Inference for the Normal Mean with Known Coefficient of Variation

Yuejiao Fu, Hangjing Wang, Augustine Wong

Department of Mathematics and Statistics, York University, Toronto, Canada.

DOI: 10.4236/ojs.2013.36A005                

Abstract

Inference for the mean of a normal distribution with known coefficient of variation is of special theoretical interest be- cause the model belongs to the curved exponential family with a scalar parameter of interest and a two-dimensional minimal sufficient statistic. Therefore, standard inferential methods cannot be directly applied to this problem. It is also of practical interest because this problem arises naturally in many environmental and agriculture studies. In this paper we proposed a modified signed log likelihood ratio method to obtain inference for the normal mean with known coeffi- cient of variation. Simulation studies show the remarkable accuracy of the proposed method even for sample size as small as 2. Moreover, a new point estimator for the mean can be derived from the proposed method. Simulation studies show that new point estimator is more efficient than most of the existing estimators.

Keywords

Canonical Parameter; Coverage Probability; Curved Exponential Family; Modified Signed Log Likelihood Ratio Statistic

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Niwitpong, “Confidence Intervals for the Normal Mean with Known Coefficient of Variation,” World Academy of Science, Engineering and Technology, Vol. 69, 2012, pp. 677-680.
[2]	K. Bhat and K. A. Rao, “On Tests for a Normal Mean with Known Coefficient of Variation,” International Statistical Review, Vol. 75, No. 2, 2007, pp. 170-182. http://dx.doi.org/10.1111/j.1751-5823.2007.00019.x
[3]	V. Brazauskas and J. Ghorai, “Estimating the Common Parameter of Normal Models with Known Coefficients of Variation: A Sensitivity Study of Asymptotically Efficient Estimators,” Journal of Statistical Computation and Simulation, Vol. 77, No. 8, 2007, pp. 663-681. http://dx.doi.org/10.1080/10629360600578221
[4]	B. Efron, “Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency),” Annals of Statistics, Vol. 3, No. 6, 1975, pp. 1189-1242. http://dx.doi.org/10.1214/aos/1176343282
[5]	D. T. Searls, “The Utilization of a Known Coefficient of Variation in the Estimation Procedure,” Journal of the American Statistical Association, Vol. 59, No. 308, 1964, pp. 1225-1226. http://dx.doi.org/10.1080/01621459.1964.10480765
[6]	R. A. Khan, “A Note on Estimating the Mean of a Normal Distribution with Known Coefficient of Variation,” Journal of the American Statistical Association, Vol. 63, No. 323, 1968, pp. 1039-1041. http://dx.doi.org/10.2307/2283896
[7]	L. J. Gleser and J. D. Healy, “Estimating the Mean of Normal Distribution with Known Coefficient of Variation,” Journal of the American Statistical Association, Vol. 71, No. 356, 1976, pp. 977-981. http://dx.doi.org/10.1080/01621459.1976.10480980
[8]	A. R. Sen, “Relative Efficiency of Estimators of the Mean of a Normal Distribution When Coefficient of Variation Is Known,” Biometrical Journal, Vol. 21, No. 2, 1979, pp. 131-137. http://dx.doi.org/10.1002/bimj.4710210206
[9]	H. Guo and N. Pal, “On a Normal Mean with Known Coefficient of Variation,” Calcutta Statistical Association Bulletin, Vol. 54, 2003, pp. 17-29.
[10]	A. Chaturvedi and S. K. Tomer, “Three-Stage and ‘Accelerated’ Sequential Procedures for the Mean of a Normal Population with Known Coefficient of Variation,” Statistics, Vol. 37, No. 1, 2003, pp. 51-64. http://dx.doi.org/10.1080/0233188031000065433
[11]	R. Singh, “Sequential Estimation of the Mean of Normal Population with Known Coefficient of Variation,” Metron, Vol. 56, 1998, pp. 73-90.
[12]	M. Z. Anis, “Estimating the Mean of Normal Distribution with Known Coefficient of Variation,” American Journal of Mathematical and Management Sciences, Vol. 28, No. 3-4, 2008, pp. 469-487. http://dx.doi.org/10.1080/01966324.2008.10737739
[13]	W. Srisodaphol and N. Tongmol, “Improved Estimators of the Mean of a Normal Distribution with a Known Coefficient of Variation,” Journal of Probability and Statistics, Vol. 2012, 2012, Article ID: 807045. http://dx.doi.org/10.1155/2012/807045
[14]	D. V. Hinkley, “Conditional Inference about a Normal Mean with Known Coefficient of Variation,” Biometrika, Vol. 64, No. 1, 1977, pp. 105-108. http://dx.doi.org/10.1093/biomet/64.1.105
[15]	D. A. S. Fraser, N. Reid and J. Wu, “A Simple General Formula for Tail Probabilities for Frequentist and Bayesian Inference,” Biometrika, Vol. 86, No. 2, 1991, pp. 249264. http://dx.doi.org/10.1093/biomet/86.2.249
[16]	O. E. Barndorff-Nielsen, “Inference on Full or Partial Parameters, Based on the Standardized Signed Log-Likelihood Ratio,” Biometrika, Vol. 73, No. 2, 1986, pp. 307322.
[17]	O. E. Barndorff-Nielsen, “Modified Signed Log-Likelihood Ratio,” Biometrika, Vol. 78, No. 3, 1991, pp. 557563. http://dx.doi.org/10.1093/biomet/78.3.557
[18]	D. A. S Fraser, N. Reid and A. Wong, “Simple and Accurate Inference for the Mean Parameter of the Gamma Model,” Canadian Journal of Statistics, Vol. 25, No. 1, 1997, pp. 91-99. http://dx.doi.org/10.2307/3315359
[19]	Fraser, D.A.S. and Reid, N., “Ancillaries and Third Order Significance,” Utilitas Mathematica, Vol. 7, 1995, pp. 3355.