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Construct a
tolIntNpar(x, coverage, conf.level, cov.type = "content",
ltl.rank = ifelse(ti.type == "upper", 0, 1),
n.plus.one.minus.utl.rank = ifelse(ti.type == "lower", 0, 1),
lb = -Inf, ub = Inf, ti.type = "two-sided")
numeric vector of observations. Missing (NA
), undefined (NaN
), and
infinite (Inf
, -Inf
) values are allowed but will be removed.
a scalar between 0 and 1 indicating the desired coverage of the coverage=0.95
. If cov.type="content"
, you must
supply a value for coverage
or a value for conf.level
, but not both.
If cov.type="expectation"
, this argument is ignored.
a scalar between 0 and 1 indicating the confidence level associated with the conf.level=0.95
. If
cov.type="content"
,
you must supply a value for coverage
or a value for conf.level
, but not both.
If cov.type="expectation"
, this argument is ignored.
character string specifying the coverage type for the tolerance interval.
The possible values are "content"
("expectation"
(
positive integer indicating the rank of the order statistic to use for the lower bound
of the tolerance interval. If ti.type="two-sided"
or ti.type="lower"
,
the default value is ltl.rank=1
(implying the minimum value of x
is used
as the lower bound of the tolerance interval). If ti.type="upper"
, this argument
is set equal to 0
and the value of lb
is used as the lower bound of the
tolerance interval.
positive integer related to the rank of the order statistic to use for
the upper bound of the toleracne interval. A value of
n.plus.one.minus.utl.rank=1
(the default) means use the
first largest value of x
, and in general a value of
n.plus.one.minus.utl.rank=
ti.type="lower"
,
this argument is set equal to 0
and the value of ub
is used as the upper
bound of the tolerance interval.
scalars indicating lower and upper bounds on the distribution. By default, lb=-Inf
and
ub=Inf
. If you are constructing a tolerance interval for a distribution
that you know has a lower bound other than -Inf
(e.g., 0
), set lb
to this
value. Similarly, if you know the distribution has an upper bound other than Inf
, set
ub
to this value. The argument lb
is ignored if ti.type="two-sided"
or
ti.type="lower"
. The argument ub
is ignored if ti.type="two-sided"
or
ti.type="upper"
.
character string indicating what kind of tolerance interval to compute.
The possible values are "two-sided"
(the default), "lower"
, and
"upper"
.
A list of class "estimate"
containing the estimated parameters,
the tolerance interval, and other information. See estimate.object
for details.
A tolerance interval for some population is an interval on the real line constructed so as to
contain
There are two kinds of tolerance intervals (Guttman, 1970):
A
A
Note: A
The Form of a Nonparametric Tolerance Interval
Let tolIntNpar
, the argument
ltl.rank
corresponds to n.plus.one.minus.utl.rank
corresponds to
If we allow
Let shape1=
shape2=
Computations for a shape1=
shape2=
Similarly, if the confidence level associated with the tolerance interval is specified as
shape1=
shape2=
Computations for a shape1=
shape2=
size=
prob=
predIntNpar
for more information on constructing
a nonparametric prediction interval.
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.
eqnpar
, estimate.object
,
tolIntNparN
, Tolerance Intervals,
Estimating Distribution Parameters, Estimating Distribution Quantiles.
# NOT RUN {
# Generate 20 observations from a lognormal mixture distribution
# with parameters mean1=1, cv1=0.5, mean2=5, cv2=1, and p.mix=0.1.
# The exact two-sided interval that contains 90% of this distribution is given by:
# [0.682312, 13.32052]. Use tolIntNpar to construct a two-sided 90%
# \eqn{\beta}-content tolerance interval. Note that the associated confidence level
# is only 61%. A larger sample size is required to obtain a larger confidence
# level (see the help file for tolIntNparN).
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(23)
dat <- rlnormMixAlt(20, 1, 0.5, 5, 1, 0.1)
tolIntNpar(dat, coverage = 0.9)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: None
#
#Data: dat
#
#Sample Size: 20
#
#Tolerance Interval Coverage: 90%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact
#
#Tolerance Interval Type: two-sided
#
#Confidence Level: 60.8253%
#
#Tolerance Limit Rank(s): 1 20
#
#Tolerance Interval: LTL = 0.5035035
# UTL = 9.9504662
#----------
# Clean up
rm(dat)
#----------
# Reproduce Example 17-4 on page 17-21 of USEPA (2009). This example uses
# copper concentrations (ppb) from 3 background wells to set an upper
# limit for 2 compliance wells. The maximum value from the 3 wells is set
# to the 95% confidence upper tolerance limit, and we need to determine the
# coverage of this tolerance interval. The data are stored in EPA.92c.copper2.df.
# Note that even though these data are Type I left singly censored, it is still
# possible to compute an upper tolerance interval using any of the uncensored
# observations as the upper limit.
EPA.92c.copper2.df
# Copper.orig Copper Censored Month Well Well.type
#1 <5 5.0 TRUE 1 1 Background
#2 <5 5.0 TRUE 2 1 Background
#3 7.5 7.5 FALSE 3 1 Background
#...
#9 9.2 9.2 FALSE 1 2 Background
#10 <5 5.0 TRUE 2 2 Background
#11 <5 5.0 TRUE 3 2 Background
#...
#17 <5 5.0 TRUE 1 3 Background
#18 5.4 5.4 FALSE 2 3 Background
#19 6.7 6.7 FALSE 3 3 Background
#...
#29 6.2 6.2 FALSE 5 4 Compliance
#30 <5 5.0 TRUE 6 4 Compliance
#31 7.8 7.8 FALSE 7 4 Compliance
#...
#38 <5 5.0 TRUE 6 5 Compliance
#39 5.6 5.6 FALSE 7 5 Compliance
#40 <5 5.0 TRUE 8 5 Compliance
with(EPA.92c.copper2.df,
tolIntNpar(Copper[Well.type=="Background"],
conf.level = 0.95, lb = 0, ti.type = "upper"))
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: None
#
#Data: Copper[Well.type == "Background"]
#
#Sample Size: 24
#
#Tolerance Interval Coverage: 88.26538%
#
#Coverage Type: content
#
#Tolerance Interval Method: Exact
#
#Tolerance Interval Type: upper
#
#Confidence Level: 95%
#
#Tolerance Limit Rank(s): 24
#
#Tolerance Interval: LTL = 0.0
# UTL = 9.2
#----------
# Repeat the last example, except compute an upper
# \eqn{\beta}-expectation tolerance interval:
with(EPA.92c.copper2.df,
tolIntNpar(Copper[Well.type=="Background"],
cov.type = "expectation", lb = 0, ti.type = "upper"))
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: None
#
#Data: Copper[Well.type == "Background"]
#
#Sample Size: 24
#
#Tolerance Interval Coverage: 96%
#
#Coverage Type: expectation
#
#Tolerance Interval Method: Exact
#
#Tolerance Interval Type: upper
#
#Tolerance Limit Rank(s): 24
#
#Tolerance Interval: LTL = 0.0
# UTL = 9.2
# }
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