Simple Formulas for Profile- and Estimated-likelihood Based Confidence Intervals for the Mean of Inverse Gaussian Distribution
Main Article Content
Abstract
In general, the construction of likelihood-based confidence intervals requires programming in a certain software package or programming language such as R and Matlab. This could be one reason why likelihoodbased confidence intervals are not often employed. The inverse Gaussian distribution is one of the important distributions as it is widely applied in many areas. In this research, the simple formulas for profile- and estimatedlikelihood based confidence intervals for the mean of Inverse Gaussian distribution with an unknown shape parameter are proposed so that constructing an interval can be calculated by hand. Also, the estimated likelihood function is mathematically proved that it does not converge to zero when the mean approaches infinity. Instead, it converges to a certain quantity depending on a sample. Comparisons of confidence intervals are achieved by using the length of intervals and coverage probabilities as criteria., and the result shows that the length of confidence intervals using the profile likelihood is greater than that produced by the estimated likelihood for all cases in the simulation study. This results in a higher coverage probability for the profile likelihood than the estimated likelihood. However, as the sample size increases, both methods produce about the same length of confidence interval and coverage probabilities.
Article Details
The articles published are the opinion of the author only. The author is responsible for any legal consequences. That may arise from that article.
References
[2] A. Onar and W. J. Padgett, “Accelerated test models with the inverse Gaussian distribution,” Journal of Statistical Planning and Inference, vol. 89, pp. 119–133, 2000.
[3] R. K. Jain and Sudha Jain, “Inverse Gaussian distribution and its application to reliability,” Microelectronics Reliability, vol. 36, no. 10, pp. 1323–1335, 1996.
[4] E. Schrödinger, “Zur theorie der fall- und steigversuche an teilchen mit brownscher bewegung,” Physikalische Zeitschrift, vol. 16, pp. 289–295, 1915.
[5] M. C. K. Tweedie, “Statistical properties of inverse Gaussian distributions I,” Annals Mathematical Statistics, vol. 28, pp. 362–377, 1957.
[6] J. L. Folks and R. S. Chhikara, “The inverse Gaussian distribution and its statistical application —A review,” Journal of the Royal Statistical Society. Series B (Methodological), vol. 40, no. 3, pp. 263–289, 1978.
[7] J. Neyman, “Outline of a theory of statistical estimation based on the classical theory of probability,” Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. 236, no. 767, pp. 333–380, 1937.
[8] Y. Pawitan, In All Likelihood: Statistical Modeling and Inference Using Likelihood, 1st ed. Oxford: Oxford University Press, 2012.
[9] A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin, Bayesian Data Analysis, 3rd ed. Florida: Chapman & Hall, 2014.
[10] C. A. Rohde, Introductory Statistical Inference with the Likelihood Function, 1st ed. London: Springer, 2014.
[11] R. A. Fisher, Statistical Methods and Scientific Inference, New York: Macmillan, 1973.
[12] S. S. Wilk, “The large sample distribution of the likelihood ratio for testing composite hypotheses,” Annals Mathematical Statistics, vol. 9, pp. 60–62, 1938.
[13] M. Arefi, G. R. Mohtashami Borzadaran, and Y. Vaghei, “A note on interval estimation for the mean of inverse Gaussian distribution,” Statistics and Operations Research Transactions, vol. 32, no. 1, 2008.
[14] P. Srisuradetchai, “Profile-likelihood based confidence intervals for the mean of inverse Gaussian distribution,” Journal of King Mongkut’s University of Technology North Bangkok, vol. 27, no. 2, 2016 (in Thai).
