eqlnorm3 {EnvStats}  R Documentation 
Estimate Quantiles of a ThreeParameter Lognormal Distribution
Description
Estimate quantiles of a threeparameter lognormal distribution.
Usage
eqlnorm3(x, p = 0.5, method = "lmle", digits = 0)
Arguments
x 
a numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a threeparameter lognormal distribution
(e.g., 
p 
numeric vector of probabilities for which quantiles will be estimated.
All values of 
method 
character string specifying the method of estimating the distribution parameters.
Possible values are

digits 
an integer indicating the number of decimal places to round to when printing out
the value of 
Details
If x
contains any missing (NA
), undefined (NaN
) or
infinite (Inf
, Inf
) values, they will be removed prior to
performing the estimation.
Quantiles are estimated by 1) estimating the distribution parameters by
calling elnorm3
, and then 2) calling the function
qlnorm3
and using the estimated distribution
parameters.
Value
If x
is a numeric vector, eqlnorm3
returns a
list of class "estimate"
containing the estimated quantile(s) and other
information. See estimate.object
for details.
If x
is the result of calling an estimation function, eqlnorm3
returns a list whose class is the same as x
. The list
contains the same components as x
, as well as components called
quantiles
and quantile.method
.
Note
The problem of estimating the parameters of a threeparameter lognormal distribution has been extensively discussed by Aitchison and Brown (1957, Chapter 6), Calitz (1973), Cohen (1951), Cohen (1988), Cohen and Whitten (1980), Cohen et al. (1985), Griffiths (1980), Harter and Moore (1966), Hill (1963), and Royston (1992b). Stedinger (1980) and Hoshi et al. (1984) discuss fitting the threeparameter lognormal distribution to hydrologic data.
The global maximum likelihood estimates are inadmissible. In the past, several
researchers have found that the local maximum likelihood estimates (lmle's)
occasionally fail because of convergence problems, but they were not using the
likelihood profile and reparameterization of Griffiths (1980). Cohen (1988)
recommends the modified methods of moments estimators over lmle's because they are
easy to compute, they are unbiased with respect to \mu
and \sigma^2
(the
mean and standard deviation on the logscale), their variances are minimal or near
minimal, and they do not suffer from regularity problems.
Because the distribution of the lmle of the threshold parameter \gamma
is far
from normal for moderate sample sizes (Griffiths, 1980), it is questionable whether
confidence intervals for \gamma
or the median based on asymptotic variances
and covariances will perform well. Cohen and Whitten (1980) and Cohen et al. (1985),
however, found that the asymptotic variances and covariances are reasonably close to
corresponding simulated variances and covariances for the modified method of moments
estimators (method="mmme"
). In a simulation study (5000 monte carlo trials),
Royston (1992b) found that the coverage of confidence intervals for \gamma
based on the likelihood profile (ci.method="likelihood.profile"
) was very
close the nominal level (94.1% for a nominal level of 95%), although not
symmetric. Royston (1992b) also found that the coverage of confidence intervals
for \gamma
based on the skewness method (ci.method="skewness"
) was also
very close (95.4%) and symmetric.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
Aitchison, J., and J.A.C. Brown (1957). The Lognormal Distribution (with special references to its uses in economics). Cambridge University Press, London, Chapter 5.
Calitz, F. (1973). Maximum Likelihood Estimation of the Parameters of the ThreeParameter Lognormal Distribution–a Reconsideration. Australian Journal of Statistics 15(3), 185–190.
Cohen, A.C. (1951). Estimating Parameters of LogarithmicNormal Distributions by Maximum Likelihood. Journal of the American Statistical Association 46, 206–212.
Cohen, A.C. (1988). ThreeParameter Estimation. In Crow, E.L., and K. Shimizu, eds. Lognormal Distributions: Theory and Applications. Marcel Dekker, New York, Chapter 4.
Cohen, A.C., and B.J. Whitten. (1980). Estimation in the ThreeParameter Lognormal Distribution. Journal of the American Statistical Association 75, 399–404.
Cohen, A.C., B.J. Whitten, and Y. Ding. (1985). Modified Moment Estimation for the ThreeParameter Lognormal Distribution. Journal of Quality Technology 17, 92–99.
Crow, E.L., and K. Shimizu. (1988). Lognormal Distributions: Theory and Applications. Marcel Dekker, New York, Chapter 2.
Griffiths, D.A. (1980). Interval Estimation for the ThreeParameter Lognormal Distribution via the Likelihood Function. Applied Statistics 29, 58–68.
Harter, H.L., and A.H. Moore. (1966). LocalMaximumLikelihood Estimation of the Parameters of ThreeParameter Lognormal Populations from Complete and Censored Samples. Journal of the American Statistical Association 61, 842–851.
Heyde, C.C. (1963). On a Property of the Lognormal Distribution. Journal of the Royal Statistical Society, Series B 25, 392–393.
Hill, .B.M. (1963). The ThreeParameter Lognormal Distribution and Bayesian Analysis of a PointSource Epidemic. Journal of the American Statistical Association 58, 72–84.
Hoshi, K., J.R. Stedinger, and J. Burges. (1984). Estimation of LogNormal Quantiles: Monte Carlo Results and FirstOrder Approximations. Journal of Hydrology 71, 1–30.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.
Royston, J.P. (1992b). Estimation, Reference Ranges and Goodness of Fit for the ThreeParameter LogNormal Distribution. Statistics in Medicine 11, 897–912.
Stedinger, J.R. (1980). Fitting Lognormal Distributions to Hydrologic Data. Water Resources Research 16(3), 481–490.
See Also
elnorm3
, Lognormal3, Lognormal,
LognormalAlt, Normal.
Examples
# Generate 20 observations from a 3parameter lognormal distribution
# with parameters meanlog=1.5, sdlog=1, and threshold=10, then use
# Cohen and Whitten's (1980) modified moments estimators to estimate
# the parameters, and estimate the 90th percentile.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat < rlnorm3(20, meanlog = 1.5, sdlog = 1, threshold = 10)
eqlnorm3(dat, method = "mmme", p = 0.9)
#Results of Distribution Parameter Estimation
#
#
#Assumed Distribution: 3Parameter Lognormal
#
#Estimated Parameter(s): meanlog = 1.5206664
# sdlog = 0.5330974
# threshold = 9.6620403
#
#Estimation Method: mmme
#
#Estimated Quantile(s): 90'th %ile = 18.72194
#
#Quantile Estimation Method: Quantile(s) Based on
# mmme Estimators
#
#Data: dat
#
#Sample Size: 20
# Clean up
#
rm(dat)