Skip to main content
SpringerLink
Log in

A Discussion of Some Basic Concepts in Statistical Inference

  • Chapter
Book cover Transactions of the Eighth Prague Conference

Part of the book series: Czechoslovak Academy of Sciences ((TPCI,volume 8A))

  • 102 Accesses

Abstract

Despite its’ distinctive past history, the problem of statistical inference in a broad sense, besides being controversial among statisticians, is continuously stimulating vigorous and rigorous research. There are some interesting recent advances but many problems either remain open and unresolved or need improvements. In order to give a reasonably comprehensive discussion of the basic concepts, ideas and principles, in this paper we concentrate on the various underlying assumptions, statistical decision error structure, various measures of information, sufficiency and ancillarity, and some important and most commonly used principles of statistical inference. It is suggested that sphericity and exchangeability, besides being weaker than normality, are more plausible and tangible, at least from a practical viewpoint. Thus they are perhaps more natural and appealing as a basis for statistical inference. For a large class of standard decisions it is proposed that instead of the classical testing approach, one should minimize a linear combination, with nonnegative weights, of the first and second type errors. After discussing various measures of information, it is pointed out that the family of likelihood functions of various types essentially contain relevant information for statistical problems. Next, along with sufficiency and ancillarity concepts, various inferential principles are discussed. Without assuming discreteness or finiteness of underlying universes, with due regard to the measure-theoretic difficulties, implications, interplay and consequences of various principles such as conditionality, invariance, sufficiency, likelihoods (all types) and their weak versions are clarified.

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

Access this chapter

Log in via an institution
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. Aczél, J. et al. (1975). Entropy and Ergodic Theory. Selecta Statistica Canadiana Vol. 2. Delhi Univ. Press.

    Google Scholar 

  2. Aczél, J. - Daroczy, Z. (1975). On Measures of Information and their Characterizations. Math. Sc. & Eng. Vol. 115. Academic Press.

    Google Scholar 

  3. Ahmad, R. (1972). Extension of the normal theory to spherical families. Trabajos de Estadistica, 23, 51–60.

    MATH  Google Scholar 

  4. Ahmad, R. (1974). On the structure of symmetric sample testing: a distribution-free approach. Ann. Inst. Statist. Math., 26, 233–245.

    Article  MATH  MathSciNet  Google Scholar 

  5. Ahmad, R. (1975a). Some characterizations of the exchangeable processes and distribution-free tests. G.P. Patil et al. (eds.), Statistical Distributions in Scientific Work, Vol. 3, 237–248. D. Reidel, Dordrecht- Holland.

    Google Scholar 

  6. Ahmad, R. (1975b). The Neyman-Pearson lemma for Choquet capacities in Bayesian problems. Bull. Inter. Statist. Inst., 46(3), 13–13.

    Google Scholar 

  7. Ahmad, R. - Abouammoh, A.M. (1977). On the structure and applications of infinite divisibility, stability and symmetry in stochastic inference. Recent Developments in Statistics, J.P. Barra et al. (eds.), 303–317. North- Holland.

    Google Scholar 

  8. Bahadur, R. (1954). Sufficiency and statistical decision functions. Ann. Math. Statist., 25, 423–462.

    Article  MathSciNet  Google Scholar 

  9. Bahadur, R. (1955). Statistics and subfields. Ibid, 26, 490–497.

    MATH  MathSciNet  Google Scholar 

  10. Bahadur, R. - Lehmann, E.L. (1955). Two comments on “Sufficiency and statistical decision functions”. Ibid, 26, 139–142.

    MATH  MathSciNet  Google Scholar 

  11. Barnard, G.A. (1967). The use of the likelihood function in statistical practice. 5th Berkeleymp. Math. Stat. Prob., 1, 27–40.

    MathSciNet  Google Scholar 

  12. Barndorff-Nielsen, O. - Blaesild, P. (1975). S-ancillarity in exponential families. Sankhya Ser. A, 37, 354–385.

    MATH  MathSciNet  Google Scholar 

  13. Basu, D. (1959). The family of ancillary statistics. Sankhyā Ser. A, 21, 247–256.

    MATH  Google Scholar 

  14. Basu, D. (1975). Statistical information and likelihood. Sankhyā Ser. A, 37, 1–71.

    MATH  Google Scholar 

  15. Birnbaum, A. (1962). On the foundations of statistical inference. JASA, 57, 269–326.

    MATH  MathSciNet  Google Scholar 

  16. Burkholder, D.L. (1961). Sufficiency in the undominated case. Ann. Math. Statist., 32, 1191–1200.

    Article  MATH  MathSciNet  Google Scholar 

  17. Csiszar, I. (1969). On generalized entropy. Stud. Sci. Math. Hung., 4, 401–419.

    MATH  MathSciNet  Google Scholar 

  18. Dawid, A.P. (1977). Conformity of inference patterns. Recent Developments in Statistics, J.R. Barra et al. (eds.), 245–256. North-Holland.

    Google Scholar 

  19. Fisher, R.A. (1921). On the mathematical foundations of theoretical statistics. Philos. Trans. Roy. Soc. London Ser. A, 222, 309–368.

    Google Scholar 

  20. Fisher, R.A. (1925). Theory of statistical estimation. Proc. Camb. Philos. Soc., 22, 700–725.

    Article  MATH  Google Scholar 

  21. Fraser, D.A.S. (1968). The Structure of Inference. Wiley.

    MATH  Google Scholar 

  22. Godambe, V.P. - Sprott, D.A. (1971). Foundations of Statistical Inference (eds.) Holt, Rinehart and Winston, Toronto.

    Google Scholar 

  23. Hajek, J. (1967). On basic concepts of statistics. 5th Berkeley Symp. Math. Stat. Prob., 1, 139–162.

    MathSciNet  Google Scholar 

  24. Hewitt, E. - Savage, L.J. (1955). Symmetric measures on Cartesian products. Trans. Amer. Math. Soc., 80, 470–501.

    MATH  MathSciNet  Google Scholar 

  25. Ingarden, R.S. - Urbanik, K. (1962). Information without probability. Colloq. Math., 9, 131–150.

    MATH  MathSciNet  Google Scholar 

  26. Johansen, S. (1977). Homomorphism and general exponential families. Recent Developments in Statistics, J.R. Barra et al. (eds.), 489–499. North-Holland.

    Google Scholar 

  27. Kullback, S. (1959). Information Theory and Statistics. Wiley.

    MATH  Google Scholar 

  28. Landers, D. (1972). Sufficient and minimal sufficient σ-fields. Z. Wahrsch., 23, 197–207.

    Article  MATH  MathSciNet  Google Scholar 

  29. LeCam, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist., 35, 1419–1455.

    Article  MathSciNet  Google Scholar 

  30. LeCam, L. (1974). On the information contained in additional observations. Ann. Statist., 4, 630–649.

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  32. Loève, M. (1963). Probability Theory. 3rd ed. D. van Nostrand, Princeton.

    MATH  Google Scholar 

  33. Neyman, J. (1954). Sur une famille de tests asymptotiques des hypotheses statistiques composées. Trabajos de Estadistica, 5, 161–168.

    Article  MATH  MathSciNet  Google Scholar 

  34. Parthasarathy, K. (1967). Probability Measures on Metric Spaces. Academic Press.

    MATH  Google Scholar 

  35. Rényi, A; (1961). On measures of entropy and information. 4th Berkeley Symp. Math. Stat. Prob., 1, 547–561.

    Google Scholar 

  36. Rényi, A. (1966). On the amount of missing information and the Neyman-Pearson lemma., Research Papers in Statistics for Neyman, F.N. David (ed.). Wiley.

    Google Scholar 

  37. Rogge, L. (1972). The relations between minimal sufficient statistics and minimal sufficient σ-fields. Z. Wahrsch., 23, 208–215.

    Article  MATH  MathSciNet  Google Scholar 

  38. Savage, L.J. (1976). On rereading R.A. Fisher. Ann. Statist., 3, 441–500.

    MathSciNet  Google Scholar 

  39. Shannon, C.E. (1948). A mathematical theory of Communication. Bell System Tech. J., 27, 379–423, 623–656.

    MATH  MathSciNet  Google Scholar 

  40. Shannon, C.E. - Weaver, W. (1949). The Mathematical Theory of Communication. Univ. of Illinois Press.

    Google Scholar 

  41. Urbanik, K. (1974). On the concept of information. Colloq. Math. Soc. J. Bolyai, 9, 863–868.

    MathSciNet  Google Scholar 

  42. Andersen, E.B. (1967). On partial sufficiency and partial ancillarity. Skand. Aktuar., 50, 137–152.

    Google Scholar 

  43. Dynkin, E.B. (1959). The Foundations of the Theory of Markov Processes. Moscow, Fizmatgiz. (In Russian).

    Google Scholar 

  44. Halmos, P.R. (1950). Measure Theory. Van Nostrand, Princeton.

    MATH  Google Scholar 

  45. Halmos, P.R. - Savage, L.J. (1949). Applications of the Radon-Nikodym theorem to the theory of sufficient statistics. Ann. Math. Statist., 20, 225–241.

    MATH  MathSciNet  Google Scholar 

  46. Hewitt, E. - Stromberg, K. (1967). Real and Abstract Analysis. Springer.

    Google Scholar 

  47. Kullback, S. - Leibler, R.A. (1951). On information and sufficiency. Ann. Math. Statist., 22, 79–86.

    Article  MATH  MathSciNet  Google Scholar 

  48. 48.Kuratowski, C. (1933). Topologie I. Warsrawa - twow.

    Google Scholar 

  49. Lehmann, E.L. - Scheffé, H. (1950;1955). Completeness, similar regions and unbiased estimation. Sankhya 10, 305–340; 15, 219–236.

    MATH  MathSciNet  Google Scholar 

  50. Meyer, P.A. (1966). Probability and Potentials. Blaisdell, New York.

    MATH  Google Scholar 

  51. Neveu, J. (1975). Discrete-Parameter Martingales. North-Holland.

    MATH  Google Scholar 

  52. Schwartz, L., (1973). Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Tata Institute of Fundamental Research, India.

    MATH  Google Scholar 

  53. Soler, J.L. (1977). Infinite dimensional exponential type statistical spaces. Recent Developments in Statistics, J.R. Barra et al. (eds.) 269–284 North-Holland.

    Google Scholar 

  54. Urbanik, K. (1975). Decomposability properties of probability measures. Sankhyer. A, 37, 530–537.

    MATH  MathSciNet  Google Scholar 

  55. Wilkinson, G.N. (1977). On resolving the controversy in statistical inference (with Discussion). J. Roy. Statist. Soc. B, 39, 119–171.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Strathclyde, 26 Richmond Street, G1 1XH, Glasgow, Scotland

    Rashid Ahmad

Authors
  1. Rashid Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 1978 ACADEMIA, Publishing House of the Czechoslovak Academy of Sciences, Prague

About this chapter

Cite this chapter

Ahmad, R. (1978). A Discussion of Some Basic Concepts in Statistical Inference. In: Transactions of the Eighth Prague Conference. Czechoslovak Academy of Sciences, vol 8A. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9857-5_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-9857-5_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-9859-9

  • Online ISBN: 978-94-009-9857-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation