Description Usage Arguments Details Value Note Author(s) References See Also Examples

Estimate quantiles of a zero-modified normal distribution.

1 | ```
eqzmnorm(x, p = 0.5, method = "mvue", digits = 0)
``` |

`x` |
a numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a zero-modified normal distribution
(e.g., |

`p` |
numeric vector of probabilities for which quantiles will be estimated.
All values of |

`method` |
character string specifying the method of estimating the disribution parameters.
Currently, the only possible
value is |

`digits` |
an integer indicating the number of decimal places to round to when printing out
the value of |

The function `eqzmnorm`

returns estimated quantiles as well as
estimates of the distribution parameters.

Quantiles are estimated by 1) estimating the distribution parameters by
calling `ezmnorm`

, and then 2) calling the function
`qzmnorm`

and using the estimated values for
the distribution parameters.

If `x`

is a numeric vector, `eqzmnorm`

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, `eqzmnorm`

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`

.

The zero-modified normal distribution is sometimes used to model chemical concentrations for which some observations are reported as “Below Detection Limit”. See, for example USEPA (1992c, pp.27-34). In most cases, however, the zero-modified lognormal (delta) distribution will be more appropriate, since chemical concentrations are bounded below at 0 (e.g., Gilliom and Helsel, 1986; Owen and DeRouen, 1980).

Once you estimate the parameters of the zero-modified normal distribution, it is often useful to characterize the uncertainty in the estimate of the mean. This is done with a confidence interval.

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`

).

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.

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.

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.

ZeroModifiedNormal, Normal,
`ezmlnorm`

, ZeroModifiedLognormal, `estimate.object`

.

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 33 34 35 36 37 38 39 40 41 42 43 | ```
# Generate 100 observations from a zero-modified normal distribution
# with mean=4, sd=2, and p.zero=0.5, then estimate the parameters and
# the 80th and 90th percentiles.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat <- rzmnorm(100, mean = 4, sd = 2, p.zero = 0.5)
eqzmnorm(dat, p = c(0.8, 0.9))
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Zero-Modified Normal
#
#Estimated Parameter(s): mean = 4.037732
# sd = 1.917004
# p.zero = 0.450000
# mean.zmnorm = 2.220753
# sd.zmnorm = 2.465829
#
#Estimation Method: mvue
#
#Estimated Quantile(s): 80'th %ile = 4.706298
# 90'th %ile = 5.779250
#
#Quantile Estimation Method: Quantile(s) Based on
# mvue Estimators
#
#Data: dat
#
#Sample Size: 100
#----------
# Compare the estimated quantiles with the true quantiles
qzmnorm(mean = 4, sd = 2, p.zero = 0.5, p = c(0.8, 0.9))
#[1] 4.506694 5.683242
#----------
# Clean up
rm(dat)
``` |

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