A general theory of purely sequential minimum risk point estimation (MRPE) of a function of the mean in a normal distribution

References

• Anscombe, F. J. (1952). Large-Sample Theory of Sequential Estimation, Proceedings of Cambridge Philosophical Society 48: 600–607. doi:10.1017/S0305004100076386
   Web of Science ®Google Scholar
• Banerjee, S. and Mukhopadhyay, N. (2016). A General Sequential Fixed-Accuracy Confidence Interval Estimation Methodology for a Positive Parameter: Illustrations Using Health and Safety Data, Annals of Institute of Statistical Mathematics 68: 541–570. doi:10.1007/s10463-015-0504-2
   Web of Science ®Google Scholar
• Basu, D. (1955). On Statistics Independent of a Complete Sufficient Statistic, Sankhyā 15: 377–380.
   Google Scholar
• Chaturvedi, A., Bapat, S. R., and Joshi, N. (2019). Multi-Stage Point Estimation of the Mean of an Inverse Gaussian Distribution, Sequential Analysis 38: 1–25. doi:10.1080/07474946.2019.1574438
   Web of Science ®Google Scholar
• Chow, Y. S., Hsiung, C., and Lai, T. L. (1979). Extended Renewal Theory and Moment Convergence in Anscombe’s Theorem, Annals of Probability 7: 304–318. doi:10.1214/aop/1176995090
   Web of Science ®Google Scholar
• Chow, Y. S. and Robbins, H. (1965). On the Asymptotic Theory of Fixed-Width Sequential Confidence Intervals for the Mean, Annals of Mathematical Statistics 36: 457–462. doi:10.1214/aoms/1177700156
   Google Scholar
• 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. doi:10.1214/aoms/1177731912
   Google Scholar
• Ghosh, B. K. and Sen, P. K. (1991). Handbook of Sequential Analysis, New York: Dekker.
   Google Scholar
• Ghosh, M. and Mukhopadhyay, N. (1975). Asymptotic Normality of Stopping Times in Sequential Analysis, unpublished. doi:10.1080/07474946.2018.1548847
   Google Scholar
• Ghosh, M. and Mukhopadhyay, N. (1976). On Two Fundamental Problems in Sequential Estimation, Sankhyā, Series B 38: 203–218.
   Google Scholar
• Ghosh, M. and Mukhopadhyay, N. (1981). Consistency and Asymptotic Efficiency of Two-Stage and Sequential Procedures, Sankhyā, Series A 43: 220–227.
   Google Scholar
• Ghosh, M., Mukhopadhyay, N., and Sen, P. K. (1997). Sequential Estimation, New York: Wiley.
   Google Scholar
• Ghosh, M., Sinha, B. K., and Mukhopadhyay, N. (1976). Multivariate Sequential Point Estimation, Journal of Multivariate Analysis 6: 281–294. doi:10.1016/0047-259X(76)90037-3
   Web of Science ®Google Scholar
• Gut, A. (2009). Stopped Random Walks: Limit Theorems and Applications, second edition, New York: Springer.
   Google Scholar
• Gut, A. (2012). Anscombe’s Theorem 60 Years Later, Sequential Analysis 31: 368–396.
   Web of Science ®Google Scholar
• Hu, J. and Mukhopadhyay, N. (2019). Second-Order Asymptotics in a Class of Purely Sequential Minimum Risk Point Estimation (MRPE) Methodologies, Japanese Journal of Statistics and Data Science 2: 81–104. doi:10.1007/s42081-018-0028-0
   Google Scholar
• Lai, T. L. and Siegmund, D. (1977). A Nonlinear Renewal Theory with Applications to Sequential Analysis I, Annals of Statistics 5: 946–954. doi:10.1214/aos/1176343950
   Web of Science ®Google Scholar
• Lai, T. L. and Siegmund, D. (1979). A Nonlinear Renewal Theory with Applications to Sequential Analysis II, Annals of Statistics 7: 60–76. doi:10.1214/aos/1176344555
   Web of Science ®Google Scholar
• Lalehzari, L., Mahmoudi, E., and Khalifeh, A. (2018). Sequential Fixed-Width Confidence Interval for the rth Power of the Exponential Scale Parameter: Two-Stage and Sequential Sampling Procedures, Sequential Analysis 37: 293–310. doi:10.1080/07474946.2018.1548841
   Web of Science ®Google Scholar
• Mukhopadhyay, N. (1988). Sequential Estimation Problems for Negative Exponential Populations, Communications in Statistics-Theory & Methods, Series A 17: 2471–2506. doi:10.1080/03610928808829758
   Web of Science ®Google Scholar
• Mukhopadhyay, N. (2000). Probability and Statistical Inference, New York: Dekker.
   Google Scholar
• Mukhopadhyay, N. and Chattopadhyay, B. (2012). A Tribute to Frank Anscombe and Random Central Limit Theorem from 1952, Sequential Analysis 31: 265–277.
   Web of Science ®Google Scholar
• Mukhopadhyay, N., Chaturvedi, A., and Malhotra, A. (2018). Two-Stage Procedures for the Bounded Risk Point Estimation of the Parameter and Hazard Rate in Two Families of Distributions, Sequential Analysis 37: 69–89. doi:10.1080/07474946.2018.1427978
   Web of Science ®Google Scholar
• Mukhopadhyay, N., Datta, S., and Chattopadhyay, S. (2004). Applied Sequential Methdologies, New York: Dekker.
   Google Scholar
• Mukhopadhyay, N. and de Silva, B. M. (2009). Sequential Methods and Their Applications, Boca Raton, FL: CRC.
   Google Scholar
• Mukhopadhyay, N. and Hu, J. (2017). Confidence Intervals and Point Estimators for a Normal Mean Under Purely Sequential Strategies Involving Gini’s Mean Difference and Mean Absolute Deviation, Sequential Analysis 36: 210–239. doi:10.1080/07474946.2017.1319684
   Web of Science ®Google Scholar
• Mukhopadhyay, N. and Solanky, T. K. S. (1994). Multistage Selection and Ranking Procedures, New York: Dekker.
   Google Scholar
• Mukhopadhyay, N. and Zhuang, Y. (2017). Purely Sequential and Two-Stage Bounded-Length Confidence Interval Estimation Problems in Fisher’s “Nile” Example, Journal of Japan Statistical Society 47: 237–272. doi:10.14490/jjss.47.237
   Google Scholar
• Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd edition, New York: Wiley.
   Google Scholar
• Ray, W. D. (1957). Sequential Confidence Intervals for the Mean of a Normal Population with Unknown Variance, Journal of Royal Statistical Society, Series B 10: 133–143. doi:10.1111/j.2517-6161.1957.tb00248.x
   Google Scholar
• Robbins, H. (1959). Sequential Estimation of the Mean of a Normal Population, in Probability and Statistics, Harald Cramér Volume, U. Grenander, ed., pp. 235–245, Uppsala, Sweden: Almquist and Wiksell.
   Google Scholar
• Sen, P. K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference, New York: Wiley.
   Google Scholar
• Siegmund, D. (1985). Sequential Analysis: Tests and Confidence Intervals, New York: Springer.
   Google Scholar
• Starr, N. (1966). On the Asymptotic Efficiency of a Sequential Procedure for Estimating the Mean, Annals of Mathematical Statistics 37: 1173–1185. doi:10.1214/aoms/1177699263
   Google Scholar
• Starr, N. and Woodroofe, M. (1969). Remarks on Sequential Point Estimation, Proceedings of National Academy of Science, USA 61: 1215–1218. doi:10.1073/pnas.61.4.1215
   Google Scholar
• Wiener, N. (1939). The Ergodic Theorem, Duke Mathematical Journal 5: 1–18. doi:10.1215/S0012-7094-39-00501-6
   Google Scholar
• Woodroofe, M. (1977). Second Order Approximations for Sequential Point and Interval Estimation, Annals of Statistics 5: 984–995. doi:10.1214/aos/1176343953
   Web of Science ®Google Scholar
• Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis, CBMS Lecture Notes #39, Philadelphia: SIAM.
   Google Scholar
• Zacks, S. (2009). Stage-Wise Adaptive Designs, New York: Wiley.
   Google Scholar
• Zacks, S. (2017). Sample Path Analysis and Distributions of Boundary Crossing Times, Lecture Notes in Mathematics, New York: Springer.
   Google Scholar