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Density, distribution function, quantile function, and random generation
for the zero-modified lognormal distribution with parameters meanlog
,
sdlog
, and p.zero
.
The zero-modified lognormal (delta) distribution is the mixture of a lognormal distribution with a positive probability mass at 0.
dzmlnorm(x, meanlog = 0, sdlog = 1, p.zero = 0.5)
pzmlnorm(q, meanlog = 0, sdlog = 1, p.zero = 0.5)
qzmlnorm(p, meanlog = 0, sdlog = 1, p.zero = 0.5)
rzmlnorm(n, meanlog = 0, sdlog = 1, p.zero = 0.5)
vector of quantiles.
vector of quantiles.
vector of probabilities between 0 and 1.
sample size. If length(n)
is larger than 1, then length(n)
random values are returned.
vector of means of the normal (Gaussian) part of the distribution on the
log scale. The default is meanlog=0
.
vector of (positive) standard deviations of the normal (Gaussian)
part of the distribution on the log scale. The default is sdlog=1
.
vector of probabilities between 0 and 1 indicating the probability the random
variable equals 0. For rzmlnorm
this must be a single, non-missing number.
dzmlnorm
gives the density, pzmlnorm
gives the distribution function,
qzmlnorm
gives the quantile function, and rzmlnorm
generates random
deviates.
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
Let meanlog=
sdlog=
meanlog=
sdlog=
p.zero=
|
|
for |
Note that
Let p.zero=
=0
, the zero-modified lognormal
distribution simplifies to the lognormal distribution.
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),
Lognormal, LognormalAlt,
Zero-Modified Normal,
ezmlnorm
, Probability Distributions and Random Numbers.
# NOT RUN {
# 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))
#[1] 0.50000000 0.31374804 0.19947114 0.12248683
#[5] 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)
#[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)
#[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)
#[1] 0.000000 0.000000 3.146641
# }
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