Last chance! 50% off unlimited learning
Sale ends in
Construct a
tolIntPois(x, coverage = 0.95, cov.type = "content", ti.type = "two-sided",
conf.level = 0.95)
numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a Poisson distribution
(i.e., epois
or epoisCensored
).
If cov.type="content"
then x
must be a numeric vector.
If x
is a numeric vector,
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 tolerance interval.
The default value is coverage=0.95
. If cov.type="expectation"
,
this argument is ignored.
character string specifying the coverage type for the tolerance interval.
The possible values are "content"
("expectation"
(
character string indicating what kind of tolerance interval to compute.
The possible values are "two-sided"
(the default), "lower"
, and
"upper"
.
a scalar between 0 and 1 indicating the confidence level associated with the tolerance
interval. The default value is conf.level=0.95
.
If x
is a numeric vector, tolIntPois
returns a list of class
"estimate"
containing the estimated parameters, a component called
interval
containing the tolerance interval information, and other
information. See estimate.object
for details.
If x
is the result of calling an estimation function, tolIntPois
returns a list whose class is the same as x
. The list contains the same
components as x
. If x
already has a component called
interval
, this component is replaced with the tolerance interval
information.
If x
contains any missing (NA
), undefined (NaN
) or
infinite (Inf
, -Inf
) values, they will be removed prior to
performing the estimation.
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
Because of the discrete nature of the Poisson distribution,
even true tolerance intervals (tolerance intervals based on the true value of
lambda=2
, the
interval [0, 4] contains 94.7% of this distribution and the interval [0, 5]
contains 98.3% of this distribution. Thus, no interval can contain exactly 95%
of this distribution.
It can be shown that all distributions that are one parameter exponential families have the MLR property, and the Poisson distribution is a one-parameter exponential family, so the method of Zacks (1970) can be applied to a Poisson distribution.
Let lambda=
Let lambda=
ti.type="upper"
, the first step is to compute the one-sided upper
epois
). Denote this upper
confidence limit by ti.type="lower"
, the first step is to compute the one-sided
lower ti.type="two-sided"
, the first step is to compute the two-sided
tolIntPois
uses the exact confidence limits for
epois
).
size=
prob=
predIntPois
for information on how these intervals are constructed.
Gibbons, R.D. (1987b). Statistical Models for the Analysis of Volatile Organic Compounds in Waste Disposal Sites. Ground Water 25, 572--580.
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.
Hahn, G.J., and W.Q. Meeker. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley and Sons, New York.
Johnson, N. L., S. Kotz, and A. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, Chapter 4.
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.
Zacks, S. (1970). Uniformly Most Accurate Upper Tolerance Limits for Monotone Likelihood Ratio Families of Discrete Distributions. Journal of the American Statistical Association 65, 307--316.
Poisson
, epois
, eqpois
,
estimate.object
, Tolerance Intervals,
Estimating Distribution Parameters, Estimating Distribution Quantiles.
# NOT RUN {
# Generate 20 observations from a Poisson distribution with parameter
# lambda=2. The interval [0, 4] contains 94.7% of this distribution and
# the interval [0,5] contains 98.3% of this distribution. Thus, because
# of the discrete nature of the Poisson distribution, no interval contains
# exactly 95% of this distribution. Use tolIntPois to estimate the mean
# parameter of the true distribution, and construct a one-sided upper 95%
# beta-content tolerance interval with associated confidence level 90%.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat <- rpois(20, 2)
tolIntPois(dat, conf.level = 0.9)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Poisson
#
#Estimated Parameter(s): lambda = 1.8
#
#Estimation Method: mle/mme/mvue
#
#Data: dat
#
#Sample Size: 20
#
#Tolerance Interval Coverage: 95%
#
#Coverage Type: content
#
#Tolerance Interval Method: Zacks
#
#Tolerance Interval Type: two-sided
#
#Confidence Level: 90%
#
#Tolerance Interval: LTL = 0
# UTL = 6
#------
# Clean up
rm(dat)
# }
Run the code above in your browser using DataLab