Confidence Intervals for The Ratio of Two Means Using the Distribution of the Quotient of Two Normals

• Authors: Galeone Carlotta , Pollastri Angiola
• Languages of publication: EN

Abstracts

EN

In various scientific fields such as medicine, biology and bioassay, several ratio quantities assumed to be Normal, are of potential interest. The estimator of the ratio of two means is a ratio of two random variables normally or asymptotically normally distributed. The present paper shows the importance of considering the real distribution of the estimator of the ratio of two means, because generally the approximation to Normal is not satisfied. The estimated asymptotic cumulative and density function of the estimator of the ratio is presented, with several considerations on the skewness. Finally, a new method for building confidence intervals for the ratio of two means was proposed. In contrast to other parametric methods, this new method is worthy to be preferred because it considers the skewness in the distribution of the ratio estimator, and the confidence intervals are always bounded.

Keywords

EN

estimator of the ratio of two means distribution of the ratio of two correlated normals; skewness; confidence intervals for the ratio Fieller’s theorem

• Publisher: Główny Urząd Statystyczny
• Journal: Statistics in Transition new series
• Year: 2012
• Volume: 13
• Issue: 3
• Pages: 451-472

Contributors

author

Galeone Carlotta

• Università degli Studi di Milano

author

Pollastri Angiola

• Università degli Studi di Milano-Bicocca

References

• AROIAN L. A. (1986). The distribution of the quotient of two correlated random variables. Proceedings of the Am. Stat. Ass. Business and Economic Section.
• BRENTARI E. (1990). Asimmetria e misure di Asimmetria, Giappichelli Ed., Torino.
• COCHRAN W. C. (1997). Sampling Techniques, John Wiley and Sons, New York, Wiley; 3rd ed.
• EVERITT B. S. (1996). An Introduction to finite mixture distributions, Stat Methods Med Rev, 5(2): 107-127.
• FIELLER E. C. (1932). The distribution of the index in a Normal Bivariate population. Biometrika 24(3/4): 428-440.
• FIELLER E. C. (1954). Some Problems in Interval Estimation. Journal of the Royal Statistical Society. Series B (Methodological) 16(2): 175-185.
• FINNEY D. J. (1964). Statistical Method in Biological Assay, Hafner, New York.
• FROSINI B. V. (1970). La stima di un quoziente nei grandi campioni. Giornale degli economisti e annali di economia : 381-400.
• FROSINI B. V. (1971). Le distribuzioni oblique, Statistica, 31: 83-117 (in FROSINI B. V. (2011). Selected Writings, Vita e Pensiero, Milano).
• GALEONE C. (2007). On the ratio of two Normal r.v., jointly distributed as a Bivariate Normal, PhD Thesis, Università di Milano-Bicocca.
• GALEONE C., POLLASTRI A. (2008). Estimation of the quantiles of the ratio of two correlated Normals, XLIV Riunione scientifica della Società Italiana di Statistica.
• GARDINER J. C., HUEBNER M. et al. (2001). On Parametric Confidence Intervals for Cost-Effectiveness Ratio, Biometrical Journal, 43 (3): 283-296.
• GEARY R. C. (1930). The frequency distribution of the quotient of two Normal variates. J Royal Stat Society, 93(3): 442-446.
• GROENEVELD R.A. (1998). Bowley’s measures of skewness, Encyclopedia of statistical Science, Update Vol. 2, 619-621, Wiley.
• JOHNSON N. L., KOTZ S., BALAKRISHNAN N. (1994). Continuous Univariate Distributions, Wiley, New York.
• KENDALL M. G., STUART A. (1961). The advanced Theory of Statistics, Vol. 2, C. Griffin & Co., London.
• KOTZ S., BALAKRISHNAN N., JOHNSON N. L. (2000). Continuous Multivariate Distributions, Wiley, New York.
• MacGILLIVRAY H. L.(1985). Mean, Median, Mode, Skewness, Encyclopedia of Statistical Science, Vol. 5, 364-367, Wiley.
• MacGILLIVRAY H. L. (1986). Skewness and asymmetry: Measures and Orderings, The Annals Of Mathematical Statistics, Vol. 14, No. 3 :994-1011.
• MARSAGLIA, G. (1965). Ratios of Normal variables and ratio of sums of Uniform variables. J Am Statist Ass 60(309): 193-204.
• MARSAGLIA, G. (2006). Ratios of Normal variables. Journal of Statistical Software, 16(4).
• MOOD A. M, GRAYBILL F. A., BOES D. C. (1974). Introduction to the Theory of Statistics, McGraw-Hill, New York.
• NATIONAL BUREAU STANDARDS (1959). Tables of the Bivariate Normal distribution function and related functions, U.S. Government Printing Office, Washington.
• OKSOY D., AROIAN L. A. (1986). Computational Techniques and examples of the density and the distribution of the quotient of two correlated normal variables, Proceeding of the American Statistical Association Business an Economic Section.
• OKSOY D., AROIAN L. A. (1994). The quotient of two correlated normal variables with applications, Comm Stat Simula, 23 (1): 223-241.
• RAO C. R. (1965). Linear Statistical Inference and its applications, Wiley, New York.