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

Authors

Jin Xia, John Hopkins University
Jie Mi, Department of Statistics, Florida International UniversityFollow
Yan Yan Zhou, California State University

Date of this Version

6-16-2009

Document Type

Article

Abstract

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.

Comments

This article was originally published in Hindawi Journal of Probability and Statistics Volume 2009 (2009).

Creative Commons License

This work is licensed under a Creative Commons Attribution 2.0 License.

Recommended Citation

Jin Xia, Jie Mi, and YanYan Zhou, “On the Existence and Uniqueness of the Maximum Likelihood Estimators of Normal and Lognormal Population Parameters with Grouped Data,” Journal of Probability and Statistics, vol. 2009, Article ID 310575, 16 pages, 2009. doi:10.1155/2009/310575