Calculate ML Estimate of Xp and Confidence Limits
Description
Calculate the ML estimate of Xp the 100pth percentile
of the lognormal distribution, and the lower and upper 100*γ% confidence limits
LX(p
,γ) and UX(p
,γ). The upper confidence limit is used to
test the null hypothesis that the exposure profile is "unacceptable".
If UX(p
,γ) < L the null hypothesis is rejected and workplace
is considered "safe" or the object/area is not contaminated. The
Type I error is ≤ α = 1  γ. The resulting interval (LX,UX)
is an approximate 100*(2γ  1) percent confidence interval for Xp.
Usage
1  percentile.ml(dd, p = 0.95, gam = 0.95, dat = TRUE)

Arguments
dd 
An n by 2 matrix or data frame with 
p 
is probability for Xp the 100pth percentile. Default is 0.95 
gam 
onesided confidence level γ. Default is 0.95 
dat 
if 
Details
The point estimate of Yp = log(Xp) is μ + z σ where μ and σ are ML estimates and z is qnorm(p). The variance of the estimate is
var(μ + zσ ) = var(μ ) + Z^2p var (σ )+ 2z cov(μ ,σ)
The 100γ {\%} LCL and UCL for Xp are
LX(p,γ ) = exp[Yp t(γ ,(m1))var(Yp)^{1/2}],
UX(p,γ ) = exp[Yp + t(γ ,(m1))var(Yp)^{1/2}].
The ML estimates of var(μ), var(σ), and cov(μ
,σ) are obtained from the ML variancecovariance matrix using
lnorm.ml
. The null hypothesis Ho: Xp ≥ Lp is rejected at the α = (1
γ ) significance level if the 100γ\% UCL for Xp < Lp (indicating the exposure profile is acceptable).
Value
A LIST with components:
Xp 
ML estimate of the pth percentile of lognormal distribution 
Xp.LCL 
100*γ% lower confidence limit for Xp 
Xp.UCL 
100*γ% upper confidence limit for Xp 
p 
probability for Xp the 100pth percentile. Default 0.95 
gam 
onesided confidence level γ. Default is 0.95 
Note
The UCL is also referred to as an upper tolerance limit(UTL), i.e., if p = 0.95 and gam = 0.99 then Xp.UCL is the UTL95%99%.
Author(s)
E. L. Frome
References
Cohen, A. C. (1991), Truncated and Censored Samples, Marcel Decker, New York
Cox, D. R. and D. V. Hinkley (1979), Theoretical Statistics, Chapman and Hall, New York.
Frome, E. L. and Wambach, P. F. (2005), "Statistical Methods and Software for the Analysis of Occupational Exposure Data with NonDetectable Values," ORNL/TM2005/52,Oak Ridge National Laboratory, Oak Ridge, TN 37830. Available at: http://www.csm.ornl.gov/esh/aoed/ORNLTM200552.pdf
See Also
Help files for lnorm.ml
,efraction.ml
Examples
1 2 3  data(beTWA)
# calculate ML estimate of 95th percentile and CLs for Example 2 in ORNLTM200552
unlist(percentile.ml(beTWA,0.95,0.95))
