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,

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

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