Abstract
In this paper we consider a three-stage procedure that was presented by Hall (Ann Stat 9(6):1229–1238, 1981) to yield a fixed-width confidence interval for the mean with a precise confidence level using Edgeworth second-order expansion assuming the underlying continuous distribution has finite but unknown six moments. The procedure is based on expanding an asymptotic second order approximation of a differentiable and bounded function of the final stage stopping rule found in Yousef et al. (J Stat Plan Inference 143(9):1606–1618, 2013) by Edgeworth expansion. The performance of the asymptotic coverage was shown to be controlled by the performance of the Edgeworth approximation for the standardized underlying density and thus sensitive to the skewness and kurtosis of the underlying standardized distribution. The impact of several parameters on the asymptotic coverage is explored under continuous classes of distributions; normal, student’s t-distribution, uniform, beta and chi-squared. For brevity, simulation results are given for three types of underlying distributions: standard uniform, standard normal and standard exponential.
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References
Stein, C.: A two sample test for a linear hypothesis whose power is independent of the variance. Ann. Math. Stat. 16, 243–258 (1945)
Cox, D.R.: Estimation by double sampling. Biometrika, 39, 217–227 (1952)
Anscombe, F.: Sequential estimation. J. Roy. Stat. Soc. B. Met., 15, 1–21 (1953)
Chow, Y.S., Robbins, H.: On the asymptotic theory of fixed width sequential confidence intervals for the mean. Ann. Math. Stat. 36, 1203–1212 (1965)
Robbins, H.: Sequential estimation of the mean of a normal population. In: Probability and Statistics (Harold Cramer Volume), pp. 235–245. Almquist and Wiksell, Stockholm (1959)
Hall, P.: Asymptotic theory of triple sampling for sequential estimation of a mean. Ann. Stat. 9(6), 1229–1238 (1981)
Vik, G., Mukhophadhyay, N.: Triple sampling to construct fixed-width confidence intervals for estimable parameters based on U-statistics. Metron 46(1–4), 165–174 (1988)
Mukhophadhyay, N., deSilva, B.M.: Multistage fixed-width confidence intervals in the two-sample problem: the normal case. J. Stat. Res. 31(1), 1–20 (1997)
Hamdy, H.I.: Remarks on the asymptotic theory of triple stage estimation of the normal mean. Scand. J. Stat. 15, 303–310 (1988)
Yousef, A.S., Kimber, A.C., Hamdy, H.I.: Sensitivity of normal-based triple sampling sequential point estimation to the normality assumption. J. Stat. Plan. Inference 143(9), 1606–1618 (2013)
Dantzig, G.B.: On the non-existence of tests of ‘students’ hypothesis having power functions independent of σ. Ann. Math. Stat. 11, 186–192 (1940)
Anscombe, F.J.: Large-sample theory of sequential estimation. Math. Proc. Camb. Philos. Soc. 45, 600–607 (1952)
Chow, Y.S., Yu, K.F.: The performance of a sequential procedure for the estimation of the mean. Ann. Stat. 9, 184–188 (1981)
DasGupta, A.: Asymptotic Theory of Statistics and Probability. Springer, New York (2008)
Draper N.R., Tierney, D.E.: Regions of positive and unimodal series expansion of the Edgeworth and Gram-Charlier approximations. Biometrika 59, 463–465 (1972)
Bhattacharya, R.N., Ghosh, J.K.: On the validity of the formal Edgeworth expansion. Ann. Stat. 6, 434–451 (1978)
Barndorff-Nielsen, O.E., Cox, D.R.: Asymptotic Techniques for Use in Statistics. Chapman and Hall, London (1989)
Hall, P.: The Bootstrap and Edgeworth Expansion. Springer, New York (1992)
Mukhopadhyay, N., deSilva, B.M.: Sequential Methods and Their Application. CRC Press, Boca Raton (2009)
Simons, G.: On the cost of not knowing the variance when making a fixed-width interval estimation of the mean. Ann. Math. Stat. 39, 1946–1952 (1968)
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Yousef, A. (2014). Constructing a Three-Stage Asymptotic Coverage Probability for the Mean Using Edgeworth Second-Order Approximation. In: Kilicman, A., Leong, W., Eshkuvatov, Z. (eds) International Conference on Mathematical Sciences and Statistics 2013. Springer, Singapore. https://doi.org/10.1007/978-981-4585-33-0_7
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DOI: https://doi.org/10.1007/978-981-4585-33-0_7
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