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2 results

  • 10 - Asymptotic theory

  • from Part B - Estimation And Inference
  • Karim M. Abadir, Imperial College London, Risto D. H. Heijmans, Jan R. Magnus, Vrije Universiteit, Amsterdam
  • Book:
    Statistics
    Published online:
    11 June 2020
    Print publication:
    08 November 2018, pp 371-436

  • Summary

    It is clear from the previous chapter that sample statistics have distributions depending, in general, on the sample size n. The natural question one may ask is how these distributions are affected by varying n deterministically and, in particular, as n increases. It turns out that one can get common limiting results, for a variety of settings, by analyzing what happens as n tends to infinity; this is large-sample asymptotic theory.We study convergence of distributions and of characteristic functions. For variates that converge to a single value, we study convergence in probability and in moments, and almost-sure convergence with its connection to events that happen infinitely-often. The speed of convergence and orders of magnitude are covered, as are law of large numbers (LLNs) weak and strong, limiting distributions such as central limit theorems (CLTs), stable limit theorems (SLTs), the generalized extreme-value theorem with its three special cases, and a law of iterated logarithms.

  • BERNOULLI'S LAW OF LARGE NUMBERS

  • Erwin Bolthausen, Mario V. Wüthrich
  • Journal:
    ASTIN Bulletin: The Journal of the IAA / Volume 43 / Issue 2 / May 2013
    Published online by Cambridge University Press:
    18 June 2013, pp. 73-79
    Print publication:
    May 2013
    • Article
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  • This year we celebrate the 300th anniversary of Jakob Bernoulli's path-breaking work Ars conjectandi, which appeared in 1713, eight years after his death. In Part IV of his masterpiece, Bernoulli proves the law of large numbers which is one of the fundamental theorems in probability theory, statistics and actuarial science. We review and comment on his original proof.