A survey of the [60′s] Monte Carlo methods [2]

May 17, 2011

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

The 24 questions asked by John Halton in the conclusion of his 1970 survey are

  1. Can we obtain a theory of convergence for random variables taking values in Fréchet spaces?
  2. Can the study of Monte Carlo estimates in separable Fréchet spaces give a theory of global approximation?
  3. When sampling functions, what constitutes a representative sample of function values?
  4. Can one apply Monte Carlo to pattern recognition?
  5. Relate Monte Carlo theory to the theory of random equations.
  6. What can be said about quasi-Monte Carlo estimates for finite-dimensional and infinite-dimensional integrals?
  7. Obtain expression, asymptotic forms or upper bounds for L² and L discrepancies of quasirandom sequences.
  8. How should one improve quasirandom sequences?
  9. How to interpret the results of statistical tests applied to pseudo- or quasirandom sequences?
  10. Can we develop a meaningful statistical theory of quasi-Monte Carlo estimates?
  11. Can existing Monte Carlo techniques be improved and applied to new classes of problems?
  12. Can the design of Monte Carlo estimators be made more systematic?
  13. How can the idea of sequential Monte Carlo be extended?
  14. Can sampling with signed probabilities be made practical?
  15. What is the best allocation effort in obtaining zeroth- and first-level estimators in algebraic problems?
  16. Examine the Monte Carlo analogues of the various matrix iterative schemes.
  17. Develop the schemes of grid refinement in continuous problems.
  18. Develop new Monte Carlo eigenvectors and eigenvalue techniques.
  19. Develop fast, reliable true canonical random generators.
  20. How is the output of a true random generator to be tested?
  21. Develop fast, efficient methods for generating arbitrary random generators.
  22. Can we really have useful general purpose pseudorandom sequences.
  23. What is the effect of the discreteness of digital computers on Monte Carlo calculations?
  24. Is there a way to estimate the accuracy of Monte Carlo estimates?

Filed under: Books, R, Statistics, University life Tagged: Fréchet, low discrepancy sequences, Monte Carlo methods, pseudo-random generator, quasi-random sequences, random generator

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