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Moments of the Noncentral Chi Distribution


Explicit closed form formulas for the integer moments of the noncentral chi distribution are given. The mean of the noncentral chi distribution is the average distance between a fixed point and a random vector with a standard multivariate normal distribution. Previous published expressions for the mean use the hypergeometric function or infinite series. In the case where the dimension, d, is even, the formula presented here can be expressed using polynomials of order (d − 1)/2, the square root function, and standard normal density and distribution functions. In the case where the dimension is odd, the formula involves two Bessel functions of the first kind. Calculation of the other positive integer moments is also discussed.

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Correspondence to John Lawrence.

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Lawrence, J. Moments of the Noncentral Chi Distribution. Sankhya A (2021).

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  • Normal distribution
  • Goodness-of-fit testing
  • Hypergeometric function
  • Bessel function