Estimate Parameters of a Lognormal Distribution (Log-Scale)

Description

Estimate the mean and standard deviation parameters of the logarithm of a lognormal distribution, and optionally construct a confidence interval for the mean.

Usage

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  elnorm(x, method = "mvue", ci = FALSE, ci.type = "two-sided", 
    ci.method = "exact", conf.level = 0.95)

Arguments

x

numeric vector of observations.

method

character string specifying the method of estimation. Possible values are "mvue" (minimum variance unbiased; the default), and "mle/mme" (maximum likelihood/method of moments). See the DETAILS section for more information on these estimation methods.

ci

logical scalar indicating whether to compute a confidence interval for the mean. The default value is FALSE.

ci.type

character string indicating what kind of confidence interval to compute. The possible values are "two-sided" (the default), "lower", and "upper". This argument is ignored if ci=FALSE.

ci.method

character string indicating what method to use to construct the confidence interval for the mean or variance. The only possible value is "exact" (the default). See the DETAILS section for more information. This argument is ignored if ci=FALSE.

conf.level

a scalar between 0 and 1 indicating the confidence level of the confidence interval. The default value is conf.level=0.95. This argument is ignored if ci=FALSE.

Details

If x contains any missing (NA), undefined (NaN) or infinite (Inf, -Inf) values, they will be removed prior to performing the estimation.

Let X denote a random variable with a lognormal distribution with parameters meanlog=μ and sdlog=σ. Then Y = log(X) has a normal (Gaussian) distribution with parameters mean=μ and sd=σ. Thus, the function elnorm simply calls the function enorm using the log-transformed values of x.

Value

a list of class "estimate" containing the estimated parameters and other information. See
estimate.object for details.

Note

The normal and lognormal distribution are probably the two most frequently used distributions to model environmental data. In order to make any kind of probability statement about a normally-distributed population (of chemical concentrations for example), you have to first estimate the mean and standard deviation (the population parameters) of the distribution. Once you estimate these parameters, it is often useful to characterize the uncertainty in the estimate of the mean or variance. This is done with confidence intervals.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Aitchison, J., and J.A.C. Brown (1957). The Lognormal Distribution (with special references to its uses in economics). Cambridge University Press, London, Chapter 5.

Crow, E.L., and K. Shimizu. (1988). Lognormal Distributions: Theory and Applications. Marcel Dekker, New York, Chapter 2.

Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.

Gilbert, R.O. (1987). Statistical Methods for Environmental Pollution Monitoring. Van Nostrand Reinhold, New York, NY.

Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.

Limpert, E., W.A. Stahel, and M. Abbt. (2001). Log-Normal Distributions Across the Sciences: Keys and Clues. BioScience 51, 341–352.

Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, FL.

Ott, W.R. (1995). Environmental Statistics and Data Analysis. Lewis Publishers, Boca Raton, FL.

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.

See Also

Lognormal, LognormalAlt, Normal.

Examples

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  # Using the Reference area TcCB data in the data frame EPA.94b.tccb.df, 
  # estimate the mean and standard deviation of the log-transformed distribution, 
  # and construct a 95% confidence interval for the mean.

  with(EPA.94b.tccb.df, elnorm(TcCB[Area == "Reference"], ci = TRUE))  

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Lognormal
  #
  #Estimated Parameter(s):          meanlog = -0.6195712
  #                                 sdlog   =  0.4679530
  #
  #Estimation Method:               mvue
  #
  #Data:                            TcCB[Area == "Reference"]
  #
  #Sample Size:                     47
  #
  #Confidence Interval for:         mean
  #
  #Confidence Interval Method:      Exact
  #
  #Confidence Interval Type:        two-sided
  #
  #Confidence Level:                95%
  #
  #Confidence Interval:             LCL = -0.7569673
  #                                 UCL = -0.4821751

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