The Zero-Modified Lognormal (Delta) Distribution (Alternative Parameterization)

Density, distribution function, quantile function, and random generation for the zero-modified lognormal distribution with parameters mean, cv, and p.zero.

The zero-modified lognormal (delta) distribution is the mixture of a lognormal distribution with a positive probability mass at 0.

dzmlnormAlt(x, mean = exp(1/2), cv = sqrt(exp(1) - 1), p.zero = 0.5)
  pzmlnormAlt(q, mean = exp(1/2), cv = sqrt(exp(1) - 1), p.zero = 0.5)
  qzmlnormAlt(p, mean = exp(1/2), cv = sqrt(exp(1) - 1), p.zero = 0.5)
  rzmlnormAlt(n, mean = exp(1/2), cv = sqrt(exp(1) - 1), p.zero = 0.5)

Arguments

x: vector of quantiles.
q: vector of quantiles.
p: vector of probabilities between 0 and 1.
n: sample size. If length(n) is larger than 1, then length(n) random values are returned.
mean: vector of means of the lognormal part of the distribution on the. The default is mean=exp(1/2).
cv: vector of (positive) coefficients of variation of the lognormal part of the distribution. The default is cv=sqrt(exp(1) - 1).
p.zero: vector of probabilities between 0 and 1 indicating the probability the random variable equals 0. For rzmlnormAlt this must be a single, non-missing number.

Details

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 $\Delta$-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 $f(x; \theta, \tau)$ denote the density of a lognormal random variable $X$ with parameters mean=$\theta$ and cv=$\tau$. The density function of a zero-modified lognormal (delta) random variable $Y$ with parameters mean=$\theta$, cv=$\tau$, and p.zero=$p$, denoted $h(y; \theta, \tau, p)$, is given by:

$h(y; \theta, \tau, p) =$	$p$	for $y = 0$
	$(1 - p) f(y; \theta, \tau)$	for $y > 0$

Note that $\theta$ is not the mean of the zero-modified lognormal distribution; it is the mean of the lognormal part of the distribution. Similarly, $\tau$ is not the coefficient of variation of the zero-modified lognormal distribution; it is the coefficient of variation of the lognormal part of the distribution.

Let $\gamma$, $\delta$, and $\omega$ denote the mean, standard deviation, and coefficient of variation of the overall zero-modified lognormal distribution. Let $\eta$ denote the standard deviation of the lognormal part of the distribution, so that $\eta = \theta \tau$. Aitchison (1955) shows that: $$E(Y) = \gamma = (1 - p) \theta$$ $$Var(Y) = \delta^2 = (1 - p) \eta^2 + p (1-p) \theta^2$$ so that $$\omega = \sqrt{(\tau^2 + p) / (1 - p)}$$ Note that when p.zero=$p$=0, the zero-modified lognormal distribution simplifies to the lognormal distribution.

Value

dzmlnormAlt gives the density, pzmlnormAlt gives the distribution function, qzmlnormAlt gives the quantile function, and rzmlnormAlt generates random deviates.

References

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.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

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 plots (see qqPlotCensored).

Examples

  # Density of the zero-modified 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
  #[1] 0.02552685 0.02197043 0.01891924

  #----------

  # The cdf of the zero-modified 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
  #[1] 0.709009

  #----------

  # The median of the zero-modified lognormal (delta) distribution with 
  # parameters mean=10, cv=2, and p.zero=0.1:

  qzmlnormAlt(0.5, 10, 2, 0.1) 
#> [1] 3.74576
  #[1] 3.74576

  #----------

  # Random sample of 3 observations from the zero-modified 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
  #[1] 0.000000 0.000000 4.907131

\(h(y; \theta, \tau, p) =\)	\(p\)	for \(y = 0\)
	\((1 - p) f(y; \theta, \tau)\)	for \(y > 0\)