Abstract
We consider fixed-size estimation for a linear function of means from independent and normally distributed populations having unknown and respective variances. We construct a fixed-width confidence interval with required accuracy about the magnitude of the length and the confidence coefficient. We propose a two-stage estimation methodology having the asymptotic second-order consistency with the required accuracy. The key is the asymptotic second-order analysis about the risk function. We give a variety of asymptotic characteristics about the estimation methodology, such as asymptotic sample size and asymptotic Fisher-information. With the help of the asymptotic second-order analysis, we also explore a number of generalizations and extensions of the two-stage methodology to such as bounded risk point estimation, multiple comparisons among components between the populations, and power analysis in equivalence tests to plan the appropriate sample size for a study.
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References
Aoshima M. (2005) Statistical inference in two-stage sampling. American Mathematical Society Translations Series 2 215: 125–145
Aoshima M., Kushida T. (2005) Asymptotic second-order efficiency for two-stage multiple comparisons with components of a linear function of mean vectors. In: Balakrishnan N., Kannan N., Nagaraja H.N. (eds) Advances in Ranking and Selection, Multiple Comparisons, and Reliability: Methodology and Applications. Birkhäuser, Boston, pp 191–211
Aoshima M., Mukhopadhyay N. (2002) Two-stage estimation of a linear function of normal means with second-order approximations. Sequential Analysis 21: 109–144
Aoshima M., Takada Y. (2000) Second-order properties of a two-stage procedure for comparing several treatments with a control. Journal of The Japan Statistical Society 30: 27–41
Aoshima M., Takada Y. (2002) Bounded risk point estimation of a linear function of k multinormal mean vectors when covariance matrices are unknown. In: Balakrishnan N. (eds) Advances on Theoretical and Methodological Aspects of Probability and Statistics. Taylor & Francis, New York, pp 279–287
Banerjee S. (1967) Confidence interval of preassigned length for the Behrens–Fisher problem. Annals of Mathematical Statistics 38: 1175–1179
Dantzig G.B. (1940) On the non-existence of tests of “Student’s” hypothesis having power functions independent of σ. Annals of Mathematical Statistics 11: 186–192
Ghosh M., Mukhopadhyay N., Sen P.K. (1997) Sequential estimation. Wiley, New York
Hall P. (1981) Asymptotic theory of triple sampling for sequential estimation of a mean. Annals of Statistics 9: 1229–1238
Liu W. (2003) Some sequential equivalence tests that control both the size and power. South African Statistical Journal 37: 105–125
Liu W., Wang N. (2007) A note on Hall’s triple sampling procedure: A multiple sample second order sequential analogue of the Behrens-Fisher problem. Journal of Statistical Planning and Inference 137: 331–343
Mukhopadhyay N. (2005) A new approach to determine the pilot sample size in two-stage sampling. Communications in Statistics-Theory and Methods 34: 1275–1295
Mukhopadhyay N., Duggan W.T. (1997) Can a two-stage procedure enjoy second-order properties?. Sankhyā, Series A 59: 435–448
Mukhopadhyay N., Duggan W.T. (1999) On a two-stage procedure having second-order properties with applications. Annals of the Institute of Statistical Mathematics 51: 621–636
Schwabe R. (1995) Some two-stage procedures for treating the Behrens–Fisher problem. In: Müller W.G., Wynn H.P., Zhigljavsky A.A. (eds) Model-oriented data analysis. Physica, Heidelberg, pp 81–89
Stein C. (1945) A two-sample test for a linear hypothesis whose power is independent of the variance. Annals of Mathematical Statistics 16: 243–258
Takada Y. (2004) Asymptotic second-order efficiency of a two-stage procedure for estimating a linear function of normal means. Sequential Analysis 23: 103–120
Takada Y., Aoshima M. (1996) Two-stage procedures for the difference of two multinormal means with covariance matrices different only by unknown scalar multipliers. Communications in Statistics-Theory and Methods 25: 2371–2379
Takada Y., Aoshima M. (1997) Two-stage procedures for estimating a linear function of multinormal mean vectors. Sequential Analysis 16: 353–362
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Research of the first author was partially supported by Grant-in-Aid for Scientific Research (B), Japan Society for the Promotion of Science (JSPS), under Contract Number 18300092.
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Aoshima, M., Yata, K. Asymptotic second-order consistency for two-stage estimation methodologies and its applications. Ann Inst Stat Math 62, 571–600 (2010). https://doi.org/10.1007/s10463-008-0188-y
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DOI: https://doi.org/10.1007/s10463-008-0188-y