## wrapped Normal distribution

Posted in Books, R, Statistics with tags , , , , , on April 14, 2020 by xi'an

One version of the wrapped Normal distribution on (0,1) is expressed as a sum of Normal distributions with means shifted by all relative integers

$\psi(x;\mu,\sigma)=\sum_{k\in\mathbb Z}\varphi(x;\mu+k,\sigma)\mathbb I_{(0,1)}(x)$

which, while a parameterised density, has imho no particular statistical appeal over the use of other series. It was nonetheless the centre of a series of questions on X validated in the past weeks. Curiously used as the basis of a random walk type move over the unit cube along with a uniform component. Simulating from this distribution is easily done when seeing it as an infinite mixture of truncated Normal distributions, since the weights are easily computed

$\sum_{k\in\mathbb Z}\overbrace{[\Phi_\sigma(1-\mu-k)-\Phi_\sigma(-\mu-k)]}^{p_k(\mu,\sigma)}\times$

$\dfrac{\varphi_\sigma(x-\mu-k)\mathbb I_{(0,1)}(y)}{\Phi_\sigma(1-\mu-k)-\Phi_\sigma(-\mu-k)}$

Hence coding simulations as

wrap<-function(x, mu, sig){
ter = trunc(5*sig + 1)
return(sum(dnorm(x + (-ter):ter, mu, sig)))}
siw = function(N=1e4,beta=.5,mu,sig){
unz = (runif(N)<beta)
ter = trunc(5*sig + 1)
qrbz = diff(prbz<-pnorm(-mu + (-ter):ter, sd=sig))
ndx = sample((-ter+1):ter,N,rep=TRUE,pr=qrbz)+ter
z = sig*qnorm(prbz[ndx]+runif(N)*qrbz[ndx])-ndx+mu+ter+1
return(c(runif(sum(unz)),z[!unz]))}


and checking that the harmonic mean estimator was functioning for this density, predictably since it is lower bounded on (0,1). The prolix originator of the question was also wondering at the mean of the wrapped Normal distribution, which I derived as (predictably)

$\mu+\sum_{k\in\mathbb Z} kp_k(x,\mu,\sigma)$

but could not simplify any further except for x=0,½,1, when it is ½. A simulated evaluation of the mean as a function of μ shows a vaguely sinusoidal pattern, also predictably periodic and unsurprisingly antisymmetric, and apparently independent of the scale parameter σ…