A normal approximation for the chi-square distribution

Abstract

An accurate normal approximation for the cumulative distribution function of the chi-square distribution with n degrees of freedom is proposed. This considers a linear combination of appropriate fractional powers of chi-square. Numerical results show that the maximum absolute error associated with the new transformation is substantially lower than that found for other power transformations of a chi-square random variable for all the degrees of freedom considered (1 6 n 6 1000).

Adaptive evolution of animal proteins over development: support for the Darwin selection opportunity hypothesis of Evo-Devo

“…This approach makes the distribution of non-zero ∆lnL approximate normal distribution (Canal 2005;Roux et al 2014;Daub et al 2017). So, with fourth-root transformation, we limit the risk that the significant pathways we found be due to a few outlier genes with extremely high ∆lnL.…”

Spectrum sensing using energy detectors with performance computation capabilities

“…In order to further reduce the approximation error of the simplified performance analysis, we propose other new transformations obtained by linearly combining different power transformations [13], [14]. Basically, this second proposed approach approximates as Gaussian the linear combination of different powers of a chi-squared random variable.…”

“…This paper investigates power transformations with generic exponents in the context of low-complexity approximation of the performance of the ED. In addition, linear combinations of power transformations are also considered [13], [14], because of their potentially increased accuracy. Specifically, this paper derives new closed-form expressions for the probability of detection as a function of the probability of false alarm, of the signal-to-noise ratio (SNR) and of the sample size, for power transformations and for suitable linear combinations of power transformations.…”

Chi-squared distribution approximation for probabilistic energy equalizer implementation in impulse-radio UWB receiver

“…In the literature, there are some proposals for the calculation of the non-central chi-squared distribution [15] and the use of the normal approximation to the chi-squared distribution [9], [10], [11], but those approximations require high bit precision and are therefore too complex for digital design. An intuitive approximation can be found by considering the Remark in [11] which stands that when a variable Σ is used to approximate a variable Ω, it is equivalent to match the mean and variance of Σ and Ω.…”

“…The problem comes mainly from the energy expression, which follows a chi-squared (χ 2 ) distribution [8]. Several papers have looked for an approximation of the chi-squared distribution [9], [10]. Nevertheless the proposed approximations remain complex for digital implementations.…”