Description

Construct a β-content or β-expectation tolerance interval nonparametrically without making any assumptions about the form of the distribution except that it is continuous.

Usage

Arguments

Details

A tolerance interval for some population is an interval on the real line constructed so as to contain 100 β \% of the population (i.e., 100 β \% of all future observations), where 0 < β < 1. The quantity 100 β \% is called the coverage.

There are two kinds of tolerance intervals (Guttman, 1970):

• A β-content tolerance interval with confidence level 100(1-α)\% is constructed so that it contains at least 100 β \% of the population (i.e., the coverage is at least 100 β \%) with probability 100(1-α)\%, where 0 < α < 1. The quantity 100(1-α)\% is called the confidence level or confidence coefficient associated with the tolerance interval.

• A β-expectation tolerance interval is constructed so that the average coverage of the interval is 100 β \%.

Note: A β-expectation tolerance interval with coverage 100 β \% is equivalent to a prediction interval for one future observation with associated confidence level 100 β \%. Note that there is no explicit confidence level associated with a β-expectation tolerance interval. If a β-expectation tolerance interval is treated as a β-content tolerance interval, the confidence level associated with this tolerance interval is usually around 50% (e.g., Guttman, 1970, Table 4.2, p.76).

The Form of a Nonparametric Tolerance Interval
Let \underline{x} denote a random sample of n independent observations from some continuous distribution and let x_{(i)} denote the i'th order statistic in \underline{x}. A two-sided nonparametric tolerance interval is constructed as:

[x_{(u)}, x_{(v)}] \;\;\;\;\;\; (1)

where u and v are positive integers between 1 and n, and u < v. That is, u denotes the rank of the lower tolerance limit, and v denotes the rank of the upper tolerance limit. To make it easier to write some equations later on, we can also write the tolerance interval (1) in a slightly different way as:

[x_{(u)}, x_{(n+1-w)}] \;\;\;\;\;\; (2)

where

w = n + 1 - v \;\;\;\;\;\; (3)

so that w is a positive integer between 1 and n-1, and u < n+1-w. In terms of the arguments to the function tolIntNpar, the argument ltl.rank corresponds to u, and the argument n.plus.one.minus.utl.rank corresponds to w.

If we allow u=0 and w=0 and define lower and upper bounds as:

x_{(0)} = lb \;\;\;\;\;\; (4)

x_{(n+1)} = ub \;\;\;\;\;\; (5)

then equation (2) above can also represent a one-sided lower or one-sided upper tolerance interval as well. That is, a one-sided lower nonparametric tolerance interval is constructed as:

[x_{(u)}, x_{(n+1)}] = [x_{(u)}, ub] \;\;\;\;\;\; (6)

and a one-sided upper nonparametric tolerance interval is constructed as:

[x_{(0)}, x_{(v)}] = [lb, x_{(v)}] \;\;\;\;\;\; (7)

Usually, lb = -∞ or lb = 0 and ub = ∞.

Let C be a random variable denoting the coverage of the above nonparametric tolerance intervals. Wilks (1941) showed that the distribution of C follows a beta distribution with parameters shape1=v-u and shape2=w+u when the unknown distribution is continuous.

Computations for a β-Content Tolerance Interval
For a β-content tolerance interval, if the coverage C = β is specified, then the associated confidence level (1-α)100\% is computed as:

1 - α = 1 - F(β, v-u, w+u) \;\;\;\;\;\; (8)

where F(y, δ, γ) denotes the cumulative distribution function of a beta random variable with parameters shape1=δ and shape2=γ evaluated at y.

Similarly, if the confidence level associated with the tolerance interval is specified as (1-α)100\%, then the coverage C = β is computed as:

β = B(α, v-u, w+u) \;\;\;\;\;\; (9)

where B(p, δ, γ) denotes the p'th quantile of a beta distribution with parameters shape1=δ and shape2=γ.

Computations for a β-Expectation Tolerance Interval
For a β-expectation tolerance interval, the expected coverage is simply the mean of a beta random variable with parameters shape1=v-u and shape2=w+u, which is given by:

E(C) = \frac{v-u}{n+1} \;\;\;\;\;\; (10)

As stated above, a β-expectation tolerance interval with coverage β 100\% is equivalent to a prediction interval for one future observation with associated confidence level β 100\%. This is because the probability that any single future observation will fall into this interval is β 100\%, so the distribution of the number of N future observations that will fall into this interval is binomial with parameters size=N and prob=β. Hence the expected proportion of future observations that fall into this interval is β 100\% and is independent of the value of N. See the help file for predIntNpar for more information on constructing a nonparametric prediction interval.

Value

A list of class "estimate" containing the estimated parameters, the tolerance interval, and other information. See estimate.object for details.

Note

Tolerance intervals have long been applied to quality control and life testing problems (Hahn, 1970b,c; Hahn and Meeker, 1991; Krishnamoorthy and Mathew, 2009). References that discuss tolerance intervals in the context of environmental monitoring include: Berthouex and Brown (2002, Chapter 21), Gibbons et al. (2009), Millard and Neerchal (2001, Chapter 6), Singh et al. (2010b), and USEPA (2009).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley and Sons, New York.

Danziger, L., and S. Davis. (1964). Tables of Distribution-Free Tolerance Limits. Annals of Mathematical Statistics 35(5), 1361–1365.

Davis, C.B. (1994). Environmental Regulatory Statistics. In Patil, G.P., and C.R. Rao, eds., Handbook of Statistics, Vol. 12: Environmental Statistics. North-Holland, Amsterdam, a division of Elsevier, New York, NY, Chapter 26, 817–865.

Davis, C.B., and R.J. McNichols. (1994a). Ground Water Monitoring Statistics Update: Part I: Progress Since 1988. Ground Water Monitoring and Remediation 14(4), 148–158.

Gibbons, R.D. (1991b). Statistical Tolerance Limits for Ground-Water Monitoring. Ground Water 29, 563–570.

Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.

Guttman, I. (1970). Statistical Tolerance Regions: Classical and Bayesian. Hafner Publishing Co., Darien, CT, Chapter 2.

Hahn, G.J., and W.Q. Meeker. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley and Sons, New York, 392pp.

Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY, pp.88-90.

Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.

Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton.

USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.

Wilks, S.S. (1941). Determination of Sample Sizes for Setting Tolerance Limits. Annals of Mathematical Statistics 12, 91–96.

See Also

eqnpar, estimate.object, tolIntNparN, Tolerance Intervals, Estimating Distribution Parameters, Estimating Distribution Quantiles.

Examples