Advertisement

The Order of the Remainder in Derivatives of Composition and Inverse Operators for p-Variation Norms

  • R. M. Dudley
Chapter
Part of the Selected Works in Probability and Statistics book series (SWPS)

Abstract

The theory of differentiable statisticals began with work of von Mises [e.g., von Mises (1936, 1947) and Filippova (1961)]. A nonlinear funtional T is defined, for examble, on distribution functions.Von Mises differentiated T at a distribution funtion F along lines. For T to have a (Gâteaux) derivative at F means that, in the direction of a function h,
$$T(F + th) = T(f) + tT^\prime(F)(h) + 0 (|t|) \quad {\rm as} \; t\rightarrow 0.$$
(1.1)
Here \(T^\prime(F)(.)\) is a bounded linear operator on funtions h, for example, of the form
$$T^\prime(F) (h) = \int g \quad dh \hbox{for some function} g (\hbox{depending on} F).$$
(1.2)

Key words and phrases

Fréchet derivative compact derivativ Hadamard derivative Gâteaux derivative Bahadur–Kiefer theorems Orlicz variation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Appell, J. and Zabrejko, P. P. (1990). Nonlinear Superposition Operators. Cambridge Univ. Press.zbMATHGoogle Scholar
  2. Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37 577–580.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Estimation for Semiparametric Models. Johns Hopkins Univ. Press.zbMATHGoogle Scholar
  4. Brokate, M. and Colonius, F. (1990). Linearizing equations with state-dependent delays. Appl. Math. Optim. 21 45–52.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Csörgő, M. and Révész, P. (1978). Strong approximations of the quantile process. Ann. Statist. 6 882–894.CrossRefMathSciNetGoogle Scholar
  6. Deheuvels, P. and Mason, D. M. (1990). Bahadur–Kiefer–type processes. Ann. Probab. 18 669–697.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Dollard, J. D. and Friedman, C. N. (1979). Product Integration with Applications to Differential Equations. Addison-Wesley, Reading, MA.zbMATHGoogle Scholar
  8. Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA.zbMATHGoogle Scholar
  9. Dudley, R. M. (1991). Differentiability of the composition and inverse operators for regulated and a.e. continuous functions. Unpublished manuscript.Google Scholar
  10. Dudley, R. M. (1992a). Fréchet differentiability, p-variation and uniform Donsker classes. Ann. Probab. 20 1968–1982.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Dudley, R. M. (1992b). Empirical processes: p-variation for p ≤ 2 and the quantile-quantile and \(\int F \ dG\) operators. Unpublished manuscript.Google Scholar
  12. Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Statist. 27 642–669.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Esty, W., Gillette, R., Hamilton, M. and Taylor, D. (1985). Asymptotic distribution theory of statistical functionals. Ann. Inst. Statist. Math. 37 109–129.zbMATHCrossRefMathSciNetGoogle Scholar
  14. Fernholz, L. T. (1983). Von Mises Calculus for Statistical Functionals. Lecture Notes in Statist. 19. Springer, New York.Google Scholar
  15. Filippova, A. (1961). Mises’ theorem on the asymptotic behavior of functionals of empirical distribution functions and its statistical applications. Theory Probab. Appl. 7 24–57.CrossRefGoogle Scholar
  16. Freedman, M. A. (1983). Operators of p-variation and the evolution representation problem. Trans. Amer. Math. Soc. 279 95–112.zbMATHMathSciNetGoogle Scholar
  17. Gill, R. D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part 1). Scand. J. Statist. 16 97–128.zbMATHMathSciNetGoogle Scholar
  18. Gill, R. D. and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18 1501–1555.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Gray, A. (1975). Differentiation of composites with respect to a parameter. J. Austral. Math. Soc. Ser. A 19 121–128.zbMATHCrossRefGoogle Scholar
  20. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York.zbMATHGoogle Scholar
  21. Holt, R. J. (1986). Computation of gamma and beta tail probabilities. Technical report, Dept. Mathematics, MIT.Google Scholar
  22. Huber, P J. (1981). Robust Statistics. Wiley, New York.zbMATHCrossRefGoogle Scholar
  23. Kendall, M. G. and Stuart, A. (1977). The Advanced Theory of Statistics 1 . Distribution Theory, 4th ed. Macmillan, New York.zbMATHGoogle Scholar
  24. Kiefer, J. (1967). On Bahadur’s representation of sample quantiles. Ann. Math. Statist. 38 1323–1342.zbMATHCrossRefMathSciNetGoogle Scholar
  25. Kiefer, J. (1970). Deviations between the sample quantile process and the sample df. In Non-parametric Techniques in Statistical Inference (M. L. Puri, ed.) 299–319. Cambridge Univ. Press.Google Scholar
  26. Krabbe, G. L. (1961). Integration with respect to operator-valued functions. Bull. Amer. Math. Soc. 67 214–218.zbMATHCrossRefMathSciNetGoogle Scholar
  27. Krasnosel’skii, M.A. and Rutickii, Ya. B. (1961). Convex Functions and Orlicz Spaces. Noordhoff, Groningen. (Translated by L. F. Boron.)Google Scholar
  28. Lepingle, D. (1976). La variation d’ordre p des semi-martingales. Z. Wahrsch. Verw. Gebiete 36 295–316.zbMATHCrossRefMathSciNetGoogle Scholar
  29. Molenaar, W. (1970). Approximations to the Poisson, Binomial and Hypergeometric Distribution Functions. Math. Centrum, Amsterdam.zbMATHGoogle Scholar
  30. Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 1293–1311.zbMATHCrossRefMathSciNetGoogle Scholar
  31. Monroe, I. (1976). Almost sure convergence of the quadratic variation of martingales: A counterexample. Ann. Probab. 4 133–138.zbMATHCrossRefMathSciNetGoogle Scholar
  32. Musielak, J. and Orlicz, W. (1959). On generalized variations (I). Studia Math. 18 11–41.zbMATHMathSciNetGoogle Scholar
  33. Pratt, J. W. (1968). A normal approximation for binomial, F, beta, and other common, related tail probabilities II. J. Amer. Statist. Assoc. 63 1457–1483.zbMATHCrossRefMathSciNetGoogle Scholar
  34. Reeds, J. A., III (1976). On the definition of von Mises functionals. Ph.D. dissertation, Dept. Statistics, Harvard Univ.Google Scholar
  35. Rényi, A. (1953). On the theory of order statistics. Acta Math. Acad Sci. Hungar. 4 191–227.zbMATHCrossRefMathSciNetGoogle Scholar
  36. Sebastiāo e Silva, J. (1956). Le calcul différential et intégral dans les espaces localement convexes, réels ou complexes I, II. Rend. Accad. Lincei Sci. Fis. Mat. (8) 20 743–750; 21 40–46.zbMATHGoogle Scholar
  37. Shorack, G. R. (1982). Kiefer’s theorem via the Hungarian construction. Z. Wahrsch. Verw. Gebiete 61 369–373.zbMATHCrossRefMathSciNetGoogle Scholar
  38. Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.zbMATHGoogle Scholar
  39. Taylor, S. J. (1972). Exact asymptotic estimates of Brownian path variation. Duke Math. J. 39 219–241.zbMATHCrossRefMathSciNetGoogle Scholar
  40. van der Vaart, A. (1991). Efficiency and Hadamard differentiability. Scand. J. Statist. 18 63–75.zbMATHMathSciNetGoogle Scholar
  41. van der Vaart, A., and Wellner, J. (1994). Weak Convergence and Empirical Processes. IMS, Hayward, CA. To appear.Google Scholar
  42. Vervaat, W. (1972). Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrsch. Verw. Gebiete 12 245–253.CrossRefMathSciNetGoogle Scholar
  43. von Mises, R. (1936). Les lois de probabilité pour les fonctions statistiques. Ann. Inst. H. Poincaré 6 185–212.Google Scholar
  44. von Mises, R. (1947). On the asymptotic distribution of differentiable statistical functions. Ann. Math. Statist. 18 309–348.zbMATHCrossRefGoogle Scholar
  45. Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res. 5 67–85.zbMATHCrossRefMathSciNetGoogle Scholar
  46. Wong, W H. and Severini, T. A. (1991). On maximum likelihood estimation in infinite dimensional parameter spaces. Ann. Statist. 19 603–632.zbMATHCrossRefMathSciNetGoogle Scholar
  47. Young, L. C. (1936). An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 251–282.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • R. M. Dudley
    • 1
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations