R/lognormalsd.R

In fuel: Framework for Unified Estimation in Lognormal Models

#' @name lognormalsd
#' @title Standard Deviation Estimation for Lognormal Distribution
#'
#' @description Lognormal models are also widely applied in various branches of natural, social and applied sciences.
#' Given a pair of known constants in the parametric function for the statistics in the lognormal distribution, sample size, degree of freedom of the variance estimation of the log-transformed data, standardized variance of the sampling distribution of the log-transformed data, mean of the log-transformed data and standard deviation of the log-transformed data, this function returns an estimation for the lognormal distribution, including a total of thirty-nine different estimation methods, under a newly proposed unified framework in Zhang and Gou (2020).
#'
#' @param data original data vector
#' @param estimator a total of thirty-eight different estimation methods. See more descriptions in Section Details.
#' @param base the base with respect to which logarithms are computed. Defaults to \emph{e}.
#' @param n sample size.
#' @param m degree of freedom of the variance estimation of the log-transformed data.
#' @param d standardized variance of the sampling distribution of the log-transformed data.
#' @return estimated standard deviation. .
#' @author Jiangtao Gou
#' @references
#' Finney, D. J. (1941). On the distribution of a variate whose logarithm is normally distributed. Supplement to the \emph{Journal of the Royal Statistical Society}, \bold{7}: 155-161. <https://doi.org/10.2307/2983663>
#'
#' Zellner, A. (1971). Bayesian and non-Bayesian analysis of the log-normal distribution and log-normal regression. \emph{Journal of the American Statistical Association}, \bold{66}: 327-330. <https://doi.org/10.1080/01621459.1971.10482263>
#'
#' Evans, I. G. and Shaban, S. A. (1974). A note on estimation in lognormal models. \emph{Journal of the American Statistical Association}, \bold{69}: 779-781. <https://doi.org/10.2307/2286017>
#'
#' Rukhin, A. L. (1986). Improved estimation in lognormal models. \emph{Journal of the American Statistical Association}, \bold{81}: 1046-1049. <https://doi.org/10.1080/01621459.1986.10478371>
#'
#' El-Shaarawi, A. H. and Viveros, R. (1997). Inference about the mean in log-regression with environmental applications. \emph{Environmetrics}, \bold{8}: 569-582. <https://doi.org/10.1002/(SICI)1099-095X(199709/10)8:5<569::AID-ENV274>3.0.CO;2-I>
#'
#' Shen, H. and Zhu, Z. (2008). Efficient mean estimation in log-normal linear models. \emph{Journal of Statistical Planning and Inference}, \bold{138}: 552-567. <https://doi.org/10.1016/j.jspi.2006.10.016>
#'
#' Longford, N. T. (2009). Inference with the lognormal distribution. \emph{Journal of Statistical Planning and Inference}, \bold{139}: 2329-2340. <https://doi.org/10.1016/j.jspi.2008.10.015>
#'
#' Fabrizi, E. and Trivisano, C. (2012). Bayesian estimation of log-normal means with finite quadratic expected loss. \emph{Bayesian Analysis}, \bold{7}: 975-996. <https://doi.org/10.1214/12-BA733>
#'
#' Gou, J. and Tamhane, A. C. (2017). Estimation of a parametric function associated with the lognormal distribution. \emph{Communications in Statistics - Theory and Methods} \bold{46}: 8134-8154. <https://doi.org/10.1080/03610926.2016.1175628>
#'
#' Zhang, F. and Gou, J. (2020). A unified framework for estimation in lognormal models. Technical Report.
#'
#' @details
#'Consider a parametric function in the original scale we are interested in estimating \eqn{\theta(a,b) = exp(a\mu + b\sigma^2/2)},where constants \emph{a} and \emph{b} are known.
#' Specifically, \eqn{\theta(1,1)} is the mean of the lognormal distribution, \eqn{\theta(2,4)} is the second moment, \eqn{\theta(2,4)-\theta(2,2)} is the variance, and \eqn{(\theta(0,2) - 1)^{1/2}} is the coeficient of variation.
#' \enumerate{
#'  \item \code{unbiased}: Unbiased estimator (Finney, 1941)
#'  \item \code{qml}: Quasi maximum likelihood estimator
#'  \item \code{ml}: Maximum likelihood estimator
#'  \item \code{sa}:  Simple adjustment estimator
#'  \item \code{f}: Finney's unbiased estimator (Finney, 1941)
#'  \item \code{z}: Zellner's estimator (Zellner, 1971)
#'  \item \code{es}: Evans and Shaban’s estimator (Evans and Shaban, 1974, 1976)
#'  \item \code{r-s}: Rukhin’s simple estimator (Rukhin, 1986)
#'  \item \code{r-f}: Rukhin’s estimator using Finney's function (Rukhin, 1986)
#'  \item \code{r-lo}: Rukhin’s locally optimal estimator (Rukhin, 1986)
#'  \item \code{r-b}: Rukhin’s Bayes estimator (Rukhin, 1986)
#'  \item \code{ev}: El-Shaarawi and Viveros' estimator (El-Shaarawi and Viveros, 1997)
#'  \item \code{zh}: Zhou's estimator (Zhou, 1998)
#'  \item \code{sz-mm}: Shen and Zhu's MM estimator (Shen and Zhu, 2008)
#'  \item \code{sz-mb}: Shen and Zhu's MB estimator (Shen and Zhu, 2008)
#'  \item \code{l-ub}: Longford's UB estimator (Longford, 2009)
#'  \item \code{l-ms}: Longford's MS estimator (Longford, 2009)
#'  \item \code{ft}: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)
#'  \item \code{ft-s}: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)
#'  \item \code{ft-b}: Fabrizi and Trivisano's Bayes estimator (Fabrizi and Trivisano, 2012)
#'  \item \code{gt-f}: Gou and Tamhane's estimator using Finney's function (Gou and Tamhane, 2017)
#'  \item \code{gt-es}: Gou and Tamhane's estimator using Evans and Shaban's function (Gou and Tamhane, 2017)
#'  \item \code{gt-r}: Gou and Tamhane's estimator using Rukhin's function (Gou and Tamhane, 2017)
#'  \item \code{zg-1}: Zhang and Gou's first estimator (Zhang and Gou, 2020)
#'  \item \code{zg-2}: Zhang and Gou's second estimator (Zhang and Gou, 2020)
#'  \item \code{zg-3}: Zhang and Gou's third estimator (Zhang and Gou, 2020)
#'  \item \code{zg-4}: Zhang and Gou's fourth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-5}: Zhang and Gou's fifth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-6}: Zhang and Gou's sixth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-7}: Zhang and Gou's seventh estimator (Zhang and Gou, 2020)
#'  \item \code{zg-8}: Zhang and Gou's eighth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-9}: Zhang and Gou's ninth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-10}: Zhang and Gou's tenth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-11}: Zhang and Gou's eleventh estimator (Zhang and Gou, 2020)
#'  \item \code{zg-12}: Zhang and Gou's twelveth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-13}: Zhang and Gou's thirteenth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-14}: Zhang and Gou's fourteenth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-15}: Zhang and Gou's fifteenth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-16}: Zhang and Gou's sixteenth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-17}: Zhang and Gou's seventeenth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-18}: Zhang and Gou's eighteenth estimator (Zhang and Gou, 2020)
#'  \item \code{zg-19}: Zhang and Gou's nineteenth estimator (Zhang and Gou, 2020)
#'  }
#' @examples
#' library(fuel)
#' # Unbiased Estimation (Finney, 1941)
#' fuel::lognormalsd(data=c(1,4,6,7), estimator='unbiased')
#' # Longford's estimator, minimize the mean squared error (Longford, 2009)
#' fuel::lognormalsd(data=c(1,4,6,7), estimator='l-ms')
#' # Gou and Tamhane's estimator, Rukhin type (Gou and Tamhane, 2017)
#' fuel::lognormalsd(data=c(1,4,6,7), estimator='gt-r')
#' # Zhang and Gou's No.4 estimator (Zhang and Gou, 2020)
#' fuel::lognormalsd(data=c(1,4,6,7), estimator='zg-4')
#'
#' @export
#' @import stats
#'
lognormalsd <- function (data, estimator, base=exp(1), n=length(data), m=n-1, d=1/n) {
  #
  rn <- log(data,base=base)
  mean.rn <- mean(rn)
  sd.rn <- sd(rn)
  #
  a <- 2
  b <- 4
  partA <- fuel::lognormalest(n, m, d, mean.rn, sd.rn, a, b, estimator)
  #
  a <- 2
  b <- 2
  partB <- fuel::lognormalest(n, m, d, mean.rn, sd.rn, a, b, estimator)
  #
  result <- sqrt(max(0,partA - partB))
  return (result)
}