# lognormalest: Lognormal Estimators In fuel: Framework for Unified Estimation in Lognormal Models

## 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

 `1` ```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 + bσ^2/2),where constants a and b are known. Specifically, θ(1,1) is the mean of the lognormal distribution, θ(2,4) is the second moment, θ(2,4)-θ(2,2) is the variance, and (θ(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

 ```1 2 3 4 5 6 7 8 9``` ```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') ```

fuel documentation built on July 2, 2020, 12:23 a.m.