Generalized Pivotal Quantity for Confidence Interval for the Mean of a Normal Distribution Based on Censored Data
Generate a generalized pivotal quantity (GPQ) for a confidence interval for the mean of a Normal distribution based on singly or multiply censored data.
gpqCiNormSinglyCensored(n, n.cen, probs, nmc, method = "mle",
censoring.side = "left", seed = NULL, names = TRUE)
gpqCiNormMultiplyCensored(n, cen.index, 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 |
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 functions gpqCiNormSinglyCensored and gpqCiNormMultiplyCensored
are called by enormCensored when ci.method="gpq". They are
used to construct generalized pivotal quantities to create confidence intervals
for the mean μ of an assumed normal distribution.
This idea was introduced by Schmee et al. (1985) in the context of Type II singly
censored data. The function
gpqCiNormSinglyCensored generates GPQs using a modification of
Algorithm 12.1 of Krishnamoorthy and Mathew (2009, p. 329). Algorithm 12.1 is
used to generate GPQs for a tolerance interval. The modified algorithm for
generating GPQs for confidence intervals for the mean μ is as follows:
Generate a random sample of n observations from a standard normal (i.e., N(0,1)) distribution and let z_{(1)}, z_{(2)}, …, z_{(n)} denote the ordered (sorted) observations.
Set the smallest n.cen observations as censored.
Compute the estimates of μ and σ by calling
enormCensored using the method
specified by the method argument, and denote these estimates as
\hat{μ}^*, \; \hat{σ}^*.
Compute the t-like pivotal quantity \hat{t} = \hat{μ}^*/\hat{σ}^*.
Repeat steps 1-4 nmc times to produce an empirical distribution of
the t-like pivotal quantity.
A two-sided (1-α)100\% confidence interval for μ is then computed as:
[\hat{μ} - \hat{t}_{1-(α/2)} \hat{σ}, \; \hat{μ} - \hat{t}_{α/2} \hat{σ}]
where \hat{t}_p denotes the p'th empirical quantile of the
nmc generated \hat{t} values.
Schmee at al. (1985) derived this method in the context of Type II singly censored data (for which these limits are exact within Monte Carlo error), but state that according to Regal (1982) this method produces confidence intervals that are close apporximations to the correct limits for Type I censored data.
The function
gpqCiNormMultiplyCensored is an extension of this idea to multiply censored
data. The algorithm is the same as for singly censored data, except
Step 2 changes to:
2. Set observations as censored for elements of the argument cen.index
that have the value TRUE.
The functions gpqCiNormSinglyCensored and gpqCiNormMultiplyCensored are
computationally intensive and provided to the user to allow you to create your own
tables.
a numeric vector containing the GPQ(s).
Steven P. Millard (EnvStats@ProbStatInfo.com)
Krishnamoorthy K., and T. Mathew. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. John Wiley and Sons, Hoboken.
Regal, R. (1982). Applying Order Statistic Censored Normal Confidence Intervals to Time Censored Data. Unpublished manuscript, University of Minnesota, Duluth, Department of Mathematical Sciences.
Schmee, J., D.Gladstein, and W. Nelson. (1985). Confidence Limits for Parameters of a Normal Distribution from Singly Censored Samples, Using Maximum Likelihood. Technometrics 27(2) 119–128.
# Reproduce the entries for n=10 observations with n.cen=6 in Table 4
# of Schmee et al. (1985, p.122).
#
# Notes:
# 1. This table applies to right-censored data, and the
# quantity "r" in this table refers to the number of
# uncensored observations.
#
# 2. 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.
# Here are the values given in Schmee et al. (1985):
Schmee.values <- c(-3.59, -2.60, -1.73, -0.24, 0.43, 0.58, 0.73)
probs <- c(0.025, 0.05, 0.1, 0.5, 0.9, 0.95, 0.975)
names(Schmee.values) <- paste(probs * 100, "%", sep = "")
Schmee.values
# 2.5% 5% 10% 50% 90% 95% 97.5%
#-3.59 -2.60 -1.73 -0.24 0.43 0.58 0.73
gpqs <- gpqCiNormSinglyCensored(n = 10, n.cen = 6, probs = probs,
nmc = 100, censoring.side = "right", seed = 529)
round(gpqs, 2)
# 2.5% 5% 10% 50% 90% 95% 97.5%
#-2.46 -2.03 -1.38 -0.14 0.54 0.65 0.84
# This is what you get if you specify nmc = 1000 with the
# same value for seed:
#-----------------------------------------------
# 2.5% 5% 10% 50% 90% 95% 97.5%
#-3.50 -2.49 -1.67 -0.25 0.41 0.57 0.71
# Clean up
#---------
rm(Schmee.values, probs, gpqs)
#==========
# Example of using gpqCiNormMultiplyCensored
#-------------------------------------------
# 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 gpqCiNormMultiplyCensored.
# Here we'll generate a GPQs to use to create a
# 95% confidence interval 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.
gpqCiNormMultiplyCensored(n = 5, cen.index = cen.index,
probs = c(0.025, 0.975), nmc = 100, seed = 237)
# 2.5% 97.5%
#-1.315592 1.848513
#----------
# Clean up
#---------
rm(dat, censored, cen.index)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.