Skip to main content
SpringerLink
Log in

Revisiting Local Asymptotic Normality (LAN) and Passing on to Local Asymptotic Mixed Normality (LAMN) and Local Asymptotic Quadratic (LAQ) Experiments

  • Chapter
  • First Online:
Advances in Directional and Linear Statistics

Abstract

Let X 1, …, X n be a random sample of size n from an underlying parametric statistical model. Then the basic statistical problem may be stated as follows: On the basis of a random sample, whose probability law depends on a parameter θ, discriminate between two values θ and θ ∗ (θ≠θ∗). When the parameters are sufficiently far apart, any decent statistical procedure will do the job. A problem arises when the parameter points are close together, and yet the corresponding probability measures are substantially or even vastly different. The present paper revolves around ways of resolving such a problem. The concepts and methodology used are those of contiguity, Local Asymptotic Normality (LAN), Local Asymptotic Mixed Normality (LAMN), and Local Asymptotic Quadratic (LAQ) experiments.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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. Akritas MG (1978) Contiguity of probability measures associated with continuous time stochastic processes. PhD thesis, Department of statistics, University of Wisconsin, Madison

    Google Scholar 

  2. Akritas MG, Puri ML, Roussas GG (1979) Sample size, parameter rates and contiguity–The iid case. Commun Stat Theory Methods 8(1):71–83

    MathSciNet  Google Scholar 

  3. Akritas MG, Roussas GG (1978) A characterization of contiguity by sample size and parameter rates. Symposia Mathematica, Instituto Nazionale di Alta Mathematica, vol 25. Academic Press, New York, pp 155–170

    Google Scholar 

  4. Akritas MG, Roussas GG (1979) Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process. Ann Inst Stat Math 31(Part A):103–120

    Google Scholar 

  5. Akritas MG, Roussas GG, Stamatelos GD (1984) Asymptotic expansion of the log-likelihood function based on stopping times defined on stochastic processes. Mathematical structures – Computational mathematics – Mathematical modelling, Bulgarian Academy of Sciences, Sofia, 90–96

    Google Scholar 

  6. Basawa IV, Brockwell PJ (1984) Asymptotic conditional inference for regular nonergodic models with an application to autoregressive processes. Ann Stat 12:161–171

    Article  MATH  MathSciNet  Google Scholar 

  7. Basawa IV, Scott DJ (1983) Asymptotic optimal inference for non-ergodic models. Springer, New York

    MATH  Google Scholar 

  8. Basu AK, Bhattacharya D (1988) Local asymptotic mixed normality of log-likelihood based on stopping times. Calcutta Stat Assoc Bull 37:143–159

    MATH  MathSciNet  Google Scholar 

  9. Basu AK, Bhattacharya D (1990) Weak convergence of randomly stopped log-likelihood ratio statistics to mixed Gaussian process. Calcutta Stat Assoc Bull 39:137–149

    MATH  MathSciNet  Google Scholar 

  10. Basu AK, Bhattacharya D (1992) On the asymptotic non-null distribution of randomly stopped log-likelihood ratio statistic. Calcutta Stat Assoc Bull 42:255–260

    MATH  MathSciNet  Google Scholar 

  11. Bhattacharya D, Roussas GG (2001) Exponential approximation for randomly stopped locally asymptotically mixture of normal experiments. Stoch Model Appl 4(2):56–71

    Google Scholar 

  12. DasGupta A (2008) Asymptotic theory of statistics and probability. Springer, New York

    MATH  Google Scholar 

  13. Davies RB (1985) Asymptotic inference when the amount of information is random. In: Le Cam L, Olson RA (eds) Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, vol 2. Wadsworth, California, pp 841–864

    Google Scholar 

  14. Efron B (1975) Defining the curvature of a statistical problem (with applications to second order efficiency). Ann Stat 3(6):1189–1242

    Article  MATH  MathSciNet  Google Scholar 

  15. Greenwood PE, Shiryayev AN (1985) Contiguity and the statistical invariance principle. Gordon and Breach, New York

    MATH  Google Scholar 

  16. Greenwood PE, Wefelmeyer W (1993) Asymptotic minimax results for stochastic process families with critical points. Stoch Process Appl 44:107–116

    Article  MATH  MathSciNet  Google Scholar 

  17. Hájek JA (1970) A characterization of limiting distributions of regular estimates. Z Wahrschelnlichkeitstheoric Und verw Gebiete 14:323–330

    Article  MATH  Google Scholar 

  18. Inagaki N (1970) On the limiting distribution of sequence of estimators with uniformity property. Ann Inst Stat Math 22:1–13

    Article  MATH  MathSciNet  Google Scholar 

  19. Jeganathan P (1982) On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal. Sankhyā 44(Ser. A):173–212

    Google Scholar 

  20. Jeganathan P (1995) Some aspects of asymptotic theory with applications to time series model. Econom Theory 2:818–887

    Article  MathSciNet  Google Scholar 

  21. Johnson RA, Roussas GG (1969) Asymptotically most powerful tests in Markov processes. Ann Math Stat 40:1207–1215

    Article  MATH  MathSciNet  Google Scholar 

  22. Johnson RA, Roussas GG (1970) Asymptotically optimal tests in Markov processes. Ann Math Stat 41:918–38

    Article  MATH  MathSciNet  Google Scholar 

  23. Johnson RA, Roussas GG (1972) Applications of contiguity to multiparameter hypothesis testing. Proceedings of the sixth Berkeley symposium on probability theory and mathematical statistics, vol 1, pp 195–226

    MathSciNet  Google Scholar 

  24. Le Cam L (1960) Locally asymptotically normal families of distributions. Univ Calif Publ Stat 3:37–98

    Google Scholar 

  25. Le Cam L (1986) Asymptotic methods in statistical decision theory. Springer series in statistics. Springer, New York

    MATH  Google Scholar 

  26. Le Cam L, Yang GL (2000) Asymptotics in statistics, some basic concepts. Lecture notes in statistics. Springer, New York

    MATH  Google Scholar 

  27. Lind B, Roussas GG (1972) A remark on quadratic mean differentiability. Ann Math Stat 43:1030–1034

    Article  MATH  MathSciNet  Google Scholar 

  28. Lind B, Roussas GG (1977) Cramér-type conditions and quadratic mean differentiability. Ann Inst Stat Math 29:189–201

    Article  MATH  MathSciNet  Google Scholar 

  29. Philippou A, Roussas GG (1973) Asymptotic distribution of the likelihood function in the independent not identically distributed case. Ann Stat 1:454–471

    Article  MATH  MathSciNet  Google Scholar 

  30. Phillips PCB (1987) Asymptotic minimax results for stochastic process families with critical points. Econometrica 55(2):277–301

    Article  MATH  MathSciNet  Google Scholar 

  31. Roussas GG (1965) Asymptotic inference in Markov processes. Ann Math Stat 36:978–993

    Article  MATH  MathSciNet  Google Scholar 

  32. Roussas GG (1968) Some applications of the asymptotic distribution of likelihood functions to the asymptotic efficiency of estimates. Z Wahrscheinlichkeitstheorie Und verw Gebeite 10:252–260

    Article  MATH  MathSciNet  Google Scholar 

  33. Roussas GG (1972) Contiguity of probability measures: Some applications in statistics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  34. Roussas GG (1975) Asymptotic properties of maximum probability estimates in the iid case. In: Puri ML (ed) Statistical inference and related topics, vol 2. Academic Press, New York, pp 211–234

    Google Scholar 

  35. Roussas GG (1977) Asymptotic properties of maximum probability estimates in Markov processes. Ann Inst Stat Math 29:203–219

    Article  MATH  MathSciNet  Google Scholar 

  36. Roussas GG (1979) Asymptotic distribution of the log-likelihood function for stochastic processes. Z Wahrschelnlichkeitstheoric Und verw Gebiete 47:31–46

    Article  MATH  MathSciNet  Google Scholar 

  37. Roussas GG (2005) An introduction to measure-theoretic probability. Elsevier, Burlington

    Google Scholar 

  38. Roussas GG (2008) Contiguity of probability measures: Some applications in statistics. Cambridge University Press, paperback reprint of the book #33

    Google Scholar 

  39. Roussas GG, Akritas MG (1980) Asymptotic inference in continuous time semi-Markov processes. Scand J Stat 7:73–79

    MATH  MathSciNet  Google Scholar 

  40. Roussas GG, Bhattacharya D (1999a) Asymptotic behavior of the log-likelihood function in stochastic processes when based on a random number of random variables. In: Janssen J, Limnios N (eds) Semi-Markov models and applications. Kluwer, Dordrecht, pp 119–147

    Google Scholar 

  41. Roussas GG, Bhattacharya D (2002) Exponential approximation of distributions. In Teoriya Imovirnostey ta Matematichna Statystika, 66:109–120. Also, in Theory of probability and mathematical statistics, 66:119–132 (2003) (English version)

    Google Scholar 

  42. Roussas GG, Bhattacharya D (2007) Asymptotic expansions, exponential approximation and Hájek-Inagaki representation theorem under a general dependence set-up. In: Proceedings of the 20th Panhellenic statistical conference, Nikosia, Cyprus, April 11–15, 45–65 (Invited keynote lecture for the annual meeting)

    Google Scholar 

  43. Roussas GG, Bhattacharya D (2008) Hájek–Inagaki representation theorem, under general stochastic processes framework, based on stopping times. Stat Probab Lett 78:2503–2510

    Article  MATH  MathSciNet  Google Scholar 

  44. Roussas GG, Bhattacharya D (2009) Hájek–Inagaki convolution representation theorem for randomly stopped locally asymptotically mixed normal experiments. Stat Inference Stoch Process (In Print). DOI 10.1007/s 11203-008-9029-0. Published online: 16 December 2008

    Google Scholar 

  45. Roussas GG, Soms A (1973) On the exponential approximation of a family of probability measures and representation theorem of Hájek–Inagaki. Ann Inst Stat Math 25:27–39

    Article  MathSciNet  Google Scholar 

  46. Sweeting T (1992) Asymptotic ancillarity and conditional inference for stochastic processes. Ann Stat 20(1):580–589

    Article  MATH  MathSciNet  Google Scholar 

  47. Taniguchi M, Kakizawa Y (2000) Asymptotic theory of statistical inference for time series. Springer series in statistics. Springer, New York

    MATH  Google Scholar 

  48. van der Vaart AW (1998) Asymptotic statistics. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  49. Wald A (1943) Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans Am Math Soc 54:426–482

    MATH  MathSciNet  Google Scholar 

  50. Weiss L, Wolfowitz J (1966) Generalized maximum likelihood estimators. Theor Probab Appl 11:58–81

    Article  MATH  MathSciNet  Google Scholar 

  51. Weiss L, Wolfowitz J (1967) Maximum probability estimators. Ann Inst Stat Math 19:193–206

    Article  MATH  MathSciNet  Google Scholar 

  52. Weiss L, Wolfowitz J (1968) Generalized maximum likelihood estimators in a particular case. Theor Probab Appl 13:622–627

    Article  MathSciNet  Google Scholar 

  53. Weiss L, Wolfowitz J (1970) Maximum probability estimators and asymptotic sufficiency. Ann Inst Stat Math 22:225–244

    Article  MATH  MathSciNet  Google Scholar 

  54. Wolfowitz J (1965) Asymptotic efficiency of the maximum likelihood estimator. Theor Probab Appl 10:247–260

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, University of California, Davis, CA, 95616, USA

    George G. Roussas

Authors
  1. George G. Roussas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Debasis Bhattacharya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George G. Roussas .

Editor information

Editors and Affiliations

  1. , Statistical Science, Cornell University, Comstock Hall 1190, Ithaca, NY 14853, 14853, USA

    Martin T. Wells

  2. , Applied Statistics Unit, Indian Statistical Institute, Barrackpore Trunk Road 203, Kolkata, 700035, India

    Ashis SenGupta

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Roussas, G.G., Bhattacharya, D. (2011). Revisiting Local Asymptotic Normality (LAN) and Passing on to Local Asymptotic Mixed Normality (LAMN) and Local Asymptotic Quadratic (LAQ) Experiments. In: Wells, M., SenGupta, A. (eds) Advances in Directional and Linear Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2628-9_17

Download citation

Publish with us

Policies and ethics

Search

Navigation