The Annals of Statistics

Deficiency distance between multinomial and multivariate normal experiments

Andrew V. Carter

Full-text: Open access


The deficiency distance between a multinomial and a multivariate normal experiment is bounded under a condition that the parameters are bounded away from zero. This result can be used as a key step in establishing asymptotic normal approximations to nonparametric density estimation experiments. The bound relies on the recursive construction of explicit Markov kernels that can be used to reproduce one experiment from the other. The distance is then bounded using classic local-limit bounds between binomial and normal distributions. Some extensions to other appropriate normal experiments are also presented.

Article information

Ann. Statist., Volume 30, Number 3 (2002), 708-730.

First available in Project Euclid: 6 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62B15: Theory of statistical experiments
Secondary: 62G20: Asymptotic properties 62G07: Density estimation

Deficiency distance nonparametric experiments density estimation multinomial distribution


Carter, Andrew V. Deficiency distance between multinomial and multivariate normal experiments. Ann. Statist. 30 (2002), no. 3, 708--730. doi:10.1214/aos/1028674839.

Export citation


  • BROWN, L. D. and LOW, M. G. (1996). Asy mptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398.
  • BROWN, L. D. and ZHANG, C.-H. (1998). Asy mptotic nonequivalence of nonparametric experiments when the smoothness index is 1/2. Ann. Statist. 26 279-287.
  • CARTER, A. V. (2001). Deficiency distance between multinomial and multivariate normal experiments under smoothness constraints on the parameter set. Technical Report, UCSB. Available at
  • FELLER, W. (1968). An Introduction to Probability Theory and Its Applications 1, 3rd ed. Wiley, New York.
  • GOLUBEV, G. K. and NUSSBAUM, M. (1998). Asy mptotic equivalence of spectral density and regression estimation. Technical Report 420, Weierstrass Institute, Berlin.
  • GRAMA, I. and NUSSBAUM, M. (1998). Asy mptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167-214.
  • JOHNSON, N. L. and KOTZ, S. (1969). Distributions in Statistics: Discrete Distributions. Houghton Mifflin, Boston.
  • KLEMELÄ, J. and NUSSBAUM, M. (1998). Constructive asy mptotic equivalence of density estimation and Gaussian white noise. Discussion paper 53, Sonderforschungsbereich 373, Humboldt Univ., Berlin.
  • KULLBACK, S. (1967). A lower bound for discrimination in terms of variation. IEEE Trans. Inform. Theory 13 126-127.
  • LE CAM, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist. 35 1419-1455.
  • LUCKHAUS, S. and SAUERMANN, W. (1989). Multinomial approximations for nonparametric experiments which minimize the maximal loss of Fisher information. Probab. Theory Related Fields 81 159-184.
  • MILSTEIN, G. and NUSSBAUM, M. (1998). Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 535-543.
  • MÜLLER, D. W. (1979). Asy mptotically multinomial experiments and the extension of a theorem of Wald. Z. Wahrsch. Verw. Gebiete 50 179-204.
  • NUSSBAUM, M. (1996). Asy mptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399-2430.
  • PROHOROV, YU. V. (1961). Asy mptotic behavior of the binomial distribution. Select. Transl. Math.
  • Statist. Probab. 1 87-96. Amer. Math. Soc., Providence, RI [(1953). Uspekhi. Mat. Nauk. 8 135-142 (in Russian)].
  • TORGERSEN, E. (1991). Comparison of Statistical Experiments. Cambridge Univ. Press.