence probability
the interval
contains at least a proportion
of the probability mass of the normal distribution of the variables
.
Assuming the existence of a probability density function , the probability of the event
is independent of
if and only if
and
are order statistics (cf. Order statistic). Precisely this fact is the basis of a general method for constructing non-parametric, or distribution-free, tolerance intervals. Let
be the vector of order statistics constructed from the sample
and let
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Since the random variable has the beta-distribution with parameters
and
, the probability of the event
can be calculated as the integral
, where
is the incomplete beta-function, and hence in this case instead of (1) one obtains the relation
![]() | (2) |
which allows one, for given ,
and
, to define numbers
and
so that the order statistics
and
are the tolerance bounds of the desired tolerance interval. Moreover, for given
,
,
, relation (2) allows one to determine the size
of the collection
necessary for the relation (2) to hold. There are statistical tables available for solving such problems.
References
[1] | 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) |
[2] | S.S. Wilks, "Mathematical statistics" , Wiley (1962) |
[3] | H.H. David, "Order statistics" , Wiley (1981) |
[4] | R.B. Murphy, "Non-parametric tolerance limits" Ann. Math. Stat. , 19 (1948) pp. 581–589 |
[5] | P.N. Somerville, "Tables for obtaining non-parametric tolerance limits" Ann. Math. Stat. , 29 (1958) pp. 599–601 |
[6] | H. Scheffé, J.W. Tukey, "Non-parametric estimation I. Validation of order statistics" Ann. Math. Stat. , 16 (1945) pp. 187–192 |
[7] | D.A.S. Fraser, "Nonparametric methods in statistics" , Wiley (1957) |
[8] | A. Wald, J. Wolfowitz, "Tolerance limits for a normal distribution" Ann. Math. Stat. , 17 (1946) pp. 208–215 |
[9] | H. Robbins, "On distribution-free tolerance limits in random sampling" Ann. Math. Stat. , 15 (1944) pp. 214–216 |
Tolerance intervals. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tolerance_intervals&oldid=48983