Journal of Probability and Statistics
Volume 2009 (2009), Article ID 310575, 16 pages
Research Article

On the Existence and Uniqueness of the Maximum Likelihood Estimators of Normal and Lognormal Population Parameters with Grouped Data

1Center on Aging and Health, John Hopkins University, Baltimore, MD 21287, USA
2Department of Statistics, Florida International University, Miami, FL 33139, USA
3Department of Statistics & Biostatistics, California State University, Hayward, CA 94542, USA

Received 11 March 2009; Accepted 16 June 2009

Academic Editor: A. Thavaneswaran

Copyright © 2009 Jin Xia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Lognormal distribution has abundant applications in various fields. In literature, most inferences on the two parameters of the lognormal distribution are based on Type-I censored sample data. However, exact measurements are not always attainable especially when the observation is below or above the detection limits, and only the numbers of measurements falling into predetermined intervals can be recorded instead. This is the so-called grouped data. In this paper, we will show the existence and uniqueness of the maximum likelihood estimators of the two parameters of the underlying lognormal distribution with Type-I censored data and grouped data. The proof was first established under the case of normal distribution and extended to the lognormal distribution through invariance property. The results are applied to estimate the median and mean of the lognormal population.