Computational Statistics

, Volume 24, Issue 2, pp 333–344 | Cite as

Numerical approximation of conditional asymptotic variances using Monte Carlo simulation

  • Tak K. MakEmail author
  • Fassil Nebebe
Original Paper


We consider in this paper the use of Monte Carlo simulation to numerically approximate the asymptotic variance of an estimator of a population parameter. When the variance of an estimator does not exist in finite samples, the variance of its limiting distribution is often used for inferences. However, in this case, the numerical approximation of asymptotic variances is less straightforward, unless their analytical derivation is mathematically tractable. The method proposed does not assume the existence of variance in finite samples. If finite sample variance does exist, it provides a more efficient approximation than the one based on the convergence of finite sample variances. Furthermore, the results obtained will be potentially useful in evaluating and comparing different estimation procedures based on their asymptotic variances for various types of distributions. The method is also applicable in surveys where the sample size required to achieve a fixed margin of error is based on the asymptotic variance of the estimator. The proposed method can be routinely applied and alleviates the complex theoretical treatment usually associated with the analytical derivation of the asymptotic variance of an estimator which is often managed on a case by case basis. This is particularly appealing in view of the advance of modern computing technology. The proposed numerical approximation is based on the variances of a certain truncated statistic for two selected sample sizes, using a Richardson extrapolation type formulation. The variances of the truncated statistic for the two sample sizes are computed based on Monte Carlo simulations, and the theory for optimizing the computing resources is also given. The accuracy of the proposed method is numerically demonstrated in a classical errors-in-variables model where analytical results are available for the purpose of comparisons.


Bootstrap variance estimator Existence of variances Sample size determination Statistical computing 


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  1. Agresti A (1990) Categorical data analysis. Wiley, New YorkzbMATHGoogle Scholar
  2. Bickel PJ, Yahav JA (1988) Richardson extrapolation and the bootstrap. J Am Stat Assoc 83:387–393. doi: 10.2307/2288854 Google Scholar
  3. Booth JG, Sarkar S (1998) Monte Carlo approximation of bootstrap variances. Am Stat 52:354–357. doi: 10.2307/2685441 Google Scholar
  4. Booth JG, Hobert JP, Wolfgang J (2001) A survey of Monte Carlo algorithms for maximizing the likelihood of a two stage hierarchical model. Stat Model 1:333–349. doi: 10.1191/147108201128249 zbMATHCrossRefGoogle Scholar
  5. Browne RH (1995) On the use of a pilot sample for sample size determination. Stat Med 14:1933–1940. doi: 10.1002/sim.4780141709 CrossRefGoogle Scholar
  6. Chan LK, Mak TK (1979) Maximum likelihood estimation of a linear structural relationship with replication. J R Stat Soc B 41: 263–268zbMATHMathSciNetGoogle Scholar
  7. Chan KS, Ledolter J (1995) Monte Carlo EM estimation for time series models involving counts. J Am Stat Assoc 90:242–252. doi: 10.2307/2291149 Google Scholar
  8. Cheng CL, Van Ness J (1997) Structural and functional models revisited. In: Van Haffet S (ed) Proceedings of the 2nd international workshop on total least squares techniques and errors-in-variables modeling. SIAM, Philadelphia, pp 37–50Google Scholar
  9. Cox DR, Hinkley DV (1974) Theoretical statistics. Chapman and Hall, LondonzbMATHGoogle Scholar
  10. Fuller WA (1987) Measurement error models. Wiley, New YorkzbMATHCrossRefGoogle Scholar
  11. Karim MR, Zeger SL (1992) Generalized linear models with random effects; salamander mating revisited. Biometrics 48:631–644. doi: 10.2307/2532317 CrossRefGoogle Scholar
  12. Mak TK (2004) Estimating variances for all sample sizes by the bootstrap. Comput Stat Data Anal 46:459–467. doi: 10.1016/j.csda.2003.08.004 zbMATHCrossRefMathSciNetGoogle Scholar
  13. Moran PAP (1971) Estimating structural and functional relationships. J Multivar Anal 1:232–255. doi: 10.1016/0047-259X(71)90013-3 zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Decision Sciences and M.I.S.JMSB, Concordia UniversityMontrealCanada

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