Package 'fuel'

Title: Framework for Unified Estimation in Lognormal Models
Description: Lognormal models have broad applications in various research areas such as economics, actuarial science, biology, environmental science and psychology. The estimation problem in lognormal models has been extensively studied. This R package 'fuel' implements thirty-nine existing and newly proposed estimators. See Zhang, F., and Gou, J. (2020), A unified framework for estimation in lognormal models, Technical report.
Authors: Jiangtao Gou and Fengqing (Zoe) Zhang
Maintainer: Jiangtao Gou <[email protected]>
License: GPL-3
Version: 1.2.0
Built: 2024-12-14 06:38:50 UTC
Source: CRAN

Help Index

Lognormal Estimators

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).

Usage

lognormalest(n, m = n - 1, d = 1/n, mean.rn, sd.rn, a, b, estimator)

Arguments

n

sample size.
m

degree of freedom of the variance estimation of the log-transformed data.
d

standardized variance of the sampling distribution of the log-transformed data.
mean.rn

mean of the log-transformed data.
sd.rn

standard deviation of the log-transformed data.
a

the first known constants in the parametric function for the statistics.
b

the second known constants in the parametric function for the statistics.
estimator

a total of thirty-eight different estimation methods. See more descriptions in Section Details.

Details

Consider a parametric function in the original scale we are interested in estimating θ(a,b)=exp(aμ+bσ2/2)\theta(a,b) = exp(a\mu + b\sigma^2/2),where constants a and b are known. Specifically, θ(1,1)\theta(1,1) is the mean of the lognormal distribution, θ(2,4)\theta(2,4) is the second moment, θ(2,4)−θ(2,2)\theta(2,4)-\theta(2,2) is the variance, and (θ(0,2)−1)1/2(\theta(0,2) - 1)^{1/2} is the coeficient of variation.

  1. unbiased: Unbiased estimator (Finney, 1941)

  2. qml: Quasi maximum likelihood estimator

  3. ml: Maximum likelihood estimator

  4. sa: Simple adjustment estimator

  5. f: Finney's unbiased estimator (Finney, 1941)

  6. z: Zellner's estimator (Zellner, 1971)

  7. es: Evans and Shaban’s estimator (Evans and Shaban, 1974, 1976)

  8. r-s: Rukhin’s simple estimator (Rukhin, 1986)

  9. r-f: Rukhin’s estimator using Finney's function (Rukhin, 1986)

  10. r-lo: Rukhin’s locally optimal estimator (Rukhin, 1986)

  11. r-b: Rukhin’s Bayes estimator (Rukhin, 1986)

  12. ev: El-Shaarawi and Viveros' estimator (El-Shaarawi and Viveros, 1997)

  13. zh: Zhou's estimator (Zhou, 1998)

  14. sz-mm: Shen and Zhu's MM estimator (Shen and Zhu, 2008)

  15. sz-mb: Shen and Zhu's MB estimator (Shen and Zhu, 2008)

  16. l-ub: Longford's UB estimator (Longford, 2009)

  17. l-ms: Longford's MS estimator (Longford, 2009)

  18. ft: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)

  19. ft-s: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)

  20. ft-b: Fabrizi and Trivisano's Bayes estimator (Fabrizi and Trivisano, 2012)

  21. gt-f: Gou and Tamhane's estimator using Finney's function (Gou and Tamhane, 2017)

  22. gt-es: Gou and Tamhane's estimator using Evans and Shaban's function (Gou and Tamhane, 2017)

  23. gt-r: Gou and Tamhane's estimator using Rukhin's function (Gou and Tamhane, 2017)

  24. zg-1: Zhang and Gou's first estimator (Zhang and Gou, 2020)

  25. zg-2: Zhang and Gou's second estimator (Zhang and Gou, 2020)

  26. zg-3: Zhang and Gou's third estimator (Zhang and Gou, 2020)

  27. zg-4: Zhang and Gou's fourth estimator (Zhang and Gou, 2020)

  28. zg-5: Zhang and Gou's fifth estimator (Zhang and Gou, 2020)

  29. zg-6: Zhang and Gou's sixth estimator (Zhang and Gou, 2020)

  30. zg-7: Zhang and Gou's seventh estimator (Zhang and Gou, 2020)

  31. zg-8: Zhang and Gou's eighth estimator (Zhang and Gou, 2020)

  32. zg-9: Zhang and Gou's ninth estimator (Zhang and Gou, 2020)

  33. zg-10: Zhang and Gou's tenth estimator (Zhang and Gou, 2020)

  34. zg-11: Zhang and Gou's eleventh estimator (Zhang and Gou, 2020)

  35. zg-12: Zhang and Gou's twelveth estimator (Zhang and Gou, 2020)

  36. zg-13: Zhang and Gou's thirteenth estimator (Zhang and Gou, 2020)

  37. zg-14: Zhang and Gou's fourteenth estimator (Zhang and Gou, 2020)

  38. zg-15: Zhang and Gou's fifteenth estimator (Zhang and Gou, 2020)

  39. zg-16: Zhang and Gou's sixteenth estimator (Zhang and Gou, 2020)

  40. zg-17: Zhang and Gou's seventeenth estimator (Zhang and Gou, 2020)

  41. zg-18: Zhang and Gou's eighteenth estimator (Zhang and Gou, 2020)

  42. zg-19: Zhang and Gou's nineteenth estimator (Zhang and Gou, 2020)

Value

estimation using a specific estimating method.

Author(s)

Jiangtao Gou

Fengqing (Zoe) Zhang

References

Finney, D. J. (1941). On the distribution of a variate whose logarithm is normally distributed. Supplement to the Journal of the Royal Statistical Society, 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. Journal of the American Statistical Association, 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. Journal of the American Statistical Association, 69: 779-781. <https://doi.org/10.2307/2286017>

Rukhin, A. L. (1986). Improved estimation in lognormal models. Journal of the American Statistical Association, 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. Environmetrics, 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. Journal of Statistical Planning and Inference, 138: 552-567. <https://doi.org/10.1016/j.jspi.2006.10.016>

Longford, N. T. (2009). Inference with the lognormal distribution. Journal of Statistical Planning and Inference, 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. Bayesian Analysis, 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. Communications in Statistics - Theory and Methods 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.

Examples

library(fuel)
# Unbiased Estimation (Finney, 1941)
fuel::lognormalest(n=10, m=9, d=1/10, mean.rn=1, sd.rn=1, a=1, b=1, estimator='unbiased')
# Longford's estimator, minimize the mean squared error (Longford, 2009)
fuel::lognormalest(n=10, m=9, d=1/10, mean.rn=1, sd.rn=1, a=1, b=1, estimator='l-ms')
# Gou and Tamhane's estimator, Rukhin type (Gou and Tamhane, 2017)
fuel::lognormalest(n=10, m=9, d=1/10, mean.rn=1, sd.rn=1, a=1, b=1, estimator='gt-r')
# Zhang and Gou's No.4 estimator (Zhang and Gou, 2020)
fuel::lognormalest(n=10, m=9, d=1/10, mean.rn=1, sd.rn=1, a=1, b=1, estimator='zg-4')

Mean 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).

Usage

lognormalmean(
  data,
  estimator,
  base = exp(1),
  n = length(data),
  m = n - 1,
  d = 1/n
)

Arguments

data

original data vector
estimator

a total of thirty-eight different estimation methods. See more descriptions in Section Details.
base

the base with respect to which logarithms are computed. Defaults to e.
n

sample size.
m

degree of freedom of the variance estimation of the log-transformed data.
d

standardized variance of the sampling distribution of the log-transformed data.

Details

Consider a parametric function in the original scale we are interested in estimating θ(a,b)=exp(aμ+bσ2/2)\theta(a,b) = exp(a\mu + b\sigma^2/2),where constants a and b are known. Specifically, θ(1,1)\theta(1,1) is the mean of the lognormal distribution, θ(2,4)\theta(2,4) is the second moment, θ(2,4)−θ(2,2)\theta(2,4)-\theta(2,2) is the variance, and (θ(0,2)−1)1/2(\theta(0,2) - 1)^{1/2} is the coeficient of variation.

  1. unbiased: Unbiased estimator (Finney, 1941)

  2. qml: Quasi maximum likelihood estimator

  3. ml: Maximum likelihood estimator

  4. sa: Simple adjustment estimator

  5. f: Finney's unbiased estimator (Finney, 1941)

  6. z: Zellner's estimator (Zellner, 1971)

  7. es: Evans and Shaban’s estimator (Evans and Shaban, 1974, 1976)

  8. r-s: Rukhin’s simple estimator (Rukhin, 1986)

  9. r-f: Rukhin’s estimator using Finney's function (Rukhin, 1986)

  10. r-lo: Rukhin’s locally optimal estimator (Rukhin, 1986)

  11. r-b: Rukhin’s Bayes estimator (Rukhin, 1986)

  12. ev: El-Shaarawi and Viveros' estimator (El-Shaarawi and Viveros, 1997)

  13. zh: Zhou's estimator (Zhou, 1998)

  14. sz-mm: Shen and Zhu's MM estimator (Shen and Zhu, 2008)

  15. sz-mb: Shen and Zhu's MB estimator (Shen and Zhu, 2008)

  16. l-ub: Longford's UB estimator (Longford, 2009)

  17. l-ms: Longford's MS estimator (Longford, 2009)

  18. ft: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)

  19. ft-s: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)

  20. ft-b: Fabrizi and Trivisano's Bayes estimator (Fabrizi and Trivisano, 2012)

  21. gt-f: Gou and Tamhane's estimator using Finney's function (Gou and Tamhane, 2017)

  22. gt-es: Gou and Tamhane's estimator using Evans and Shaban's function (Gou and Tamhane, 2017)

  23. gt-r: Gou and Tamhane's estimator using Rukhin's function (Gou and Tamhane, 2017)

  24. zg-1: Zhang and Gou's first estimator (Zhang and Gou, 2020)

  25. zg-2: Zhang and Gou's second estimator (Zhang and Gou, 2020)

  26. zg-3: Zhang and Gou's third estimator (Zhang and Gou, 2020)

  27. zg-4: Zhang and Gou's fourth estimator (Zhang and Gou, 2020)

  28. zg-5: Zhang and Gou's fifth estimator (Zhang and Gou, 2020)

  29. zg-6: Zhang and Gou's sixth estimator (Zhang and Gou, 2020)

  30. zg-7: Zhang and Gou's seventh estimator (Zhang and Gou, 2020)

  31. zg-8: Zhang and Gou's eighth estimator (Zhang and Gou, 2020)

  32. zg-9: Zhang and Gou's ninth estimator (Zhang and Gou, 2020)

  33. zg-10: Zhang and Gou's tenth estimator (Zhang and Gou, 2020)

  34. zg-11: Zhang and Gou's eleventh estimator (Zhang and Gou, 2020)

  35. zg-12: Zhang and Gou's twelveth estimator (Zhang and Gou, 2020)

  36. zg-13: Zhang and Gou's thirteenth estimator (Zhang and Gou, 2020)

  37. zg-14: Zhang and Gou's fourteenth estimator (Zhang and Gou, 2020)

  38. zg-15: Zhang and Gou's fifteenth estimator (Zhang and Gou, 2020)

  39. zg-16: Zhang and Gou's sixteenth estimator (Zhang and Gou, 2020)

  40. zg-17: Zhang and Gou's seventeenth estimator (Zhang and Gou, 2020)

  41. zg-18: Zhang and Gou's eighteenth estimator (Zhang and Gou, 2020)

  42. zg-19: Zhang and Gou's nineteenth estimator (Zhang and Gou, 2020)

Value

estimated mean. .

Author(s)

Jiangtao Gou

References

Finney, D. J. (1941). On the distribution of a variate whose logarithm is normally distributed. Supplement to the Journal of the Royal Statistical Society, 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. Journal of the American Statistical Association, 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. Journal of the American Statistical Association, 69: 779-781. <https://doi.org/10.2307/2286017>

Rukhin, A. L. (1986). Improved estimation in lognormal models. Journal of the American Statistical Association, 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. Environmetrics, 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. Journal of Statistical Planning and Inference, 138: 552-567. <https://doi.org/10.1016/j.jspi.2006.10.016>

Longford, N. T. (2009). Inference with the lognormal distribution. Journal of Statistical Planning and Inference, 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. Bayesian Analysis, 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. Communications in Statistics - Theory and Methods 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.

Examples

library(fuel)
# Unbiased Estimation (Finney, 1941)
fuel::lognormalmean(data=c(1,4,6,7), estimator='unbiased')
# Longford's estimator, minimize the mean squared error (Longford, 2009)
fuel::lognormalmean(data=c(1,4,6,7), estimator='l-ms')
# Gou and Tamhane's estimator, Rukhin type (Gou and Tamhane, 2017)
fuel::lognormalmean(data=c(1,4,6,7), estimator='gt-r')
# Zhang and Gou's No.4 estimator (Zhang and Gou, 2020)
fuel::lognormalmean(data=c(1,4,6,7), estimator='zg-4')

Median 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).

Usage

lognormalmedian(
  data,
  estimator,
  base = exp(1),
  n = length(data),
  m = n - 1,
  d = 1/n
)

Arguments

data

original data vector
estimator

a total of thirty-eight different estimation methods. See more descriptions in Section Details.
base

the base with respect to which logarithms are computed. Defaults to e.
n

sample size.
m

degree of freedom of the variance estimation of the log-transformed data.
d

standardized variance of the sampling distribution of the log-transformed data.

Details

Consider a parametric function in the original scale we are interested in estimating θ(a,b)=exp(aμ+bσ2/2)\theta(a,b) = exp(a\mu + b\sigma^2/2),where constants a and b are known. Specifically, θ(1,1)\theta(1,1) is the mean of the lognormal distribution, θ(2,4)\theta(2,4) is the second moment, θ(2,4)−θ(2,2)\theta(2,4)-\theta(2,2) is the variance, and (θ(0,2)−1)1/2(\theta(0,2) - 1)^{1/2} is the coeficient of variation.

  1. unbiased: Unbiased estimator (Finney, 1941)

  2. qml: Quasi maximum likelihood estimator

  3. ml: Maximum likelihood estimator

  4. sa: Simple adjustment estimator

  5. f: Finney's unbiased estimator (Finney, 1941)

  6. z: Zellner's estimator (Zellner, 1971)

  7. es: Evans and Shaban’s estimator (Evans and Shaban, 1974, 1976)

  8. r-s: Rukhin’s simple estimator (Rukhin, 1986)

  9. r-f: Rukhin’s estimator using Finney's function (Rukhin, 1986)

  10. r-lo: Rukhin’s locally optimal estimator (Rukhin, 1986)

  11. r-b: Rukhin’s Bayes estimator (Rukhin, 1986)

  12. ev: El-Shaarawi and Viveros' estimator (El-Shaarawi and Viveros, 1997)

  13. zh: Zhou's estimator (Zhou, 1998)

  14. sz-mm: Shen and Zhu's MM estimator (Shen and Zhu, 2008)

  15. sz-mb: Shen and Zhu's MB estimator (Shen and Zhu, 2008)

  16. l-ub: Longford's UB estimator (Longford, 2009)

  17. l-ms: Longford's MS estimator (Longford, 2009)

  18. ft: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)

  19. ft-s: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)

  20. ft-b: Fabrizi and Trivisano's Bayes estimator (Fabrizi and Trivisano, 2012)

  21. gt-f: Gou and Tamhane's estimator using Finney's function (Gou and Tamhane, 2017)

  22. gt-es: Gou and Tamhane's estimator using Evans and Shaban's function (Gou and Tamhane, 2017)

  23. gt-r: Gou and Tamhane's estimator using Rukhin's function (Gou and Tamhane, 2017)

  24. zg-1: Zhang and Gou's first estimator (Zhang and Gou, 2020)

  25. zg-2: Zhang and Gou's second estimator (Zhang and Gou, 2020)

  26. zg-3: Zhang and Gou's third estimator (Zhang and Gou, 2020)

  27. zg-4: Zhang and Gou's fourth estimator (Zhang and Gou, 2020)

  28. zg-5: Zhang and Gou's fifth estimator (Zhang and Gou, 2020)

  29. zg-6: Zhang and Gou's sixth estimator (Zhang and Gou, 2020)

  30. zg-7: Zhang and Gou's seventh estimator (Zhang and Gou, 2020)

  31. zg-8: Zhang and Gou's eighth estimator (Zhang and Gou, 2020)

  32. zg-9: Zhang and Gou's ninth estimator (Zhang and Gou, 2020)

  33. zg-10: Zhang and Gou's tenth estimator (Zhang and Gou, 2020)

  34. zg-11: Zhang and Gou's eleventh estimator (Zhang and Gou, 2020)

  35. zg-12: Zhang and Gou's twelveth estimator (Zhang and Gou, 2020)

  36. zg-13: Zhang and Gou's thirteenth estimator (Zhang and Gou, 2020)

  37. zg-14: Zhang and Gou's fourteenth estimator (Zhang and Gou, 2020)

  38. zg-15: Zhang and Gou's fifteenth estimator (Zhang and Gou, 2020)

  39. zg-16: Zhang and Gou's sixteenth estimator (Zhang and Gou, 2020)

  40. zg-17: Zhang and Gou's seventeenth estimator (Zhang and Gou, 2020)

  41. zg-18: Zhang and Gou's eighteenth estimator (Zhang and Gou, 2020)

  42. zg-19: Zhang and Gou's nineteenth estimator (Zhang and Gou, 2020)

Value

estimated median. .

Author(s)

Jiangtao Gou

References

Finney, D. J. (1941). On the distribution of a variate whose logarithm is normally distributed. Supplement to the Journal of the Royal Statistical Society, 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. Journal of the American Statistical Association, 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. Journal of the American Statistical Association, 69: 779-781. <https://doi.org/10.2307/2286017>

Rukhin, A. L. (1986). Improved estimation in lognormal models. Journal of the American Statistical Association, 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. Environmetrics, 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. Journal of Statistical Planning and Inference, 138: 552-567. <https://doi.org/10.1016/j.jspi.2006.10.016>

Longford, N. T. (2009). Inference with the lognormal distribution. Journal of Statistical Planning and Inference, 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. Bayesian Analysis, 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. Communications in Statistics - Theory and Methods 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.

Examples

library(fuel)
# Unbiased Estimation (Finney, 1941)
fuel::lognormalmedian(data=c(1,4,6,7), estimator='unbiased')
# Longford's estimator, minimize the mean squared error (Longford, 2009)
fuel::lognormalmedian(data=c(1,4,6,7), estimator='l-ms')
# Gou and Tamhane's estimator, Rukhin type (Gou and Tamhane, 2017)
fuel::lognormalmedian(data=c(1,4,6,7), estimator='gt-r')
# Zhang and Gou's No.4 estimator (Zhang and Gou, 2020)
fuel::lognormalmedian(data=c(1,4,6,7), estimator='zg-4')

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).

Usage

lognormalsd(
  data,
  estimator,
  base = exp(1),
  n = length(data),
  m = n - 1,
  d = 1/n
)

Arguments

data

original data vector
estimator

a total of thirty-eight different estimation methods. See more descriptions in Section Details.
base

the base with respect to which logarithms are computed. Defaults to e.
n

sample size.
m

degree of freedom of the variance estimation of the log-transformed data.
d

standardized variance of the sampling distribution of the log-transformed data.

Details

Consider a parametric function in the original scale we are interested in estimating θ(a,b)=exp(aμ+bσ2/2)\theta(a,b) = exp(a\mu + b\sigma^2/2),where constants a and b are known. Specifically, θ(1,1)\theta(1,1) is the mean of the lognormal distribution, θ(2,4)\theta(2,4) is the second moment, θ(2,4)−θ(2,2)\theta(2,4)-\theta(2,2) is the variance, and (θ(0,2)−1)1/2(\theta(0,2) - 1)^{1/2} is the coeficient of variation.

  1. unbiased: Unbiased estimator (Finney, 1941)

  2. qml: Quasi maximum likelihood estimator

  3. ml: Maximum likelihood estimator

  4. sa: Simple adjustment estimator

  5. f: Finney's unbiased estimator (Finney, 1941)

  6. z: Zellner's estimator (Zellner, 1971)

  7. es: Evans and Shaban’s estimator (Evans and Shaban, 1974, 1976)

  8. r-s: Rukhin’s simple estimator (Rukhin, 1986)

  9. r-f: Rukhin’s estimator using Finney's function (Rukhin, 1986)

  10. r-lo: Rukhin’s locally optimal estimator (Rukhin, 1986)

  11. r-b: Rukhin’s Bayes estimator (Rukhin, 1986)

  12. ev: El-Shaarawi and Viveros' estimator (El-Shaarawi and Viveros, 1997)

  13. zh: Zhou's estimator (Zhou, 1998)

  14. sz-mm: Shen and Zhu's MM estimator (Shen and Zhu, 2008)

  15. sz-mb: Shen and Zhu's MB estimator (Shen and Zhu, 2008)

  16. l-ub: Longford's UB estimator (Longford, 2009)

  17. l-ms: Longford's MS estimator (Longford, 2009)

  18. ft: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)

  19. ft-s: Fabrizi and Trivisano's Simplified Bayes estimator (Fabrizi and Trivisano, 2012)

  20. ft-b: Fabrizi and Trivisano's Bayes estimator (Fabrizi and Trivisano, 2012)

  21. gt-f: Gou and Tamhane's estimator using Finney's function (Gou and Tamhane, 2017)

  22. gt-es: Gou and Tamhane's estimator using Evans and Shaban's function (Gou and Tamhane, 2017)

  23. gt-r: Gou and Tamhane's estimator using Rukhin's function (Gou and Tamhane, 2017)

  24. zg-1: Zhang and Gou's first estimator (Zhang and Gou, 2020)

  25. zg-2: Zhang and Gou's second estimator (Zhang and Gou, 2020)

  26. zg-3: Zhang and Gou's third estimator (Zhang and Gou, 2020)

  27. zg-4: Zhang and Gou's fourth estimator (Zhang and Gou, 2020)

  28. zg-5: Zhang and Gou's fifth estimator (Zhang and Gou, 2020)

  29. zg-6: Zhang and Gou's sixth estimator (Zhang and Gou, 2020)

  30. zg-7: Zhang and Gou's seventh estimator (Zhang and Gou, 2020)

  31. zg-8: Zhang and Gou's eighth estimator (Zhang and Gou, 2020)

  32. zg-9: Zhang and Gou's ninth estimator (Zhang and Gou, 2020)

  33. zg-10: Zhang and Gou's tenth estimator (Zhang and Gou, 2020)

  34. zg-11: Zhang and Gou's eleventh estimator (Zhang and Gou, 2020)

  35. zg-12: Zhang and Gou's twelveth estimator (Zhang and Gou, 2020)

  36. zg-13: Zhang and Gou's thirteenth estimator (Zhang and Gou, 2020)

  37. zg-14: Zhang and Gou's fourteenth estimator (Zhang and Gou, 2020)

  38. zg-15: Zhang and Gou's fifteenth estimator (Zhang and Gou, 2020)

  39. zg-16: Zhang and Gou's sixteenth estimator (Zhang and Gou, 2020)

  40. zg-17: Zhang and Gou's seventeenth estimator (Zhang and Gou, 2020)

  41. zg-18: Zhang and Gou's eighteenth estimator (Zhang and Gou, 2020)

  42. zg-19: Zhang and Gou's nineteenth estimator (Zhang and Gou, 2020)

Value

estimated standard deviation. .

Author(s)

Jiangtao Gou

References

Finney, D. J. (1941). On the distribution of a variate whose logarithm is normally distributed. Supplement to the Journal of the Royal Statistical Society, 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. Journal of the American Statistical Association, 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. Journal of the American Statistical Association, 69: 779-781. <https://doi.org/10.2307/2286017>

Rukhin, A. L. (1986). Improved estimation in lognormal models. Journal of the American Statistical Association, 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. Environmetrics, 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. Journal of Statistical Planning and Inference, 138: 552-567. <https://doi.org/10.1016/j.jspi.2006.10.016>

Longford, N. T. (2009). Inference with the lognormal distribution. Journal of Statistical Planning and Inference, 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. Bayesian Analysis, 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. Communications in Statistics - Theory and Methods 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.

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')