Estimate Quantiles of a Lognormal Distribution

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Description

Estimate quantiles of a lognormal distribution, and optionally construct a confidence interval for a quantile.

Usage

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  eqlnorm(x, p = 0.5, method = "qmle", ci = FALSE, 
    ci.method = "exact", ci.type = "two-sided", conf.level = 0.95, 
    digits = 0)

Arguments

x

a numeric vector of positive observations, or an object resulting from a call to an estimating function that assumes a lognormal distribution (i.e., elnorm, elnormCensored). You cannot use objects resulting from a call to estimating functions that use the alternative parameterization such as elnormAlt. If x is a numeric vector, missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed.

p

numeric vector of probabilities for which quantiles will be estimated. All values of p must be between 0 and 1. When ci=TRUE, p must be a scalar. The default value is p=0.5.

method

character string indicating what method to use to estimate the quantile(s). The possible values are "qmle" (quasi maximum likelihood; the default) and
"mvue" (minimum variance unbiased). The method "mvue" is available only when p=0.5 (i.e., when you are estimating the median). See the DETAILS section for more information.

ci

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

ci.method

character string indicating what method to use to construct the confidence interval for the quantile. The possible values are "exact" (exact method; the default) and "normal.approx" (normal approximation). See the DETAILS section for more information.

ci.type

character string indicating what kind of confidence interval for the quantile to compute. The possible values are "two-sided" (the default), "lower", and "upper". 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.

digits

an integer indicating the number of decimal places to round to when printing out the value of 100*p. The default value is digits=0.

Details

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

Quantiles and their associated confidence intervals are constructed by calling the function eqnorm using the log-transformed data and then exponentiating the quantiles and confidence limits.

In the special case when p=0.5 and method="mvue", the estimated median is computed using the method given in Gilbert (1987, p.172) and Bradu and Mundlak (1970).

Value

If x is a numeric vector, eqlnorm 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, eqlnorm 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. In addition, if ci=TRUE, the returned list contains a component called interval containing the confidence interval information. If x already has a component called interval, this component is replaced with the confidence interval information.

Note

Percentiles are sometimes used in environmental standards and regulations. For example, Berthouex and Brown (2002, p.71) note that England has water quality limits based on the 90th and 95th percentiles of monitoring data not exceeding specified levels. They also note that the U.S. EPA has specifications for air quality monitoring, aquatic standards on toxic chemicals, and maximum daily limits for industrial effluents that are all based on percentiles. Given the importance of these quantities, it is essential to characterize the amount of uncertainty associated with the estimates of these quantities. This is done with confidence intervals.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Berthouex, P.M., and L.C. Brown. (2002). Statistics for Environmental Engineers. Lewis Publishers, Boca Raton.

Bradu, D., and Y. Mundlak. (1970). Estimation in Lognormal Linear Models. Journal of the American Statistical Association 65, 198-211.

Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley and Sons, New York.

Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.

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

Helsel, D.R., and R.M. Hirsch. (1992). Statistical Methods in Water Resources Research. Elsevier, New York, NY, pp.88-90.

Johnson, N.L., and B.L. Welch. (1940). Applications of the Non-Central t-Distribution. Biometrika 31, 362-389.

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

Owen, D.B. (1962). Handbook of Statistical Tables. Addison-Wesley, Reading, MA.

Stedinger, J. (1983). Confidence Intervals for Design Events. Journal of Hydraulic Engineering 109(1), 13-27.

Stedinger, J.R., R.M. Vogel, and E. Foufoula-Georgiou. (1993). Frequency Analysis of Extreme Events. In: Maidment, D.R., ed. Handbook of Hydrology. McGraw-Hill, New York, Chapter 18, pp.29-30.

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.

USEPA. (2010). Errata Sheet - March 2009 Unified Guidance. EPA 530/R-09-007a, August 9, 2010. Office of Resource Conservation and Recovery, Program Information and Implementation Division. U.S. Environmental Protection Agency, Washington, D.C.

See Also

eqnorm, Lognormal, elnorm, estimate.object.

Examples

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  # Generate 20 observations from a lognormal distribution with 
  # parameters meanlog=3 and sdlog=0.5, then estimate the 90th 
  # percentile and create a one-sided upper 95% confidence interval 
  # for that percentile. 
  # (Note: the call to set.seed simply allows you to reproduce this 
  # example.)

  set.seed(47) 
  dat <- rlnorm(20, meanlog = 3, sdlog = 0.5) 
  eqlnorm(dat, p = 0.9, ci = TRUE, ci.type = "upper")

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Lognormal
  #
  #Estimated Parameter(s):          meanlog = 2.9482139
  #                                 sdlog   = 0.4553215
  #
  #Estimation Method:               mvue
  #
  #Estimated Quantile(s):           90'th %ile = 34.18312
  #
  #Quantile Estimation Method:      qmle
  #
  #Data:                            dat
  #
  #Sample Size:                     20
  #
  #Confidence Interval for:         90'th %ile
  #
  #Confidence Interval Method:      Exact
  #
  #Confidence Interval Type:        upper
  #
  #Confidence Level:                95%
  #
  #Confidence Interval:             LCL =  0.00000
  #                                 UCL = 45.84008

  #----------
  # Compare these results with the true 90'th percentile:

  qlnorm(p = 0.9, meanlog = 3, sdlog = 0.5)
  #[1] 38.1214

  #----------

  # Clean up
  rm(dat)
  
  #--------------------------------------------------------------------

  # Example 17-3 of USEPA (2009, p. 17-17) shows how to construct a 
  # beta-content upper tolerance limit with 95% coverage and 95% 
  # confidence using chrysene data and assuming a lognormal 
  # distribution.

  # A beta-content upper tolerance limit with 95% coverage and 95% 
  # confidence is equivalent to the 95% upper confidence limit for the 
  # 95th percentile.

  attach(EPA.09.Ex.17.3.chrysene.df)
  Chrysene <- Chrysene.ppb[Well.type == "Background"]
  eqlnorm(Chrysene, p = 0.95, ci = TRUE, ci.type = "upper")

  #Results of Distribution Parameter Estimation
  #--------------------------------------------
  #
  #Assumed Distribution:            Lognormal
  #
  #Estimated Parameter(s):          meanlog = 2.5085773
  #                                 sdlog   = 0.6279479
  #
  #Estimation Method:               mvue
  #
  #Estimated Quantile(s):           95'th %ile = 34.51727
  #
  #Quantile Estimation Method:      qmle
  #
  #Data:                            Chrysene
  #
  #Sample Size:                     8
  #
  #Confidence Interval for:         95'th %ile
  #
  #Confidence Interval Method:      Exact
  #
  #Confidence Interval Type:        upper
  #
  #Confidence Level:                95%
  #
  #Confidence Interval:             LCL =  0.0000
  #                                 UCL = 90.9247

  #----------
  # Clean up

  rm(Chrysene)
  detach("EPA.09.Ex.17.3.chrysene.df")

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