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Asymptotic expansions for the distribution of quadratic forms in normal variables

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Abstract

Higher order asymptotic expansions for the distribution of quadratic forms in normal variables are obtained. The Cornish-Fisher inverse expansions for the percentiles of the distribution are also given. The resulting formula for a definite quadratic form guarantees accuracy almost up to fourth decimal place if the distribution is not very skew. The normalizing transformation investigated by Jensen and Solomon (1972, J. Amer. Statist. Assoc., 67, 898–902) is reconsidered based on the rate of convergence to the normal distribution.

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Author information

Authors and Affiliations

  1. The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, 106, Tokyo, Japan

    Sadanori Konishi & Naoto Niki

  2. Department of Mathematics and Statistics, Bowling Green State University, 43403, Bowling Green, OH, U.S.A.

    Arjun K. Gupta

Authors
  1. Sadanori Konishi
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  2. Naoto Niki
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  3. Arjun K. Gupta
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Additional information

Faculty of Science, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812 Japan

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Konishi, S., Niki, N. & Gupta, A.K. Asymptotic expansions for the distribution of quadratic forms in normal variables. Ann Inst Stat Math 40, 279–296 (1988). https://doi.org/10.1007/BF00052345

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  • DOI: https://doi.org/10.1007/BF00052345

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