Second-order asymptotics in a class of purely sequential minimum risk point estimation (MRPE) methodologies

Abstract

Under the squared error loss plus linear cost of sampling, we revisit the minimum risk point estimation (MRPE) problem for an unknown normal mean \(\mu\) when the variance \(\sigma ^{2}\) also remains unknown. We begin by defining a new class of purely sequential MRPE methodologies based on a general estimator \(W_{n}\) for \(\sigma\) satisfying a set of conditions in proposing the requisite stopping boundary. Under such appropriate set of sufficient conditions on \(W_{n}\) and a properly constructed associated stopping variable, we show that (i) the normalized stopping time converges in law to a normal distribution (Theorem 3.3), and (ii) the square of such a normalized stopping time is uniformly integrable (Theorem 3.4). These results subsequently lead to an asymptotic second-order expansion of the associated regret function in general (Theorem 4.1). After such general considerations, we include a number of substantial illustrations where we respectively substitute appropriate multiples of Gini’s mean difference and the mean absolute deviation in the place of the general estimator \(W_{n}\). These illustrations show a number of desirable asymptotic first-order and second-order properties under the resulting purely sequential MRPE strategies. We end this discourse by highlighting selected summaries obtained via simulations.

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References

  1. Abramowitz, M., & Stegun, I. A. (1972). Handbook of mathematical functions, ninth printing. New York: Dover.

    Google Scholar 

  2. Anscombe, F. J. (1950). Sampling theory of the negative binomial and logarithmic series distributions. Biometrika, 37, 358–382.

    MathSciNet  MATH  Article  Google Scholar 

  3. Anscombe, F. J. (1952). Large-sample theory of sequential estimation. Proceedings of Cambridge Philosophical Society, 48, 600–607.

    MathSciNet  MATH  Article  Google Scholar 

  4. Anscombe, F. J. (1953). Sequential estimation. Journal of Royal Statistical Society, Series B, 15, 1–29.

    MathSciNet  MATH  Google Scholar 

  5. Aoshima, M., & Mukhopadhyay, N. (2002). Two-stage estimation of a linear function of normal means with second-order approximations. Sequential Analysis, 21, 109–144.

    MathSciNet  MATH  Article  Google Scholar 

  6. Babu, G. J., & Rao, C. R. (1992). Expansions for statistics involving the mean absolute deviations. Annals of Institute of Statistical Mathematics, 44, 387–403.

    MathSciNet  MATH  Article  Google Scholar 

  7. Basu, D. (1955). On statistics independent of a complete sufficient statistic. Sankhyā, 15, 377–380.

    MathSciNet  MATH  Google Scholar 

  8. Carroll, R. J. (1977). On the asymptotic normality of stopping times based on robust estimators. Sankhyā, Series A, 39, 355–377.

    MathSciNet  MATH  Google Scholar 

  9. Chattopadhyay, B., & Mukhopadhyay, N. (2013). Two-stage fixed-width confidence intervals for a normal mean in the presence of suspect outliers. Sequential Analysis, 32, 134–157.

    MathSciNet  MATH  Article  Google Scholar 

  10. Chow, Y. S., & Martinsek, A. T. (1982). Bounded regret of a sequential procedure for estimation of the mean. Annals of Statistics, 10, 909–914.

    MathSciNet  MATH  Article  Google Scholar 

  11. Chow, Y. S., & Robbins, H. (1965). On the asymptotic theory of fixed-width sequential confidence intervals for the mean. Annals of Mathematical Statistics, 36, 457–462.

    MathSciNet  MATH  Article  Google Scholar 

  12. Chow, Y. S., & Yu, K. F. (1981). The performance of a sequential procedure for the estimation of the mean. Annals of Statistics, 9, 184–188.

    MathSciNet  MATH  Article  Google Scholar 

  13. Ghosh, B. K., & Sen, P. K. (1991). Handbook of sequential analysis, edited volume. New York: Dekker.

    Google Scholar 

  14. Ghosh, M., & Mukhopadhyay, N. (1975). Asymptotic normality of stopping times in sequential analysis, unpublished manuscript, Indian Statistical Institute, Calcutta, India.

  15. Ghosh, M., & Mukhopadhyay, N. (1976). On Two fundamental problems of sequential estimation. Sankhyā, Series B, 38, 203–218.

    MathSciNet  MATH  Google Scholar 

  16. Ghosh, M., & Mukhopadhyay, N. (1979). Sequential point estimation of the mean when the distribution is unspecified. Communications in Statistics-Theory & Methods, Series A, 8, 637–652.

    MathSciNet  MATH  Article  Google Scholar 

  17. Ghosh, M., & Mukhopadhyay, N. (1980). Sequential point estimation of the difference of two normal means. Annals of Statistics, 8, 221–225.

    MathSciNet  MATH  Article  Google Scholar 

  18. Ghosh, M., & Mukhopadhyay, N. (1981). Consistency and asymptotic efficiency of two-stage and sequential procedures. Sankhyā, Series A, 43, 220–227.

    MathSciNet  MATH  Google Scholar 

  19. Ghosh, M., Mukhopadhyay, N., & Sen, P. K. (1997). Sequential estimation. New York: Wiley.

    Google Scholar 

  20. Gini, C. (1914). Sulla Misura della Concertrazione e della Variabilit dei Caratteri. Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti, 73, 1203–1248.

    Google Scholar 

  21. Gini, C. (1921). Measurement of inequality of incomes. Economic Journal, 31, 124–126.

    Article  Google Scholar 

  22. Gut, A. (2012). Anscombe’s theorem 60 years later. Sequential Analysis, 31, 368–396.

    MathSciNet  MATH  Google Scholar 

  23. Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Annals of Mathematical Statistics, 19, 293–325.

    MathSciNet  MATH  Article  Google Scholar 

  24. Hoeffding, W. (1961). The strong law of large numbers for u-statistics. Institute of Statistics Mimeo Series #302. University of North Carolina, Chapel Hill.

  25. Jurečkovā, J., & Sen, P. K. (1996). Robust Statistical procedures. New York: Wiley.

    Google Scholar 

  26. Lai, T. L., & Siegmund, D. (1977). A nonlinear renewal theory with applications to sequential analysis I. Annals of Statistics, 5, 946–954.

    MathSciNet  MATH  Article  Google Scholar 

  27. Lai, T. L., & Siegmund, D. (1979). A nonlinear renewal theory with applications to sequential analysis II. Annals of Statistics, 7, 60–76.

    MathSciNet  MATH  Article  Google Scholar 

  28. Lee, A. J. (1990). U-statistics. Theory and practice. New York: Dekker.

    Google Scholar 

  29. Mukhopadhyay, N. (1975). Sequential methods in estimation and prediction, Ph.D. dissertation, Indian Statistical Institute, Calcutta, India.

  30. Mukhopadhyay, N. (1978). Sequential point estimation of the mean when the distribution is unspecified, Statistics Technical Report Number 312. University of Minnesota, Minneapolis.

  31. Mukhopadhyay, N. (1982). Stein’s Two-Stage Procedure and Exact Consistency, Scandinavian Actuarial Journal 110–122.

  32. Mukhopadhyay, N. (1988). Sequential estimation problems for negative exponential populations. Communications in Statistics-Theory & Methods, Series A, 17, 2471–2506.

    MathSciNet  MATH  Article  Google Scholar 

  33. Mukhopadhyay, N. (2000). Probability and statistical inference. New York: Dekker.

    Google Scholar 

  34. Mukhopadhyay, N., & Chattopadhyay, B. (2012). A tribute to frank anscombe and random central limit theorem from 1952. Sequential Analysis, 31, 265–277.

    MathSciNet  MATH  Google Scholar 

  35. Mukhopadhyay, N., Datta, S., & Chattopadhyay, S. (2004). Applied sequential methodologies, edited volume. New York: Dekker.

    Google Scholar 

  36. Mukhopadhyay, N., & de Silva, B. M. (2009). Sequential methods and their applications. Boca Ratton: CRC.

    Google Scholar 

  37. Mukhopadhyay, N., & 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.

    MathSciNet  MATH  Article  Google Scholar 

  38. Mukhopadhyay, N., & Hu, J. (2018). Gini’s mean difference and mean absolute deviation based two-stage estimation for a normal mean with known lower bound of variance. Sequential Analysis, 37, 204–221.

    MathSciNet  MATH  Article  Google Scholar 

  39. Mukhopadhyay, N., & Solanky, T. K. S. (1994). Multistage selection and ranking procedures. New York: Dekker.

    Google Scholar 

  40. Ray, W. D. (1957). Sequential confidence intervals for the mean of a normal population with unknown variance. Journal of Royal Statistical Society, Series B, 19, 133–143.

    MathSciNet  MATH  Google Scholar 

  41. Robbins, H. (1959). Sequential estimation of the mean of a normal population. In Ulf Grenander (Ed.), Probability and statistics, H Cramér volume (pp. 235–245). Uppsala: Almquist & Wiksell.

    Google Scholar 

  42. Sen, P. K. (1981). Sequential nonparametrics: invariance principles and statistical inference. New York: Wiley.

    Google Scholar 

  43. Sen, P. K. (1985). Theory and applications of sequential nonparametrics, CBMS #49. Philadelphia: SIAM.

    Google Scholar 

  44. Sen, P. K., & Ghosh, M. (1981). Sequential point estimation of estimable parameters based on U-statistics. Sankhyā, Series A, 43, 331–344.

    MathSciNet  MATH  Google Scholar 

  45. Siegmund, D. (1985). Sequential analysis: Tests and confidence intervals. New York: Springer.

    Google Scholar 

  46. Starr, N. (1966). On the asymptotic efficiency of a sequential procedure for estimating the mean. Annals of Mathematical Statistics, 37, 1173–1185.

    MathSciNet  MATH  Article  Google Scholar 

  47. Starr, N., & Woodroofe, M. (1969). Remarks on sequential point estimation. Proceedings of National Academy of Sciences, 63, 285–288.

    MathSciNet  MATH  Article  Google Scholar 

  48. Wiener, N. (1939). The Ergodic theorem. Duke Mathematical Journal, 5, 1–18.

    MathSciNet  MATH  Article  Google Scholar 

  49. Woodroofe, M. (1977). Second order approximations for sequential point and interval estimation. Annals of Statistics, 5, 984–995.

    MathSciNet  MATH  Article  Google Scholar 

  50. Woodroofe, M. (1982). Nonlinear renewal theory in sequential analysis, CBMS lecture notes #39. Philadelphia: SIAM.

    Google Scholar 

  51. Zacks, S. (2009). Stage-wise adaptive designs. New York: Wiley.

    Google Scholar 

  52. Zacks, S. (2017). Sample path analysis and distributions of boundary crossing times, lecture notes in mathematics. New York: Springer.

    Google Scholar 

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Acknowledgements

The comments received from two anonymous reviewers, the Associate Editor, and the Executive Editor on our earlier version have genuinely helped us in preparing this revised manuscript. We express our gratitude to all of them and thank them.

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Correspondence to Nitis Mukhopadhyay.

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Hu, J., Mukhopadhyay, N. Second-order asymptotics in a class of purely sequential minimum risk point estimation (MRPE) methodologies. Jpn J Stat Data Sci 2, 81–104 (2019). https://doi.org/10.1007/s42081-018-0028-0

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Keywords

  • Asymptotic first-order properties
  • Asymptotic second-order properties
  • Linear cost
  • Regret
  • Risk efficiency
  • Sequential strategy
  • Simulations
  • Squared error loss

Mathematics Subject Classification

  • 62L10
  • 62L12
  • 62G05
  • 62G20