Statistical Inference for k (k>=3) Lognormal Means from Left Censored Data


  • Abou El-Makarim Aboueissa



multiple detection limits; left censored data; normal and lognormal distributions; maximum likelihood estimators; expectation maximization algorithm; likelihood ratio test


The occurrence of censored data due to less than detectable measurements is a common problem with environmental data such as quality and quantity monitoring applications of water, soil, and air samples. The log-normal distribution is one of the most common distributions used for modeling skewed and positive data. Over the past decades, various methods for comparing the parameters of two lognormal distributions in the presence censored data have been proposed. Some of them are differing in terms of how the statistic test adjust to accept or to reject the null hypothesis. As a model distribution of measured environmental and/or biomedical data, log-normal distribution is considered. Logmormal means can be compared either by confidence intervals or hypothesis testing procedures. In this article, a new test procedure for comparing the means of k (k >= 3) lognormal distributions in the presence of left-censored data is introduced and evaluated. Asymptotic chi-square test is used in the proposed test procedure. A simulation study was performed to examine the power and the size of the proposed test procedure introduced in this article utilizing a computer program written in the R language. We find  analytically that the considered test procedure is doing well through comparing the size and power of the statistic test.


(1) Abdollahnezhad K., Babanezhad M. and Jafari A.A. (2012). Inference on Difference of Means of two Log-Normal Distributions; A Generalized Approach, Journal of Statistical and Econometric Methods 1 (2): 125-131.

(2) Aboueissa A. A. (2015). Comparison of Two Means of Two Log-Normal Distributions When Data is Singly Censored, International Journal of Statistics and Probability 4 (2): 73-86.

(3) Cohen A. C. R. (1959). Simplified Estimators For The Normal Distribution When Samples Are Singly Censored Or Truncated, Technometrics 3: 217-237.

(4) Dempster A. P., N. Laird M. and Rubin D. B. (1977). Maximum Likelihood from Incomplete Data via the EM Algorithm, The Journal Of Royal Statistical Society B 39: 1-38.

(5) El-Shaarawi, A. H. (1989). Inferences about the Mean from Censored Water Quality Data, Water Resources Research 25: 685-690.

(6) El-Shaarawi A. H. and Dolan D. M. (1989). Maximum Likelihood Estimation Of Water Concentrations From Censored Data, Canadian Journal Of Fisheries And Aquatic Sciences

: 1033-1039.

(7) El-Shaarawi A. H. and Esterby S. R. (1992). Replacement Of Censored Observations By A Constant: An Evaluation, Water Research 26(6): 835-844.

(8) Gibbons RD. (1994). Statistical Methods For Groundwater Monitoring, John Wiley&Sons, New York.

(9) Gilbert Richard O. (1987). Statistical Methods For Environmental Pollution Monitoring, Van Nostrand Reinhold: New York.

(10) Gleit A. (1985). Estimation for small normal data sets with detection limits, Environ. Sci. Technol. 19: 1201-1206.

(11) Gupta R. C. and Li X. (2006). Statistical Inference for the Common Mean of two Log-normal Distributions and some Applications in Reliability, Computational Statistics and Data Analysis 50: 3141-3164.

(12) Harris G. A. (1991). Two-samples Comparisons in the Presence of Less-than-detectable data, Proceeding of the Section on Statistics and the Environment: American Statistical Association: 197-201.

(13) Jianrong W., Jiang G., Wong A., and Xiang S. (2002). Likelihood Analysis for the Ratio of Means of Two Independent Log-Normal Distributions, Biometrics 58: 463-469.

(14) Krishnamoorthy K., Mathew T. and Xu Z. (2014). Comparison of Means of Two Lognormal Distributions Based on Samples with Multiple Detection Limits, Journal of Occupational and Environmental Hygiene 11 (8): 538-546.

(15) Krishnamoorthy K., Mathew T. and Xu Z. (2014). Standardized Likelihood Inference for the Mean and Percentiles of a Lognormal Distribution Based on Samples with Multiple Detection Limits, Journal of Environmental Statistics 6 (5): 1-18.

(16) Krishnamoorthy K. and Mathew T. (2003). Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals, Journal of Statistical Planning and Inference 115: 103-121.

(17) Krishnamoorthy K., Mathew T. and Ramachandran, G. (2006). Generalized P-Values and Confidence Intervals: A Novel Approach for Analyzing Lognormally Distributed Exposure Data, Journal of Occupational and Environmental Hygiene 3: 642-650.

(18) Krishnamoorthy K., Avishek M. and Mathew T. (2011). Inference for the Lognormal Mean and Quantiles Based on Samples with Left and Right Type I Censoring, Technometrics 53 (1): 72-83.

(19) Marco Bee (2005). On Maximum Likelihood Estimation of Operational Loss Distributions: Universita Degli, Studi Di Trento, Discussion paper No. 3

(20) Millard S. P. and S. J. Deverel (1998). Nonparametric Statistical Methods for Comparing two Sites Based on Data with multiple Nondetect Limits, Water Resources Research 24: 2087 - 2098.

(21) Prentice R. L. (1978). Linear rank Tests with Right Censored Data, Biometrika 65: 167-179.

(22) Paul H. and Gary H. G.(2007). A Comparison of Several Methods for Analyzing Censored Data, Oxford

University Press on behalf of the British Occupational Hygiene Society 51 (7): 611-632.

(23) Schneider H. (1986). Truncated and Censored Samples from Normal Population, Marcel Dekker: New York.

(24) Shumway R. H., Azari A. S. and Johnson P. (1989). Estimating mean concentrations under transformation for environmental data with detection limits, Technometrics 31: 347356.

(25) Stoline Michael R. (1993). Comparison of Two Medians Using a Two-Sample Log-normal Model in Environmental Contexts, Environmetrics 4(3): 323-339.

(26) Stavros Pouloukas (2004). Estimation and comparison of Lognormal Parameters in the Presence of Censored Data, Journal of Statistical Computation & Simulation 74(3): 157-169.

(27) Wolynetz M. S. (1979). Maximum Likelihood Estimations from Confined and Censored Normal Data, Journal of the Royal Statistical Society. Series C (Applied Statistics) 28 (2): 185 - 195.

(28) Yan Jin, Misty J. Hein, James A. Deddens and Cynthia J. Hines (2011). Analysis of Lognormally Distributed Exposure Data with Repeated Measures and Values below the Limit of Detection Using SAS, Oxford University Press, British Occupational Hygiene Society 55 (1): 97-112.

(29) Zhou X., Sujuan G. and Hui S. L. (1997). Methods for Comparing the Means of Two Independent Log-normal Samples, Biometrics 53: 1129-1135.




How to Cite

Aboueissa, A. E.-M. (2020). Statistical Inference for k (k>=3) Lognormal Means from Left Censored Data. Transactions on Engineering and Computing Sciences, 8(1), 09–34.