Description Usage Arguments Details Value Note Author(s) References See Also Examples
Density, distribution function, quantile function, and random generation
for the zeromodified lognormal distribution with parameters mean
,
cv
, and p.zero
.
The zeromodified lognormal (delta) distribution is the mixture of a lognormal distribution with a positive probability mass at 0.
1 2 3 4 
x 
vector of quantiles. 
q 
vector of quantiles. 
p 
vector of probabilities between 0 and 1. 
n 
sample size. If 
mean 
vector of means of the lognormal part of the distribution on the.
The default is 
cv 
vector of (positive) coefficients of variation of the lognormal
part of the distribution. The default is 
p.zero 
vector of probabilities between 0 and 1 indicating the probability the random
variable equals 0. For 
The zeromodified lognormal (delta) distribution is the mixture of a lognormal distribution with a positive probability mass at 0. This distribution was introduced without a name by Aitchison (1955), and the name Δdistribution was coined by Aitchison and Brown (1957, p.95). It is a special case of a “zeromodified” distribution (see Johnson et al., 1992, p. 312).
Let f(x; θ, τ) denote the density of a
lognormal random variable X with parameters
mean=
θ and cv=
τ. The density function of a
zeromodified lognormal (delta) random variable Y with parameters
mean=
θ, cv=
τ, and p.zero=
p,
denoted h(y; θ, τ, p), is given by:
h(y; θ, τ, p) =  p  for y = 0 
(1  p) f(y; θ, τ)  for y > 0 
Note that θ is not the mean of the zeromodified lognormal distribution; it is the mean of the lognormal part of the distribution. Similarly, τ is not the coefficient of variation of the zeromodified lognormal distribution; it is the coefficient of variation of the lognormal part of the distribution.
Let γ, δ, and ω denote the mean, standard deviation, and coefficient of variation of the overall zeromodified lognormal distribution. Let η denote the standard deviation of the lognormal part of the distribution, so that η = θ τ. Aitchison (1955) shows that:
E(Y) = γ = (1  p) θ
Var(Y) = δ^2 = (1  p) η^2 + p (1p) θ^2
so that
ω = √{(τ^2 + p) / (1  p)}
Note that when p.zero=
p=0
, the zeromodified lognormal
distribution simplifies to the lognormal distribution.
dzmlnormAlt
gives the density, pzmlnormAlt
gives the distribution function,
qzmlnormAlt
gives the quantile function, and rzmlnormAlt
generates random
deviates.
The zeromodified lognormal (delta) distribution is sometimes used to model chemical concentrations for which some observations are reported as “Below Detection Limit” (the nondetects are assumed equal to 0). See, for example, Gilliom and Helsel (1986), Owen and DeRouen (1980), and Gibbons et al. (2009, Chapter 12). USEPA (2009, Chapter 15) recommends this strategy only in specific situations, and Helsel (2012, Chapter 1) strongly discourages this approach to dealing with nondetects.
A variation of the zeromodified lognormal (delta) distribution is the zeromodified normal distribution, in which a normal distribution is mixed with a positive probability mass at 0.
One way to try to assess whether a zeromodified lognormal (delta),
zeromodified normal, censored normal, or censored lognormal is the best
model for the data is to construct both censored and detectsonly probability
plots (see qqPlotCensored
).
Steven P. Millard (EnvStats@ProbStatInfo.com)
Aitchison, J. (1955). On the Distribution of a Positive Random Variable Having a Discrete Probability Mass at the Origin. Journal of the American Statistical Association 50, 901908.
Aitchison, J., and J.A.C. Brown (1957). The Lognormal Distribution (with special reference to its uses in economics). Cambridge University Press, London. pp.9499.
Crow, E.L., and K. Shimizu. (1988). Lognormal Distributions: Theory and Applications. Marcel Dekker, New York, pp.4751.
Gibbons, RD., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring. Second Edition. John Wiley and Sons, Hoboken, NJ.
Gilliom, R.J., and D.R. Helsel. (1986). Estimation of Distributional Parameters for Censored Trace Level Water Quality Data: 1. Estimation Techniques. Water Resources Research 22, 135146.
Helsel, D.R. (2012). Statistics for Censored Environmental Data Using Minitab and R. Second Edition. John Wiley and Sons, Hoboken, NJ, Chapter 1.
Johnson, N. L., S. Kotz, and A.W. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, p.312.
Owen, W., and T. DeRouen. (1980). Estimation of the Mean for Lognormal Data Containing Zeros and LeftCensored Values, with Applications to the Measurement of Worker Exposure to Air Contaminants. Biometrics 36, 707719.
USEPA (1992c). Statistical Analysis of GroundWater Monitoring Data at RCRA Facilities: Addendum to Interim Final Guidance. Office of Solid Waste, Permits and State Programs Division, US Environmental Protection Agency, Washington, D.C.
USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R09007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.
ZeroModified Lognormal, LognormalAlt,
ezmlnormAlt
, Probability Distributions and Random Numbers.
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  # Density of the zeromodified lognormal (delta) distribution with
# parameters mean=10, cv=1, and p.zero=0.5, evaluated at
# 9, 10, and 11:
dzmlnormAlt(9:11, mean = 10, cv = 1, p.zero = 0.5)
#[1] 0.02552685 0.02197043 0.01891924
#
# The cdf of the zeromodified lognormal (delta) distribution with
# parameters mean=10, cv=2, and p.zero=0.1, evaluated at 8:
pzmlnormAlt(8, 10, 2, .1)
#[1] 0.709009
#
# The median of the zeromodified lognormal (delta) distribution with
# parameters mean=10, cv=2, and p.zero=0.1:
qzmlnormAlt(0.5, 10, 2, 0.1)
#[1] 3.74576
#
# Random sample of 3 observations from the zeromodified lognormal
# (delta) distribution with parameters mean=10, cv=2, and p.zero=0.4.
# (Note: The call to set.seed simply allows you to reproduce this example.)
set.seed(20)
rzmlnormAlt(3, 10, 2, 0.4)
#[1] 0.000000 0.000000 4.907131

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