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Penalized Fieller's confidence interval for the ratio of bivariate normal means

Author

Listed:
  • Peng Wang
  • Siqi Xu
  • Yi‐Xin Wang
  • Baolin Wu
  • Wing Kam Fung
  • Guimin Gao
  • Zhijiang Liang
  • Nianjun Liu

Abstract

Constructing a confidence interval for the ratio of bivariate normal means is a classical problem in statistics. Several methods have been proposed in the literature. The Fieller method is known as an exact method, but can produce an unbounded confidence interval if the denominator of the ratio is not significantly deviated from 0; while the delta and some numeric methods are all bounded, they are only first‐order correct. Motivated by a real‐world problem, we propose the penalized Fieller method, which employs the same principle as the Fieller method, but adopts a penalized likelihood approach to estimate the denominator. The proposed method has a simple closed form, and can always produce a bounded confidence interval by selecting a suitable penalty parameter. Moreover, the new method is shown to be second‐order correct under the bivariate normality assumption, that is, its coverage probability will converge to the nominal level faster than other bounded methods. Simulation results show that our proposed method generally outperforms the existing methods in terms of controlling the coverage probability and the confidence width and is particularly useful when the denominator does not have adequate power to reject being 0. Finally, we apply the proposed approach to the interval estimation of the median response dose in pharmacology studies to show its practical usefulness.

Suggested Citation

  • Peng Wang & Siqi Xu & Yi‐Xin Wang & Baolin Wu & Wing Kam Fung & Guimin Gao & Zhijiang Liang & Nianjun Liu, 2021. "Penalized Fieller's confidence interval for the ratio of bivariate normal means," Biometrics, The International Biometric Society, vol. 77(4), pages 1355-1368, December.
  • Handle: RePEc:bla:biomet:v:77:y:2021:i:4:p:1355-1368
    DOI: 10.1111/biom.13363
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    References listed on IDEAS

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    2. Paige, Robert L. & Chapman, Phillip L. & Butler, Ronald W., 2011. "Small Sample LD50 Confidence Intervals Using Saddlepoint Approximations," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 334-344.
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    5. Michael Sherman & Arnab Maity & Suojin Wang, 2011. "Inferences for the ratio: Fieller’s interval, log ratio, and large sample based confidence intervals," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 95(3), pages 313-323, September.
    6. Yanqing Wang & Suojin Wang & Raymond J. Carroll, 2015. "The direct integral method for confidence intervals for the ratio of two location parameters," Biometrics, The International Biometric Society, vol. 71(3), pages 704-713, September.
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