# Lognormal Estimators 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). ```r 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. ## Returns estimation using a specific estimating method. ## Details Consider a parametric function in the original scale we are interested in estimating $\theta(a,b) = exp(a\mu + b\sigma^2/2)$,where constants **a** and **b** are known. Specifically, $\theta(1,1)$ is the mean of the lognormal distribution, $\theta(2,4)$ is the second moment, $\theta(2,4)-\theta(2,2)$ is the variance, and $(\theta(0,2) - 1)^{1/2}$ 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) ## Examples ```r 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') ``` ## 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. 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. Evans, I. G. and Shaban, S. A. (1974). A note on estimation in lognormal models. **Journal of the American Statistical Association**, 69 : 779-781. Rukhin, A. L. (1986). Improved estimation in lognormal models. **Journal of the American Statistical Association**, 81 : 1046-1049. El-Shaarawi, A. H. and Viveros, R. (1997). Inference about the mean in log-regression with environmental applications. **Environmetrics**, 8 : 569-582. 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. Longford, N. T. (2009). Inference with the lognormal distribution. **Journal of Statistical Planning and Inference**, 139 : 2329-2340. Fabrizi, E. and Trivisano, C. (2012). Bayesian estimation of log-normal means with finite quadratic expected loss. **Bayesian Analysis**, 7 : 975-996. 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. Zhang, F. and Gou, J. (2020). A unified framework for estimation in lognormal models. Technical Report. ## Author(s) Jiangtao Gou Fengqing (Zoe) Zhang

lognormalest function