The Zero-Modified Lognormal (Delta) Distribution
Density, distribution function, quantile function, and random generation
for the zero-modified lognormal distribution with parameters
The zero-modified lognormal (delta) distribution is the mixture of a lognormal distribution with a positive probability mass at 0.
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vector of quantiles.
vector of quantiles.
vector of probabilities between 0 and 1.
sample size. If
vector of means of the normal (Gaussian) part of the distribution on the
log scale. The default is
vector of (positive) standard deviations of the normal (Gaussian)
part of the distribution on the log scale. The default is
vector of probabilities between 0 and 1 indicating the probability the random
variable equals 0. For
The zero-modified 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 Δ-distribution was coined by Aitchison and Brown (1957, p.95). It is a special case of a “zero-modified” distribution (see Johnson et al., 1992, p. 312).
Let f(x; μ, σ) denote the density of a
lognormal random variable X with parameters
sdlog=σ. The density function of a
zero-modified lognormal (delta) random variable Y with parameters
denoted h(y; μ, σ, p), is given by:
|h(y; μ, σ, p) =||p||for y = 0|
|(1 - p) f(y; μ, σ)||for y > 0|
Note that μ is not the mean of the zero-modified lognormal distribution on the log scale; it is the mean of the lognormal part of the distribution on the log scale. Similarly, σ is not the standard deviation of the zero-modified lognormal distribution on the log scale; it is the standard deviation of the lognormal part of the distribution on the log scale.
Let γ and δ denote the mean and standard deviation of the overall zero-modified lognormal distribution on the log scale. Aitchison (1955) shows that:
E[log(Y)] = γ = (1 - p) μ
Var[log(Y)] = δ^2 = (1 - p) σ^2 + p (1-p) μ^2
Note that when
=0, the zero-modified lognormal
distribution simplifies to the lognormal distribution.
dzmlnorm gives the density,
pzmlnorm gives the distribution function,
qzmlnorm gives the quantile function, and
rzmlnorm generates random
The zero-modified 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 non-detects.
A variation of the zero-modified lognormal (delta) distribution is the zero-modified normal distribution, in which a normal distribution is mixed with a positive probability mass at 0.
One way to try to assess whether a zero-modified lognormal (delta),
zero-modified normal, censored normal, or censored lognormal is the best
model for the data is to construct both censored and detects-only probability
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, 901-908.
Aitchison, J., and J.A.C. Brown (1957). The Lognormal Distribution (with special reference to its uses in economics). Cambridge University Press, London. pp.94-99.
Crow, E.L., and K. Shimizu. (1988). Lognormal Distributions: Theory and Applications. Marcel Dekker, New York, pp.47-51.
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, 135-146.
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 Left-Censored Values, with Applications to the Measurement of Worker Exposure to Air Contaminants. Biometrics 36, 707-719.
USEPA (1992c). Statistical Analysis of Ground-Water 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/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.
Zero-Modified Lognormal (Alternative Parameterization),
ezmlnorm, Probability Distributions and Random Numbers.
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# Density of the zero-modified lognormal (delta) distribution with # parameters meanlog=0, sdlog=1, and p.zero=0.5, evaluated at # 0, 0.5, 1, 1.5, and 2: dzmlnorm(seq(0, 2, by = 0.5)) # 0.50000000 0.31374804 0.19947114 0.12248683 # 0.07843701 #---------- # The cdf of the zero-modified lognormal (delta) distribution with # parameters meanlog=1, sdlog=2, and p.zero=0.1, evaluated at 4: pzmlnorm(4, 1, 2, .1) # 0.6189203 #---------- # The median of the zero-modified lognormal (delta) distribution with # parameters meanlog=2, sdlog=3, and p.zero=0.1: qzmlnorm(0.5, 2, 3, 0.1) # 4.859177 #---------- # Random sample of 3 observations from the zero-modified lognormal # (delta) distribution with parameters meanlog=1, sdlog=2, and p.zero=0.4. # (Note: The call to set.seed simply allows you to reproduce this example.) set.seed(20) rzmlnorm(3, 1, 2, 0.4) # 0.000000 0.000000 3.146641
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