Brazilian Journal of Probability and Statistics

Contiguity and irreconcilable nonstandard asymptotics of statistical tests

Pranab K. Sen and Antonio C. Pedroso-de-Lima

Full-text: Open access


Wald-type test statistics based on asymptotically normally distributed estimators (not necessarily maximum likelihood estimation or best asymptotically normal) provides an easy access to have tests for statistical hypotheses, far beyond the parametric paradigms. The methodological perspectives rest on a basic consistent asymptotic normal (CAN) condition which is interrelated to the well-known local asymptotic normality (LAN) condition. Contiguity of probability measures facilitates the C(L)AN condition in a relatively easier way. For many regular families of distributions, when statistical hypotheses do not involve nonstandard constraints, verification of contiguity of probability measures is facilitated by the well-known LeCam’s First Lemma [see Hájek, Šidák and Sen Theory of Rank Tests (1999), Chapter 7]. For nonregular families, though contiguity may hold under different setups, CAN estimators are not fully exploitable in the Wald type testing theory. This simple feature is illustrated by a two-parameter exponential model. Guided by this simple example, mixture of distributions are appraised in the context of Wald-type tests and the so-called χ̅2- and -test theory is thoroughly appraised. A general result on counter examples is presented in detail.

Article information

Braz. J. Probab. Stat., Volume 25, Number 3 (2011), 444-470.

First available in Project Euclid: 22 August 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

CAN statistics exponential family of densities Hellinger-distance LA(M)N likelihood ratio maximum likelihood estimator nonparametrics Rao’s score statistics regular family of estimators resampling plans


Sen, Pranab K.; Pedroso-de-Lima, Antonio C. Contiguity and irreconcilable nonstandard asymptotics of statistical tests. Braz. J. Probab. Stat. 25 (2011), no. 3, 444--470. doi:10.1214/11-BJPS156.

Export citation


  • Bahadur, R. R. (1966). A note on quantiles in large samples. Ann. Math. Statist. 37, 577–580.
  • Chernoff, H. and Savage, I. R. (1958). Asymptotic normality and efficiency of certain nonparametric test statistics. Ann. Math. Statist. 29, 972–994.
  • Cox, D. R. (1972). Regression models and life-tables. J. Roy. Statist. Soc. Ser. B 34, 187–220.
  • Hájek, J. (1962). Asymptotically most powerful rank-order tests. Ann. Math. Statist. 33, 1124–1147.
  • Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39, 325–346.
  • Hájek, J. (1970). A characterization of limiting distributions of regular estimates. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 14, 323–330.
  • Hájek, J. (1972). Local asymptotic minimax and admissibility in estimation. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of Statistics 175–194. Berkeley: Univ. California Press.
  • Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests, 2nd ed. San Diego, CA: Academic Press Inc.
  • Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293–325.
  • Inagaki, N. (1970). On the limiting distribution of a sequence of estimators with uniformity property. Ann. Inst. Math. Statist. 22, 1–13.
  • Inagaki, N. (1973). Asymptotic relations between the likelihood estimating function and the maximum likelihood estimator. Inst. Statist. Math. 25, 1–26.
  • Jurečková, J. and Sen, P. K. (1996). Robust Statistical Procedures. Asymptotics and Interrelations. New York: John Wiley & Sons, Inc.
  • Kendall, M. and Stuart, A. (1977). The Advanced Theory of Statistics, Vol. 1, Distribution Theory, 4th ed. New York: Macmillan Publishing Co., Inc.
  • LeCam, L. (1960). Locally asymptotically normal families of distributions. Certain approximations to families of distributions and their use in the theory of estimation and testing hypotheses Convergence of random processes and limit theorems in probability theory. Univ. California Publ. Statist. 3, 37–98.
  • LeCam, L. (1979). On a theorem of J. Hájek. In Contributions to Statistics 119–135. Dordrecht: Reidel.
  • Lehmann, E. L. (1953). The power of rank tests. Ann. Math. Statist. 24, 119–135.
  • Perlman, M. D. (1969). One-sided testing problems in multivariate analysis. Ann. Math. Statist. 40, 549–567.
  • Pinheiro, A., Sen, P. K. and Pinheiro, H. P. (2011). A class of asymptotically normal degenerate quasi U-Statistics. Ann. Inst. Statist. Math. DOI 10.1007/s10463-010-0271-z. To appear.
  • Sen, P. K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference. New York: John Wiley & Sons Inc.
  • Sen, P. K. (2000). The Hájek convolution theorem and empirical Bayes estimation: Parametrics, semiparametrics and nonparametrics analysis. J. Statist. Plann. Inference 91, 541–556.
  • Sen, P. K., Singer, J. M. and Pedroso-de-Lima, A. C. (2010). From Finite Sample to Asymptotic Methods in Statistics. New York: Cambridge Univ. Press.
  • Silvapulle, M. J. and Sen, P. K. (2005). Constrained Statistical Inference. Inequality, Order, and Shape Restrictions. New York: Wiley-Interscience.