• Bernoulli
  • Volume 17, Number 2 (2011), 592-608.

Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators

Larry Goldstein, Yosef Rinott, and Marco Scarsini

Full-text: Open access


We compare estimators of the (essential) supremum and the integral of a function f defined on a measurable space when f may be observed at a sample of points in its domain, possibly with error. The estimators compared vary in their levels of stratification of the domain, with the result that more refined stratification is better with respect to different criteria. The emphasis is on criteria related to stochastic orders. For example, rather than compare estimators of the integral of f by their variances (for unbiased estimators), or mean square error, we attempt the stronger comparison of convex order when possible. For the supremum, the criterion is based on the stochastic order of estimators.

Article information

Bernoulli, Volume 17, Number 2 (2011), 592-608.

First available in Project Euclid: 5 April 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

convex loss convex order majorization stochastic order stratified sampling


Goldstein, Larry; Rinott, Yosef; Scarsini, Marco. Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators. Bernoulli 17 (2011), no. 2, 592--608. doi:10.3150/10-BEJ295.

Export citation


  • [1] Bai, S.K. and Durairajan, T.M. (1997). Optimal equivariant estimator with respect to convex loss function. J. Statist. Plann. Inference 64 283–295.
  • [2] Berger, J.O. (1976). Admissibility results for generalized Bayes estimators of coordinates of a location vector. Ann. Statist. 4 334–356.
  • [3] Blackwell, D. (1951). Comparison of experiments. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950 93–102. Berkeley and Los Angeles, CA: California Univ. Press.
  • [4] Blackwell, D. (1953). Equivalent comparisons of experiments. Ann. Math. Statist. 24 265–272.
  • [5] Eberl Jr., W. (1984). On unbiased estimation with convex loss functions. Statist. Decisions 1984 177–192.
  • [6] Ermakov, S.M., Zhiglyavskiĭ, A.A. and Kondratovich, M.V. (1988). Reduction of a problem of random estimation of an extremum of a function. Dokl. Akad. Nauk SSSR 302 796–798.
  • [7] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer.
  • [8] Goldstein, L., Rinott, Y. and Scarsini, M. (2010). Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators. Technical report. Available at arXiv:1005.5414v1 [math.ST].
  • [9] Karlin, S. and Novikoff, A. (1963). Generalized convex inequalities. Pacific J. Math. 13 1251–1279.
  • [10] Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10 467–498.
  • [11] Kondratovich, M. and Zhigljavsky, A. (1998). Comparison of independent and stratified sampling schemes in problems of global optimization. In Monte Carlo and Quasi-Monte Carlo Methods 1996 (Salzburg) 292–299. New York: Springer.
  • [12] Kozek, A. (1977). Efficiency and Cramér–Rao type inequalities for convex loss functions. J. Multivariate Anal. 7 89–106.
  • [13] Laycock, P.J. (1972). Convex loss applied to design in regression problems. J. Roy. Statist. Soc. Ser. B 34 148–170, 170–186.
  • [14] Laycock, P.J. and Silvey, S.D. (1968). Optimal designs in regression problems with a general convex loss function. Biometrika 55 53–66.
  • [15] Lin, P.E. and Mousa, A. (1982). Proper Bayes minimax estimators for a multivariate normal mean with unknown common variance under a convex loss function. Ann. Inst. Statist. Math. 34 441–456.
  • [16] Marshall, A.W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. New York: Academic Press.
  • [17] Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. Chichester: Wiley.
  • [18] Novak, E. (1988). Deterministic and Stochastic Error Bounds in Numerical Analysis. Berlin: Springer.
  • [19] Papageorgiou, A. (1993). Integration of monotone functions of several variables. J. Complexity 9 252–268.
  • [20] Petropoulos, C. and Kourouklis, S. (2001). Estimation of an exponential quantile under a general loss and an alternative estimator under quadratic loss. Ann. Inst. Statist. Math. 53 746–759.
  • [21] Rosenthal, J.S. (2006). A First Look at Rigorous Probability Theory, 2nd ed. Hackensack, NJ: World Scientific Publishing.
  • [22] Ross, S.M. and Schechner, Z. (1984). Some reliability applications of the variability ordering. Oper. Res. 32 679–687.
  • [23] Shaked, M. (1982). A general theory of some positive dependence notions. J. Multivariate Anal. 12 199–218.
  • [24] Shaked, M. and Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer.
  • [25] Zhigljavsky, A. and Žilinskas, A. (2008). Stochastic Global Optimization. New York: Springer.
  • [26] Zhigljavsky, A.A. and Chekmasov, M.V. (1996). Comparison of independent, stratified and random covering sample schemes in optimization problems. Math. Comput. Modelling 23 97–110.