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A central limit theorem for measurements on the logarithmic scale and its application to dimension estimates

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  • Denker, Manfred
  • Min, Aleksey

Abstract

We show consistency and asymptotic normality of certain estimators for expected exponential growth rates under i.i.d. observations. These statistical functionals are of the formT(F)=[integral operator]log[integral operator]h(x,y)F(dx)F(dy)and are applicable to dimension estimates (information dimension), entropy estimates and estimations of the growth rate of "generating" functions. We also give an affirmative answer to a question posed by Keller in 1997 [A new estimator for information dimension with standard errors and confidence intervals, Stochastic Process. Appl. 71(2):187-206] whether this estimator, specialized for dimension, is an alternative to standard procedures.

Suggested Citation

  • Denker, Manfred & Min, Aleksey, 2008. "A central limit theorem for measurements on the logarithmic scale and its application to dimension estimates," Journal of Multivariate Analysis, Elsevier, vol. 99(4), pages 665-683, April.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:4:p:665-683
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    References listed on IDEAS

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    1. Cutler, C. D. & Dawson, D. A., 1989. "Estimation of dimension for spatially distributed data and related limit theorems," Journal of Multivariate Analysis, Elsevier, vol. 28(1), pages 115-148, January.
    2. Keller, Gerhard, 1997. "A new estimator for information dimension with standard errors and confidence intervals," Stochastic Processes and their Applications, Elsevier, vol. 71(2), pages 187-206, November.
    3. Dehling, Herold, 1989. "Complete convergence of triangular arrays and the law of the iterated logarithm for U-statistics," Statistics & Probability Letters, Elsevier, vol. 7(4), pages 319-321, February.
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