Description Usage Arguments Details Value Author(s) References See Also Examples

Generate a generalized pivotal quantity (GPQ) for a confidence interval for the mean of a Normal distribution based on singly or multiply censored data.

1 2 3 4 5 | ```
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 |

`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 |

`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 ([email protected])

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.

`enormCensored`

, `estimateCensored.object`

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | ```
# 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)
``` |

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