1. Approximate Confidence Interval for the Ratio of Normal Means
with a Known Coefficient of Variation
Wararit Panichkitkosolkul
Department of Mathematics and Statistics, Faculty of Science and Technology,
Thammasat University, Thailand
Introduction
Statistical analysis for the ratio of normal means is applied in the area of bioassay and bioequivalence. The ratio of normal means is defined by
where and are the population means of X and Y, respectively. The confidence interval for the ratio of normal means has also been studied by many researchers (for example, Fieller, 1944, 1954; Koschat, 1987).
Recently, Niwitpong et al. (2011) proposed two confidence intervals for the ratio of normal means with a known coefficient of variation. Their confidence intervals can be applied in some situations, for instance when the coefficient of variation of a control group is known. One of their confidence intervals is developed based on an exact method in which this confidence interval is constructed from the pivotal statistics Z, where Z follows the standard normal distribution. The other confidence interval is constructed based on the generalized confidence interval (Weerahandi, 1993). Simulation results show that the coverage probabilities of the two confidence intervals are not significantly different. However, the confidence interval based on the exact method is shorter than the generalized confidence interval. The exact method uses Taylor series expansion to find the expectation and variance of the estimator of and uses these results for constructing the confidence interval for
The lower and upper limits of the confidence interval based on the exact method are difficult to compute since they depend on an infinite summation. Therefore, our main aim in this paper is to propose an approximate confidence interval for the ratio of normal means with a known coefficient of variation. The computation of the new proposed confidence interval is easier than the exact confidence interval proposed by Niwitpong et al. (2011). In addition, we also compare the estimated coverage probabilities and average lengths of the new proposed confidence interval and existing confidence interval using a Monte Carlo simulation.
where and is the percentile of the standard normal distribution.
Simulation Study
A Monte Carlo simulation was conducted using the R statistical software version to compare the estimated coverage probabilities and average lengths of the new proposed confidence interval and the exact confidence interval. The number of simulation runs was 10,000 and the nominal confidence level was fixed at 0.95.
Table 1. Estimated coverage probabilities and expected lengths of confidence intervals for the ratio of normal means with a known coefficient of variation
Conclusions and Discussions
In this paper, we proposed an approximate confidence interval for the ratio of normal population means with a known coefficient of variation. Normally, this arises when the scientist knows the coefficient of variation of the control group. The approximate confidence interval proposed uses the approximation of the expectation and variance of the estimator. The proposed new confidence interval is compared with the exact confidence interval constructed by Niwitpong et al. (2011) through a Monte Carlo simulation study. The approximate confidence interval performs as efficiently as the exact confidence interval in terms of coverage probability and expected length. Moreover, approximate confidence interval also is easy to compute compared with the exact confidence interval.
References
Fieller, E.C.: A Fundamental Formula in the Statistics of Biological Assay and Some Applications, Q. J. Pharm. Pharmacol. 17, (1944).
Fieller, E.C.: Some Problems in Interval Estimation, J. R. Stat. Soc. Ser B. 16, (1954).
Koschat, M.A.: A Characterization of the Fieller Solution, Ann. Statist. 15, (1987).
Niwitpong, S., Koonprasert, S., Niwitpong, S.: Confidence Intervals for the Ratio of Normal Means with a Known Coefficient of Variation, Adv. Appl. Stat. 25, (2011).
Weerahandi, S.: Generalized Confidence Intervals J.Amer. Statist. Assoc. 88, (1993).
Author
Assistant Professor Dr.Wararit Panichkitkosolkul
Department of Mathematics and Statistics, Thammasat University
Education:
Ph.D. (Applied Statistics),
King Mongkut’s University of Technology North Bangkok
M.Sc. (Statistics), Chulalongkorn University
B.Sc. (Applied Statistics),
B.B.A. (Marketing), B.Tech. (Business Information Technology),
B.Econ. (Business Economics) and B.Ed. (Educational Measurement and Evaluation) from Sukhothai Thammathirat Open University
Existing Confidence Interval
Niwitpong et al. (2011) proposed the exact confidence interval for based on an exact method and the central limit theorem. Therefore, the exact confidence interval for is
where and is the
percentile of the standard normal distribution.
Proposed Confidence Interval
Theorem 1. Let be a random sample of size n from a normal distribution with mean and variance and be a random sample of size m from a normal distribution with mean and variance The estimator of is where and The approximate expectation and variance of when a coefficient of variation, is known, are respectively
and
It is clear that is asymptotically unbiased. Therefore, the unbiased estimator of is where
We then apply the central limit theorem and Theorem 1,
Therefore, it is easily seen that the approximate confidence interval for is