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The broken sample problem
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  • Published: 12 September 2004

The broken sample problem

  • Zhidong Bai1 &
  • Tailen Hsing2 

Probability Theory and Related Fields volume 131, pages 528–552 (2005)Cite this article

  • 116 Accesses

  • 7 Citations

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Abstract.

Suppose that (X i ,Y i ),i=1,2, ... ,n, are iid. random vectors with uniform marginals and a certain joint distribution F ρ , where ρ is a parameter with ρ=ρ o corresponds to the independence case. However, the X’s and Y’s are observed separately so that the pairing information is missing. Can ρ be consistently estimated? This is an extension of a problem considered in (1980) which focused on the bivariate normal distribution with ρ being the correlation. In this paper we show that consistent discrimination between two distinct parameter values ρ1 and ρ2 is impossible if the density f ρ of F ρ is square integrable and the second largest singular value of the linear operator is strictly less than 1 for ρ=ρ1 and ρ2. We also consider this result from the perspective of a bivariate empirical process which contains information equivalent to that of the broken sample.

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References

  1. Bai, Z.D.: Methodologies in spectral analysis of large dimensional random matrices. A rev. Statistica Sinica 9, 611–677 (1999)

    MATH  Google Scholar 

  2. Chan, H.-P., Loh, W.-L.: A file linkage problem of DeGroot and Goel revisited. Statistica Sinica 11, 1031–1045 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Cramér, H.: Mathematical methods of statistics. Princeton University Press, 1946

  4. DeGroot, M.H., Goel, P.K.: Estimation of the correlation coefficient from a broken sample. Ann. Statist. 8, 264–278 (1980)

    MathSciNet  MATH  Google Scholar 

  5. Feller, W.: An Introduction to Probability and Its Applications. vol 2. Wiley, 1971

  6. Grenander, U.: Abstract Inference. Wiley, 1981

  7. Hajék, J.: On a property of normal distribution of any stochastic process (in Russian). Czechoslovak Math. J. 8(83), 610–618 (1958), (An English translation appeared in American Mathematical Society Translations in Probability and Statistics, 1961)

    Google Scholar 

  8. Harvill, D.A.: Matrix Algebra from a Statistician’s Perspective. Springer, 1997

  9. Parzen, E.: Probability density functionals and reproducing kernel Hilbert spaces. M. Rosenblatt (ed.), Proceedings of the Symposium on Time Series Analysis, Wiley, 1963, pp. 155–169

  10. Resnick, S.: Extreme Values, Regular Variation, and Point Processes. Springer, 1987

  11. Riesz, F., Sz.-Nagy, B.: Functional Analysis. Translated from the 2d French ed. by Leo F. Boron. Ungar, 1955

  12. Rozanov, J.A.: Infinite Dimensional Gaussian Distributions. English translation published by American Mathematical Society, 1971

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Author information

Authors and Affiliations

  1. North East Normal University, Department of Statistics and Applied Probability, National University of Singapore, China, Singapore

    Zhidong Bai

  2. Department of Statistics, Texas A&M University, College Station, Texas, USA

    Tailen Hsing

Authors
  1. Zhidong Bai
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  2. Tailen Hsing
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Corresponding author

Correspondence to Zhidong Bai.

Additional information

Dedicated to Professor Xiru Chen on His 70th Birthday

Mathematics Subject Classification (2000): primary: 60F99, 62F12

Research supported by NSFC Grant 201471000 and the NUS Grant R-155-000-040-112.

Research supported by the Texas Advanced Research Program.

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Cite this article

Bai, Z., Hsing, T. The broken sample problem. Probab. Theory Relat. Fields 131, 528–552 (2005). https://doi.org/10.1007/s00440-004-0384-5

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  • Received: 20 February 2002

  • Revised: 16 June 2004

  • Published: 12 September 2004

  • Issue Date: April 2005

  • DOI: https://doi.org/10.1007/s00440-004-0384-5

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Keywords

  • Consistent estimation
  • Empirical process
  • Gaussian process
  • Kulback-Leibler information
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