On the Folded Normal Distribution

Abstract

The characteristic function of the folded normal distribution and its moment function are derived. The entropy of the folded normal distribution and the Kullback-Leibler from the normal and half normal distributions are approximated using Taylor series. The accuracy of the results are also assessed using different criteria. The maximum likelihood estimates and confidence intervals for the parameters are obtained using the asymptotic theory and bootstrap method. The coverage of the confidence intervals is also examined.

A simple algorithm for calculating values for folded normal distribution

“…This validates the observation made by Tsagris et al [16] that, for the said BMI data, the fitted folded normal converges in distribution to normal distribution.…”

“…Tsagris et al [16] have shown that the data on BMI follow folded normal distribution. Thus, from the said data, we haveμ f = 26.6847 andσ f = 4.6213.…”

A double-slit proposal for quantum annealing

“…the half-normal distribution). Thus, the expectation value is precisely the imaginary part of the folded-normal characteristic function [64]:…”

A Bayesian Decision Theory Approach for Genomic Selection

“…Let $W=\left|Y\right|$, then $F_W\left(w\right)=N^f(\mu ,{\sigma }^2)$, where $N^f$ means a folded normal distribution. Its probability density function is$\begin{array}{c}leftf\left(w\right)=\frac{1}{\sigma }\left[\phi \left(\frac{w-\mu }{\sigma }\right)+\phi \left(\frac{w+\mu }{\sigma }\right)\right]\\ =\frac{1}{\sigma \sqrt{2\pi }}\left\{\exp \left(-\frac{1}{2}{\left(\frac{w+\mu }{\sigma }\right)}^2\right)+\exp \left(-\frac{1}{2}{\left(\frac{w-\mu }{\sigma }\right)}^2\right)\right\}, w\in \left[0,\mathrm{\infty }\right).\end{array}$The expectation of $W$ is$E\left(W\right)=\mu \left[1-2\Phi (-\mu /\sigma )\right]+\sigma \sqrt{\frac{2}{\pi }}\exp \left(-{\mu }^2/2{\sigma }^2\right).$Details of folded normal distribution can be found in Tsagris et al (2014). Now, we will proceed to evaluate $CRPS\left(F_{Y_o},{\mu }_S\right)=E_F\left|Y_o-{\mu }_S\right|-\frac{1}{2}{E}_F|Y_o-Y_o^{’}|$ as follows.…”