Skip to main content

Some Auxiliary Results — Monotonicity Properties of Probability Distributions

  • Chapter
Multiple Statistical Decision Theory: Recent Developments

Part of the book series: Lecture Notes in Statistics ((LNS,volume 6))

  • 165 Accesses

Abstract

It is very important to study the monotonicity properties of distributions in order to obtain inequalities useful in statistical inference. Some monotonicity properties of distributions are well known and have proved to be very useful. During the last decade, more concepts have been introduced and used by several authors in multiple decision problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alam, k. (1973). On a multiple decision rule. Ann. Statist. 1, 750–755.

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, T. W. (1955). The integral of a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6, 170–176.

    Article  MathSciNet  MATH  Google Scholar 

  3. Barndorff-Nielson, O. (1978). Information and Exponential Families in Statistical Theory. John Wiley & Sons, New York.

    Google Scholar 

  4. Bickel, P. J. (1965). Some contributions to the theory of order statistics. Proc. Berkeley Symp. Math. Statist. Prob., 1, 575–591. Univ. of California Press, Berkeley, California.

    Google Scholar 

  5. Blomqvist, N. (1950). On a measure of dependence between any two random variables. Ann. Math. Statist. 21, 593–600.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonnesen, T. and Fenchel, W. Theorie der Konvexen Kaper, Chelsea Publishing Company, New York, 1948.

    Google Scholar 

  7. Das, S. C. The numerical evaluation of a class of integrals II. Proc. Cambridge Philos. Soc. 52, 442–448, 1956.

    Article  MathSciNet  MATH  Google Scholar 

  8. Das Gupta, S., Eaton, M. L., Olkin, I., Perlman, M., Savage, I. R. and Sobel, M. Inequalities on the probability content of convex regions for elliptically contoured distributions. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2, 241–265. Univ. of California Press, Berkeley, California, 1970.

    Google Scholar 

  9. Doornbos, R. and Prins, H. J. On slippage tests. Indag. Math. 20, 38–55, 438–447, 1958.

    MathSciNet  Google Scholar 

  10. Dunnett, C. W. and Sobel, M. (1955). Approximation to the probability integral and certain percentage points of a multivariate analogue of Student’s t-distribution. Biometrika 42, 258–260.

    MathSciNet  MATH  Google Scholar 

  11. Eaton, M. L. (1967). Some optimum properties of ranking procedures. Ann. Math. Statist. 38, 124–137.

    Article  MathSciNet  MATH  Google Scholar 

  12. Efron, B. (1965). Increasing properties of Polya frequency functions. Ann. Math. Statist. 36, 272–279.

    Article  MathSciNet  MATH  Google Scholar 

  13. Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.

    Article  MathSciNet  MATH  Google Scholar 

  14. Gupta, S. S. (1956). On a decision rule for a problem in ranking means. Inst. of Statist., Univ. of North Carolina, Mimeo. Ser. No. 150. (Tables AI and AII).

    MATH  Google Scholar 

  15. Gupta, S. S. (1963). Probability integrals of multivariate normal and multivariate t. Ann. Math. Statist. 34, 792–828.

    Article  MathSciNet  MATH  Google Scholar 

  16. Gupta, S. S. and Huang, D. Y. (1980). On an essentially complete class of multiple decision procedures. To appear in JSPI.

    Google Scholar 

  17. Gupta, S. S., Nagel, K. and Panchapakesan, S. (1973). On the order statistics from equally correlated normal random variables. Biometrika 60, 403–413.

    Article  MathSciNet  MATH  Google Scholar 

  18. Gupta, S. S. and Panchapakesan, S. (1972). On a class of subset selection procedures. Ann. Math. Statist. 43, 814–822.

    Article  MathSciNet  MATH  Google Scholar 

  19. Gupta, S. S. and Hsu, J. C (1977). On the monotonicity of Bayes subset selection procedures. Proceedings of the 41st Session of the International Statistical Institute, Vol. 47, Book 4, 208–211.

    Google Scholar 

  20. Hardy, G. H., Littlewood, J. E. and Polya, G. (1934). Inequalities. Cambridge Univ. Press, Cambridge, England.

    Google Scholar 

  21. Hoel, D. G. (1970). On the monotonicity of the OC of an SPRT. Ann. Math. Statist. 41, 310–314.

    Article  MathSciNet  MATH  Google Scholar 

  22. Hollander, M., Proschan, F. and Sethuraman, J. (1977). Functions decreasing in transposition and their applications in ranking problems. Ann. Statist. 5, 722–733.

    Article  MathSciNet  MATH  Google Scholar 

  23. Hsu, J. C (1977). On some decision-theoretic contributions to the problem of subset selection. Mimeo. Ser. #491, Dept. of Statist., Purdue Univ., W. Lafayette, Indiana.

    Google Scholar 

  24. Ihm, P. (1959). Numerical evaluation of certain multivariate normal integrals. Sankhyā 21, 363–366.

    MathSciNet  MATH  Google Scholar 

  25. Jogdeo, K. (1970). A simple proof of an inequality for multivariate normal probabilities of rectangle. Ann. Math. Statist. 41, 1357–1359.

    Article  MathSciNet  MATH  Google Scholar 

  26. Karlin, S. (1968). Total Positivity, Vol. I. Stanford Univ. Press, Stanford, California.

    MATH  Google Scholar 

  27. Karlin, S. and Rubin, H. (1956). The theory of decision procedures for distributions with monotone likelihood ratio. Ann. Math. Statist. 27, 272–299.

    Article  MathSciNet  MATH  Google Scholar 

  28. Kingman, J.F.C. (1978). Uses of exchangeability. Ann. Prob. 6, 183–197.

    Article  MathSciNet  MATH  Google Scholar 

  29. Lehmann, E. L. (1955). Ordered families/of distributions. Ann. Math. Statist. 26, 399–419.

    Article  MathSciNet  MATH  Google Scholar 

  30. Lehmann, E. L. (1959). Testing Statistical Hypotheses. John Wiley.

    MATH  Google Scholar 

  31. Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.

    Article  MathSciNet  MATH  Google Scholar 

  32. Marshall, A. W. and Olkin, I. (1974). Majorization in multivariate distributions. Ann. Math. Statist. 2, 1180–1200.

    MathSciNet  Google Scholar 

  33. Marshall, A. W. and Olkin, I. (1979). Majorization and Schur Functions. Academic Press, New York.

    Google Scholar 

  34. Moran, P.A.P. (1956). The numerical evaluation of a class of integrals. Proc. Cambridge Philos. Soc. 52, 230–233.

    Article  MathSciNet  MATH  Google Scholar 

  35. Olkin, I. and Rubin, H. (1964). Multivariate beta distributions and independence properties of the Wishart distribution. Ann. Math. Statist. 35, 261–269.

    Article  MathSciNet  Google Scholar 

  36. Oosterhoff, J. (1969). Combination of One-Sided Statistical Tests. Mathematical Centre Tracks 28, Amsterdam.

    MATH  Google Scholar 

  37. Ostrowski, A. (1952). Sur quelque applications des functions convexes et concaves au sens de I. Schur. J. Math. Pure Appl. 31, 253–292.

    MathSciNet  MATH  Google Scholar 

  38. Patil, G. P. and Bosewell, M. T. (1970). A characteristic property of the multivariate normal density function and some of its applications. Ann. Math. Statist. 41, 1970–1977.

    Article  MathSciNet  MATH  Google Scholar 

  39. Ruben, H. (1954). On the moments of order statistics in samples from normal populations. Biometrika 41, 200–227.

    MathSciNet  MATH  Google Scholar 

  40. Schur, I. (1923). Über eine Klasse von Mittelbildengen mit Anwendungen auf die Determinatentheorie. Sitzber Berl. Math. Ges. 22, 9–20.

    Google Scholar 

  41. Sidak, Z. (1968). On multivariate normal probabilities of rectangles: Their dependence on correlations. Ann. Math. Statist. 39, 1425–1434.

    MathSciNet  MATH  Google Scholar 

  42. Slepian, D. (1962). The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.

    MathSciNet  Google Scholar 

  43. Stuart, A. (1958). Equally correlated variates and the multinomial integral. J. Roy. Statist. Soc. Ser. B 20, 373–378.

    MathSciNet  MATH  Google Scholar 

  44. Sherman, S. (1955). A theorem on convex sets with applications. Ann. Math. Statist. 26, 763–766.

    Article  MathSciNet  MATH  Google Scholar 

  45. Tong, Y. L. (1980). Probability Inequalities in Multivariate Distributions. Academic Press, New York.

    MATH  Google Scholar 

  46. Tukey, J. W. (1958). A problem of Berkson, and minimum variance orderly estimators. Ann. Math. Statist. 29, 588–592.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 1981 Springer-Verlag New York Inc.

About this chapter

Cite this chapter

Gupta, S.S., Huang, DY. (1981). Some Auxiliary Results — Monotonicity Properties of Probability Distributions. In: Multiple Statistical Decision Theory: Recent Developments. Lecture Notes in Statistics, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5925-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-5925-1_1

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-90572-3

  • Online ISBN: 978-1-4612-5925-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics