Generalized Pivotal Quantity for Tolerance Interval for a Normal Distribution Based on Censored Data
Generate a generalized pivotal quantity (GPQ) for a tolerance interval for a Normal distribution based on singly or multiply censored data.
gpqTolIntNormSinglyCensored(n, n.cen, p, probs, nmc, method = "mle",
censoring.side = "left", seed = NULL, names = TRUE)
gpqTolIntNormMultiplyCensored(n, cen.index, p, probs, nmc, method = "mle",
censoring.side = "left", seed = NULL, names = TRUE)n |
positive integer ≥ 3 indicating the sample size. |
n.cen |
for the case of singly censored data, a positive integer indicating the number of
censored observations. The value of |
cen.index |
for the case of multiply censored data, a sorted vector of unique integers indicating the
indices of the censored observations when the observations are “ordered”.
The length of |
p |
numeric scalar strictly greater than 0 and strictly less than 1 indicating the quantile for which to generate the GPQ(s) (i.e., the coverage associated with a one-sided tolerance interval). |
probs |
numeric vector of values between 0 and 1 indicating the confidence level(s) associated with the GPQ(s). |
nmc |
positive integer ≥ 10 indicating the number of Monte Carlo trials to run in order to compute the GPQ(s). |
method |
character string indicating the method to use for parameter estimation. |
censoring.side |
character string indicating on which side the censoring occurs. The possible values are
|
seed |
positive integer to pass to the function |
names |
a logical scalar passed to |
The function gpqTolIntNormSinglyCensored generates GPQs as described in Algorithm 12.1
of Krishnamoorthy and Mathew (2009, p. 329). The function
gpqTolIntNormMultiplyCensored is an extension of this idea to multiply censored data.
These functions are called by tolIntNormCensored when ti.method="gpq",
and also by eqnormCensored when ci=TRUE and ci.method="gpq". See
the help files for these functions for an explanation of GPQs.
Note that technically these are only GPQs if the data are Type II censored. However, Krishnamoorthy and Mathew (2009, p. 328) state that in the case of Type I censored data these quantities should approximate the true GPQs and the results appear to be satisfactory, even for small sample sizes.
The functions gpqTolIntNormSinglyCensored and gpqTolIntNormMultiplyCensored are
computationally intensive and provided to the user to allow you to create your own tables.
a numeric vector containing the GPQ(s).
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).
Steven P. Millard (EnvStats@ProbStatInfo.com)
Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.
# Reproduce the entries for n=10 observations with n.cen=1 in Table 12.2
# of Krishnamoorthy and Mathew (2009, p.331).
#
# (Note: passing a value for the argument "seed" simply allows you to
# reproduce this example.)
#
# NOTE: Here to save computing time for the sake of example, we will specify
# just 100 Monte Carlos, whereas Krishnamoorthy and Mathew (2009)
# suggest *10,000* Monte Carlos.
gpqTolIntNormSinglyCensored(n = 10, n.cen = 1, p = 0.05, probs = 0.05,
nmc = 100, seed = 529)
# 5%
#-3.483403
gpqTolIntNormSinglyCensored(n = 10, n.cen = 1, p = 0.1, probs = 0.05,
nmc = 100, seed = 497)
# 5%
#-2.66705
gpqTolIntNormSinglyCensored(n = 10, n.cen = 1, p = 0.9, probs = 0.95,
nmc = 100, seed = 623)
# 95%
#2.478654
gpqTolIntNormSinglyCensored(n = 10, n.cen = 1, p = 0.95, probs = 0.95,
nmc = 100, seed = 623)
# 95%
#3.108452
#==========
# Example of using gpqTolIntNormMultiplyCensored
#-----------------------------------------------
# Consider the following set of multiply left-censored data:
dat <- 12:16
censored <- c(TRUE, FALSE, TRUE, FALSE, FALSE)
# Since the data are "ordered" we can identify the indices of the
# censored observations in the ordered data as follow:
cen.index <- (1:length(dat))[censored]
cen.index
#[1] 1 3
# Now we can generate a GPQ using gpqTolIntNormMultiplyCensored.
# Here we'll generate a GPQ corresponding to an upper tolerance
# interval with coverage 90% with 95% confidence for
# left-censored data.
# NOTE: Here to save computing time for the sake of example, we will specify
# just 100 Monte Carlos, whereas Krishnamoorthy and Mathew (2009)
# suggest *10,000* Monte Carlos.
gpqTolIntNormMultiplyCensored(n = 5, cen.index = cen.index, p = 0.9,
probs = 0.95, nmc = 100, seed = 237)
# 95%
#3.952052
#==========
# Clean up
#---------
rm(dat, censored, cen.index)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.