Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals

Abstract

The cumulative distribution function of the non-central chi-square distribution χn′2(λ) of n degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin.

Keywords

• non-central <i>χ</i>2 distribution
• second mean-value theorem for definite integrals
• modified Bessel function of the first kind
• Marcum <i>Q</i>–function
• lower incomplete gamma function