Comment on "Comparison of Some Tests of Fit for the Inverse Gaussian Distribution"

Abstract

Albert Vexler 1 and Yang Zhao 1 and Hadi Alizadeh Noughabi 2 Academic Editor:Shelton Peiris 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; 12 January 2015 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. 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 [figure omitted; refer to PDF] .

Albert Vexler 1 and Yang Zhao 1 and Hadi Alizadeh Noughabi 2

Academic Editor:Shelton Peiris

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; 12 January 2015

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 [figure omitted; refer to PDF] , such as [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] the [figure omitted; refer to PDF] statistic is often infinite." The test statistic [figure omitted; refer to PDF] does not depend on [figure omitted; refer to PDF] . The structure of the test statistic [figure omitted; refer to PDF] , which consists of the operator "min" over [figure omitted; refer to PDF] 's, insures that the value of [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] under the "min"-operator are infinity, for all [figure omitted; refer to PDF] , and then [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] we cannot assume our sample is IG-distributed. Taking into account practical issues related to measuring errors, we can impute small values, when [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] . 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 [figure omitted; refer to PDF] and [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] . Table 1 depicts obtained results that can be compared with the outputs of Table 1(b) in [3].

Table 1: The Monte Carlo powers of the test-statistics under the alternative hypotheses: [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ).

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
15	0.69 (0.05)	0.53 (0.08)	0.74 (0.03)	0.63 (0)	0.74 (0.36)	0.75 (0.11)
20	0.79 (0.04)	0.68 (0.19)	0.84 (0.03)	0.74 (0)	0.89 (0.50)	0.87 (0.16)
25	0.87 (0.04)	0.76 (0.31)	0.92 (0.05)	0.82(0)	0.95 (0.62)	0.93 (0.21)

We cannot provide here results corresponding to [figure omitted; refer to PDF] , [figure omitted; refer to PDF] -type test statistics considered in [3], due to explanation problems in [3] that we will point out in comments below. We just can remark that in the scenarios [figure omitted; refer to PDF] and [figure omitted; refer to PDF] the [figure omitted; refer to PDF] test statistic gives the power of 0.47 and 0.06, respectively, but we suppose there is a problem in the tests presentations in [3].

Our results are different from those demonstrated in [3] and may change the conclusions shown in [3] with respect to the MC power comparisons. Also these results as well as several MC outputs of [3] raise questions about the consistency of some tests for the IG distribution.

(3) Equations [figure omitted; refer to PDF] in [3] are employed from [5]. However, these equations are different from those used in [5]. The authors of [3] used right formal notations shown in [5] to calculate the tests but provided wrong definitions.

In [figure omitted; refer to PDF] in [3], the authors of [3] used "1+" in [figure omitted; refer to PDF] , whereas in the original paper [6] "1-" is proposed.

In page 5 of [3], line 5 from the bottom, perhaps [figure omitted; refer to PDF] should be used instead of [figure omitted; refer to PDF] defined in [figure omitted; refer to PDF] .

In the "Jug Bridge Example," the value of [figure omitted; refer to PDF] should be 0.0033.

We cannot confirm the MC results of [3] regarding [figure omitted; refer to PDF] .

(4) The paper [3] considered a very interesting issue to compare the tests for the IG distribution. Perhaps, a more systematic approach to define corresponding alternatives (see, e.g., [7]) can be considered in future studies and various different sample sizes in the relevant MC evaluations can be applied.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] N. Balakrishnan Methods and Applications of Statistics in Clinical Trials , vol. 2, John Wiley & Sons, Hoboken, NJ, USA, 2014.

[2] A. Vexler, W.-M. Tsai, A. D. Hutson, "A simple density-based empirical likelihood ratio test for independence," The American Statistician , vol. 68, no. 3, pp. 158-169, 2014.

[3] D. J. Best, J. C. Rayner, O. Thas, "Comparison of some tests of fit for the inverse Gaussian distribution," Advances in Decision Sciences , vol. 2012, 2012.

[4] A. Vexler, G. Shan, S. Kim, W.-M. Tsai, L. Tian, A. D. Hutson, "An empirical likelihood ratio based goodness-of-fit test for inverse Gaussian distributions," Journal of Statistical Planning and Inference , vol. 141, no. 6, pp. 2128-2140, 2011.

[5] G. R. Ducharme, "Goodness-of-fit tests for the inverse Gaussian and related distributions," Test , vol. 10, no. 2, pp. 271-290, 2001.

[6] N. Henze, B. Klar, "Goodness-of-fit tests for the inverse Gaussian distribution based on the empirical Laplace transform," Annals of the Institute of Statistical Mathematics , vol. 54, no. 2, pp. 425-444, 2002.

[7] E. J. Dudewicz, E. C. van der Meulen, "Entropy-based tests of uniformity," The Journal of the American Statistical Association , vol. 76, no. 376, pp. 967-974, 1981.