Exact convergence rate and leading term in central limit theorem for student's t statistic

Peter Hall*, Qiying Wang

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    17 Citations (Scopus)

    Abstract

    The leading term in the normal approximation to the distribution of Student's t statistic is derived in a general setting, with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law. The form of the leading term is shown to have its origin in the way in which extreme data influence properties of the Studentized sum. The leading-term approximation is used to give the exact rate of convergence in the central limit theorem up to order n -1/2, where n denotes sample size. It is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points. Moreover, the exact rate is identical to that for the non-Studentized sum when the latter is normalized for scale using a truncated form of variance, but when the corresponding truncated centering constant is omitted. Examples of characterizations of convergence rates are also given. It is shown that, in some instances, their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points.

    Original languageEnglish
    Pages (from-to)1419-1437
    Number of pages19
    JournalAnnals of Probability
    Volume32
    Issue number2
    DOIs
    Publication statusPublished - Apr 2004

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