- Corrigendum to “Comparison of Some Tests of Fit for the Inverse Gaussian Distribution”, D. J. Best, J. C. W. Rayner, and O. Thas

Advances in Decision Sciences

Corrigendum (2 pages), Article ID 1097157, Volume 2016 (2016)

Published 9 November 2016

- Comparison of Some Tests of Fit for the Inverse Gaussian Distribution, D. J. Best, J. C. W. Rayner, and O. Thas

Advances in Decision Sciences

Research Article (9 pages), Article ID 150303, Volume 2012 (2012)

Published 11 November 2012

Advances in Decision Sciences

Volume 2015, Article ID 969245, 2 pages

http://dx.doi.org/10.1155/2015/969245

## Comment on “Comparison of Some Tests of Fit for the Inverse Gaussian Distribution”

^{1}Department of Biostatistics, The State University of New York at Buffalo, Buffalo, NY 14221, USA^{2}Department of Statistics, University of Birjand, Birjand, Iran

Received 11 November 2014; Revised 25 December 2014; Accepted 6 January 2015

Academic Editor: Shelton Peiris

Copyright © 2015 Albert Vexler 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.

The statistical literature shows that the density-based empirical likelihood (DBEL) concept (e.g., [1, pages 150-151], [2]) can be employed successfully to construct efficient non/semiparametric testing procedures. The DBEL approach implies a standard scheme to develop highly efficient procedures, approximating nonparametrically most powerful Neyman-Pearson test-rules.

The paper [3] displayed several concerns regarding the power and practical applicability of the DBEL ratio test for inverse Gaussian (IG) distributions proposed in [4].

(1) Introducing the DBEL ratio test, the authors of [3] wrote, “Observe that, for small , such as , the statistic can take an infinite value when there are tied data. Vexler et al. [4] do not appear to note this. For the Poisson alternative in Table 1 and the statistic is often infinite.” The test statistic does not depend on . The structure of the test statistic , which consists of the operator “min” over ’s, insures that the value of should not be infinity if just one observed value of the statistics under the “min”-operator is not infinity. The DBEL decision rule says to reject the null hypothesis for large values of the test statistic. If observed values of the statistics involved in under the “min”-operator are infinity, for all , and then , this implies rejecting the null hypothesis. In these cases, we observe that the data consists of too many equal observations and it is clear that the data cannot follow a continuous IG distribution. In a similar manner to the note mentioned above, we can consider data with many zero values, for example, when the Poisson alternative is evaluated. Formally speaking, even when we observe one we cannot assume our sample is IG-distributed. Taking into account practical issues related to measuring errors, we can impute small values, when , but evaluations of such techniques do not belong to the aims of this letter.

(2) Considering the power of the test statistics, the authors of [3] evaluated just samples with the size of . This and the comments above lead us to provide a limited Monte Carlo (MC) study based on 10,000 generations of samples that followed the and distributions. Using MC simulations, we compared the powers of the tests, controlling the TIE rate to be 5%. To tabulate the corresponding percentiles of the null distributions of the test statistics, we drew 75,000 replicate samples of the test statistics based on IG(1,1)-distributed data points at each sample size . Table 1 depicts obtained results that can be compared with the outputs of Table 1(b) in [3].