1. Home
  2. Vol 5, No 4 (2021)
  3. Yosboonruang

Confidence Intervals for the Coefficient of Quartile Variation of a Zero-inflated Lognormal Distribution

Noppadon Yosboonruang, Sa-Aat Niwitpong

Abstract


There are many types of skewed distribution, one of which is the lognormal distribution that is positively skewed and may contain true zero values. The coefficient of quartile variation is a statistical tool used to measure the dispersion of skewed and kurtosis data. The purpose of this study is to establish confidence and credible intervals for the coefficient of quartile variation of a zero-inflated lognormal distribution. The proposed approaches are based on the concepts of the fiducial generalized confidence interval, and the Bayesian method. Coverage probabilities and expected lengths were used to evaluate the performance of the proposed approaches via Monte Carlo simulation. The results of the simulation studies show that the fiducial generalized confidence interval and the Bayesian based on uniform and normal inverse Chi-squared priors were appropriate in terms of the coverage probability and expected length, while the Bayesian approach based on Jeffreys' rule prior can be used as alternatives. In addition, real data based on the red cod density from a trawl survey in New Zealand is used to illustrate the performances of the proposed approaches.

 

Doi: 10.28991/esj-2021-01289

Full Text: PDF


Keywords


Bayesian; Coverage Probability; Lognormal; Fiducial Generalized Confidence Interval; Prior Distribution.

References


Pennington, Michael. “Efficient Estimators of Abundance, for Fish and Plankton Surveys.” Biometrics 39, no. 1 (March 1983): 281. doi:10.2307/2530830.

Syrjala, S. “Critique on the Use of the Delta Distribution for the Analysis of Trawl Survey Data.” ICES Journal of Marine Science 57, no. 4 (August 2000): 831–842. doi:10.1006/jmsc.2000.0571.

Terceiro, M. “The statistical properties of recreational catch rate data for some fish stocks off the northeast U.S. coast.” Fishery Bulletin 101, no. 3 (Apr 2003): 653-672.

Fletcher, D. “Confidence intervals for the mean of the delta-lognormal distribution.” Environmental and Ecological Statistics 15, no. 2 (Jun 2008): 175-189. doi:10.1007/s10651-007-0046-8.

Wu, W.-H., and H.-N. Hsieh. “Generalized confidence interval estimation for the mean of delta-lognormal distribution: an application to New Zealand trawl survey data.” Journal of Applied Statistics 41, no. 7 (Jan 2014): 1471-1485. doi:10.1080/02664763.2014.881780.

Verrill, S., and R. A. Johnson. “Confidence bounds and hypothesis tests for normal distribution coefficients of variation.” Communications in Statistics - Theory and Methods 36, no. 12 (Aug 2007): 2187-2206. doi:10.1080/03610920701215126.

Mahmoudvand, R., and H. Hassani. “Two new confidence intervals for the coefficient of variation in a normal distribution.” Journal of Applied Statistics 36 no. 4 (Apr 2009): 429-442. doi:10.1080/02664760802474249.

Hayter, A. J. “Confidence bounds on the coefficient of variation of a normal distribution with applications to win-probabilities.” Journal of Statistical Computation and Simulation 85, no. 18 (Apr 2015): 3778-3791. doi:10.1080/00949655.2015.1035654.

Bonett, Douglas G. “Confidence Interval for a Coefficient of Quartile Variation.” Computational Statistics & Data Analysis 50, no. 11 (July 2006): 2953–2957. doi:10.1016/j.csda.2005.05.007.

Altunkaynak, B., and H. Gamgam. “Bootstrap confidence intervals for the coefficient of quartile variation.” Communications in Statistics - Simulation and Computation 48, no. 7 (2019): 2138-2146. doi:10.1080/03610918.2018.1435800.

Krishnamoorthy, K., and T. Mathew. “Inferences on the means of lognormal distributions using generalized p-values and generalized confidence intervals.” Journal of Statistical Planning and Inference 115, no. 1 (Jul 2003): 103-121. doi:10.1016/S0378-3758(02)00153-2.

Hannig, J., H. Iyer, and P. Patterson. “Fiducial generalized confidence intervals.” Journal of the American Statistical Association 101 no. 473 (Mar 2006): 254-269. doi:10.1198/016214505000000736.

Lin, S. H., and R. S. Wang. “Modified method on the means for several log-normal distributions.” Journal of Applied Statistics 40, no. 1 (Jan 2013): 194-208. doi:10.1080/02664763.2012.740622.

Nam, J., and D. Kwon. “Inference on the ratio of two coefficients of variation of two lognormal distributions.” Communications in Statistics - Theory and Methods 46, no. 17 (May 2016):8575-8587. doi:10.1080/03610926.2016.1185118.

Hasan, Md Sazib, and K. Krishnamoorthy. “Improved Confidence Intervals for the Ratio of Coefficients of Variation of Two Lognormal Distributions.” Journal of Statistical Theory and Applications 16, no. 3 (September 2017): 345. doi:10.2991/jsta.2017.16.3.6.

Thangjai, W., and S.-A. Niwitpong. “Confidence intervals for the signal-to-noise ratio and difference of signal-to-noise ratios of log-normal distributions.” Stats 2, no. 1 (Feb 2019): 164-173. doi:10.3390/stats2010012.

Thangjai, Warisa, Sa-Aat Niwitpong, and Suparat Niwitpong. “Bayesian Confidence Intervals for Coefficients of Variation of PM10 Dispersion.” Emerging Science Journal 5, no. 2 (April 1, 2021): 139–154. doi:10.28991/esj-2021-01264.

Callahan, C. M., J. G. Kesterson, and W. M. Tierney. “Association of symptoms of depression with diagnostic test charges among older adults.” Annals of Internal Medicine 126, no. 6 (Mar 1997): 426-432. doi:10.7326/0003-4819-126-6-199703150-00002.

Owen, W. J., and T. A. DeRouen. “Estimation of the mean for lognormal data containing zeroes and left-censored values, with applications to the measurement of worker exposure to air contaminants.” Biometrics 36 no. 4 (Dec 1980): 707-719. doi:10.2307/2556125.

Lo, N. C., L. D. Jacobson, and J. L. Squire. “Indices of relative abundance from fish spotter data based on delta-lognornial models.” Canadian Journal of Fisheries and Aquatic Sciences 49 (Dec 1992): 2515-2526. doi:10.1139/f92-278.

Yosboonruang, N., S.-A. Niwitpong, and S. Niwitpong. “Measuring the dispersion of rainfall using Bayesian confidence intervals for coefficient of variation of delta-lognormal distribution: a study from Thailand.” PeerJ 7 (Jul 2019): e7344. doi:10.7717/peerj.7344.

Maneerat, P., S.-A. Niwitpong, and S. Niwitpong “Bayesian confidence intervals for a single mean and the difference between two means of delta-lognormal distributions.” Communications in Statistics - Simulation and Computation (May 2019). doi:10.1080/03610918.2019.1616095.

Zhou, X. H., and W. Tu. “Confidence intervals for the mean of diagnostic test charge data containing zeros.” Biometrics 56 no. 4 (Dec 2000): 1118-1125. doi:10.1111/j.0006-341x.2000.01118.x.

Tian, L. “Inferences on the common coefficient of variation.” Statistics in Medicine 24, no. 14 (Apr 2005): 2213-2210. doi:10.1002/sim.2088.

Tian, L., and J. Wu. “Confidence intervals for the mean of lognormal data with excess zeros.” Biometrical Journal 48, no. 1 (Feb 2006): 149-156. doi:10.1002/bimj.200510155.

Li, X., X. Zhou, and L. Tian. “Interval estimation for the mean of lognormal data with excess zeros.” Statistics & Probability Letters 83, no. 11 (Nov 2013): 2447-2453. doi:10.1016/j.spl.2013.07.004.

Hasan, M. S., and K. Krishnamoorthy. “Confidence intervals for the mean and a percentile based on zero-inflated lognormal data.” Journal of Statistical Computation and Simulation 88, no. 8 (Feb 2018): 1499-1514. doi:10.1080/00949655.2018.1439033.

Maneerat, P., S.-A. Niwitpong, and S. Niwitpong. “A Bayesian approach to construct confidence intervals for comparing the rainfall dispersion in Thailand.” PeerJ 8 (Aug 2020): e8502. doi:10.7717/peerj.8502.

Maneerat, P., S.-A. Niwitpong, and S. Niwitpong. “Bayesian confidence intervals for the difference between variances of delta-lognormal distributions.” Biometrical Journal 62, no. 7 (Jun 2020): 1769-1790. doi:10.1002/bimj.201900079.

Yosboonruang, N., S.-A. Niwitpong, and S. Niwitpong. “The Bayesian confidence intervals for measuring the difference between dispersions of rainfall in Thailand.” PeerJ 8 (Aug 2020): e9662. doi:10.7717/peerj.9662.

Aitchison, John. “On the Distribution of a Positive Random Variable Having a Discrete Probability Mass at the Origin.” Journal of the American Statistical Association 50, no. 271 (September 1955): 901–908. doi:10.1080/01621459.1955.10501976.

Fisher, R. A. “Inverse Probability.” Mathematical Proceedings of the Cambridge Philosophical Society 26, no. 4 (October 1930): 528–535. doi:10.1017/s0305004100016297.

Hannig, J. “On generalized fiducial inference.” Statistica Sinica 19, no. 2 (Apr 2009): 491-544.

Harvey, J., and A. J. van der Merwe. “Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions.” Journal of Statistical Planning and Inference 142, no. 6 (Jun 2012): 1294-1309. doi:10.1016/j.jspi.2011.12.006.

Bolstad, William M., and James M. Curran. “Introduction to Bayesian Statistics, Third Edition” (August 25, 2016). doi:10.1002/9781118593165.

Kalkur, T. A., and A. Rao. “Bayes estimator for coefficient of variation and inverse coefficient of variation for the normal distribution.” International Journal of Statistics and Systems 12, no. 6 (2017): 721-732.


Full Text: PDF

DOI: 10.28991/esj-2021-01289

Refbacks

  • There are currently no refbacks.


Copyright (c) 2021 Noppadon Yosboonruang, Sa-Aat Niwitpong