On the Fourth Root Transformation of Chi‐square

Abstract

We show that, within the family of power transformations of a Chisquare variable, the square and fourth roots m i n i z e Pearson's index of kurtosis. Two new transtormations of the fourth root, a symmetrized-truncated version and its linear combination with the square root are also studied. The first transformation shows a considerable improvement over the fourth root while the second one turns out to be even more accurate than Hilferty-Wilson's cube root transformation.

Spectrum sensing using energy detectors with performance computation capabilities

“…The motivation for this generalization lies in the statistical investigation done in [12], which analyzes the Kullback-Leibler (KL) divergence of a power-transformed chisquared random variable from a Gaussian random variable with suitable mean and variance: the results of this analysis clearly show that the KL divergence is minimized for power exponents ranging from 3 r = to 4 r = . A similar result is obtained by [13], which compares the cumulants of a power-transformed chi-squared random variable with the cumulants of a Gaussian random variable: if we match either the skewness or the kurtosis of the two random variables, we obtain integer power exponents 2 4 r ≤ ≤ . As a consequence, herein we propose to approximate a chi-squared random variable x , normalized by the number of DOF 2N , with the rth power of a Gaussian random variable, as expressed by 2 2…”

“…where ( ) r m N and ( ) r V N are the mean and the variance of the Gaussian variable, expressed respectively by [11], [13] 2016…”

“…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.…”

“…As a consequence of the reduced accuracy of the Gaussian approximation, [9] proposed an improved approximation based on the cube transformation of a Gaussian random variable. This transformation belongs to a wider class of approximations, known as power (or root) transformations [11]- [13], where the power exponent can be different from that used in [9]. This paper investigates power transformations with generic exponents in the context of low-complexity approximation of the performance of the ED.…”

“…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.…”