eqlnorm3.RdEstimate quantiles of a three-parameter lognormal distribution.
eqlnorm3(x, p = 0.5, method = "lmle", digits = 0)a numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a three-parameter lognormal distribution
(e.g., elnorm3). If x is a numeric vector,
missing (NA), undefined (NaN), and infinite (Inf, -Inf)
values are allowed but will be removed.
numeric vector of probabilities for which quantiles will be estimated.
All values of p must be between 0 and 1. When ci=TRUE, p
must be a scalar. The default value is p=0.5.
character string specifying the method of estimating the distribution parameters.
Possible values are
"lmle" (local maximum likelihood; the default), "mme"
(method of moments), "mmue" (method of moments using an unbaised
estimate of variance), "mmme" (modified method of moments due to Cohen
and Whitten (1980)), "zero.skew" (zero-skewness estimator due to
Griffiths (1980)), and "royston.skew" (estimator based on
Royston's (1992b) index of skewness). See the DETAILS section of the help file
for elnorm3 for more information on these estimation methods.
an integer indicating the number of decimal places to round to when printing out
the value of 100*p. The default value is digits=0.
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.
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.
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 Three-Parameter Lognormal Distribution–a Reconsideration. Australian Journal of Statistics 15(3), 185–190.
Cohen, A.C. (1951). Estimating Parameters of Logarithmic-Normal Distributions by Maximum Likelihood. Journal of the American Statistical Association 46, 206–212.
Cohen, A.C. (1988). Three-Parameter 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 Three-Parameter 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 Three-Parameter 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 Three-Parameter Lognormal Distribution via the Likelihood Function. Applied Statistics 29, 58–68.
Harter, H.L., and A.H. Moore. (1966). Local-Maximum-Likelihood Estimation of the Parameters of Three-Parameter 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 Three-Parameter Lognormal Distribution and Bayesian Analysis of a Point-Source Epidemic. Journal of the American Statistical Association 58, 72–84.
Hoshi, K., J.R. Stedinger, and J. Burges. (1984). Estimation of Log-Normal Quantiles: Monte Carlo Results and First-Order 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 Three-Parameter Log-Normal Distribution. Statistics in Medicine 11, 897–912.
Stedinger, J.R. (1980). Fitting Lognormal Distributions to Hydrologic Data. Water Resources Research 16(3), 481–490.
The problem of estimating the parameters of a three-parameter 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 three-parameter 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 log-scale), 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.
# Generate 20 observations from a 3-parameter 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: 3-Parameter 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
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: 3-Parameter 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)