Skip to main content
SpringerLink
Log in

Symmetrizing and Variance Stabilizing Transformations of Sample Coefficient of Variation from Inverse Gaussian Distribution

  • Published:
Sankhya B Aims and scope Submit manuscript

Abstract

Coefficient of variation (CV) plays an important role in statistical practice; however, its sampling distribution may not be easy to compute. In this paper, the distributional properties of the sample CV from an inverse Gaussian distribution are investigated through transformations. Specifically, the symmetrizing transformation as outlined in Chaubey and Mudholkar (1983), that requires numerical techniques, is contrasted with the explicitly available variance stabilizing transformation (VST). The symmetrizing transformation scores very high as compared to the VST, especially in a power family. The usefulness of the resulting approximation is illustrated through a numerical example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Banik, S. and Kibria, B.M.G. (2011). Estimating the population coefficient of variation by confidence intervals. Communications in Statistics - Simulation and Computation, 40, 1236–1261.

    Article  MathSciNet  MATH  Google Scholar 

  • Bartlett, M.S. (1947). The use of transformations. Biometrika, 3, 39–52.

    Article  MathSciNet  Google Scholar 

  • Chaubey, Y.P. and Mudholkar, G.S. (1983). On the symmetrizing transformations of random variables. Preprint, Concordia University, Montreal. Available at http://spectrum.library.concordia.ca/973582/.

  • Chaubey, Y.P. and Mudholkar, G.S. (1984). On the almost symmetry of Fisher’s Z. Metron, 42(I/II), 165–169.

    MathSciNet  Google Scholar 

  • Chaubey, Y.P., Sarker, A. and Singh, M. (2016). Power Transformations: An Application for Symmetrizing the Distribution of Sample Coefficient of Variation from Inverse Gaussian Populations. In Applied Mathematics and Omics to Assess Crop Genetic Resources for Climate Change Adaptive Traits, Chapter 11, p127-137, Eds.: Abdallah, Bari, Ardeshir B. Damania, Kenneth Street, Michael Mackay and Selvadurai Dayanandan, CRC Press, Boca Raton, Florida.

  • Chaubey, Y.P., Sen, D. and Saha, K.K. (2014). On testing the coefficient of variation in an inverse Gaussian population. Statistics and Probability Letters, 90, 121–128.

    Article  MathSciNet  MATH  Google Scholar 

  • Chaubey, Y.P., Singh, M. and Sen, D. (2013). On symmetrizing transformation of the sample coefficient of variation from a normal population. Communications in Statistics - Simulation and Computation, 42, 2118–2134.

    Article  MathSciNet  MATH  Google Scholar 

  • Chhikara, R.S. and Folks, J.L. (1977). The inverse Gaussian distribution as a lifetime model. Technometrics, 19, 461–468.

    Article  MATH  Google Scholar 

  • Chhikara, R.S. and Folks, J.L. (1989). The Inverse Gaussian Distribution. Marcel Dekker, New York.

    MATH  Google Scholar 

  • Folks, J.L. and Chhikara, R.S. (1978). The inverse Gaussian distribution and its statistical application - a review. J. Roy. Statist. Soc., Ser. B 40, 263–289.

    MathSciNet  MATH  Google Scholar 

  • Gradshteyn, I.S. and Ryzhik, I.M. (2007). Tables of Integrals, Series, and Products, 7th Edition, Edited by A. Jeffrey and D. Zwillinger. Academic Press, New York.

  • Hall ,P. (1992). On the removal of skewness by transformation. J. Roy. Statist. Soc., Ser. B 54, 221–228.

    MathSciNet  Google Scholar 

  • Henze, N. and Klar, B. (2002). Goodness-of-fit tests for the inverse Gaussian distribution based on the empirical Laplace transform. Ann. Inst. Statist. Math. 54, 425–444.

    Article  MathSciNet  MATH  Google Scholar 

  • Hinkley, H. (1975). On power transformations to symmetry. Biometrika 62, 101–111.

    Article  MathSciNet  MATH  Google Scholar 

  • Hinkley, D. (1977). On quick choice of power transformation. Journal of the Royal Statistical Society, Series C 26, 67–69.

    Google Scholar 

  • Hsieh, H.K. (1990). Inferences on the coefficient of variation of an inverse Gaussian distribution. Communications in Statistics – Theory and Methods, 19, 1589–1605.

    Article  MathSciNet  Google Scholar 

  • Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and graphics. Journal of Computational and Graphical Statistics, 5, 299–314.

    Google Scholar 

  • Jensen, D.R. and Solomon, H. (1972). A Gaussian approximation to the distribution of a definite quadratic form. Journal of American Statistical Association, 67, 898–902.

    MATH  Google Scholar 

  • Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994). Distributions in Statistics: Continuous Univariate Distributions -1, 2nd Edition. John Wiley & Sons, New York.

    MATH  Google Scholar 

  • Johnson, N.L. and Welch, B.L. (1940). Applications of the non-central t-distribution. Biometrika, 31, 362–381.

    Article  MathSciNet  MATH  Google Scholar 

  • Koopmans, L.H., Owen, D.B. and Rosenblatt, J.I. (1964). Confidence intervals for the coefficient of variation for the normal and log normal distributions. Biometrika, 51, 25–32.

    Article  MathSciNet  MATH  Google Scholar 

  • Kumagai, S., Matsunaga, I., Kusaka, Y. and Takagi, K. (1996). Fitness of occupational exposure data to inverse Gaussian distribution. Environmental Modeling and Assessment, 1, 277–280.

    Article  Google Scholar 

  • Laubscher, N.F. (1960). Normalizing the noncentral t and F- distributions. Annals of Mathematical Statistics, 31, 1105–1112.

    Article  MathSciNet  MATH  Google Scholar 

  • Mann, H.B. and Wald, A. (1943). On stochastic limit and order relationships. The Annals of Mathematical Statistics, 14, 217–226.

    Article  MathSciNet  MATH  Google Scholar 

  • Mudholkar, G.S. and Natarajan, R. (2002). The inverse Gaussian models: analogues of symmetry, skewness and kurtosis. Annals of the Institute of Statistical Mathematics, 54, 138–154.

    Article  MathSciNet  MATH  Google Scholar 

  • Mudholkar, G.S. and Trivedi, M.C. (1981). A Gaussian approximation to the distribution of the sample variance for nonnormal population. Journal of the American Statistical Association, 76, 479–485.

    Article  MathSciNet  Google Scholar 

  • Rao, C.R. (1973). Linear Statistical Inference and Its applications. John Wiley, New York.

    Book  MATH  Google Scholar 

  • Seshadri, V. (1993). The Inverse Gaussian Distribution: A Case Study in Exponential Families. Clarenden Press, Oxford.

    Google Scholar 

  • Seshadri, V. (1998). The Inverse Gaussian Distribution: Statistical Theory and Applications. Springer Verlag, New York.

    MATH  Google Scholar 

  • Singh, M. (1993). Behavior of sample coefficient of variation drawn from several distributions. Sankhya, 55, 65–76.

    MATH  Google Scholar 

  • Takagi, K., Kumagai, S., Matsunaga, I. and Kusaka, Y. (1997). Application of inverse gaussian distribution to occupational exposure data. Ann. Occup. Hyg., 41, 505–514.

    Article  Google Scholar 

  • Taylor, J.M.G. (1985). Power transformations to symmetry. Biometrika 72, 145–152.

    Article  MathSciNet  MATH  Google Scholar 

  • Tweedie, M.C.K. (1957a). Statistical properties of inverse Gaussian distributions-I. The Annals of Mathematical Statistics, 28, 362–377.

    Article  MathSciNet  MATH  Google Scholar 

  • Tweedie, M.C.K. (1957b). Statistical properties of inverse Gaussian distributions-II. The Annals of Mathematical Statistics, 28, 696–705.

    Article  MathSciNet  MATH  Google Scholar 

  • Whitmore, G.A. and Yalovsky, M. (1978). A normalizing logarithmic transformation for inverse Ganssian random variables. Technometrics, 20, 207–208.

    Article  MATH  Google Scholar 

  • Wilson, E. B and Hilferty, M. M. (1931). The distribution of chi-square. Proc. Nat. Acad. Sc. 17, 684–688.

    Article  MATH  Google Scholar 

  • Yeo, I. and Johnson, R.A. (2000). A new family of power transformations to improve normality or symmetry. Biometirka, 87, 954–959.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Concordia University, Montreal, QC, H3G 1M8, Canada

    Yogendra P. Chaubey & Debaraj Sen

  2. International Center for Agricultural Research in the Dry Areas, Amman, Jordan

    Murari Singh

Authors
  1. Yogendra P. Chaubey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Murari Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Debaraj Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Murari Singh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chaubey, Y.P., Singh, M. & Sen, D. Symmetrizing and Variance Stabilizing Transformations of Sample Coefficient of Variation from Inverse Gaussian Distribution. Sankhya B 79, 217–246 (2017). https://doi.org/10.1007/s13571-017-0136-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13571-017-0136-z

Keywords and phrases.

AMS (2000) subject classification.

Search

Navigation