## the strange occurrence of the one bump

**W**hen answering an X validated question on running an accept-reject algorithm for the Gamma distribution by using a mixture of Beta and drifted (bt 1) Exponential distributions, I came across the above glitch in the fit of my 10⁷ simulated sample to the target, apparently displaying a wrong proportion of simulations above (or below) one.

a=.9 g<-function(T){ x=rexp(T) v=rt(T,1)<0 x=c(1+x[v],exp(-x/a)[!v]) x[runif(T)<x^a/x/exp(x)/((x>1)*exp(1-x)+a*(x<1)*x^a/x)*a]}

It took me a while to spot the issue, namely that the output of

z=g(T) while(sum(!!z)<T)z=c(z,g(T)) z[1:T]

was favouring simulations from the drifted exponential by truncating. Permuting the elements of z before returning solved the issue (as shown below for a=½)!

June 9, 2020 at 8:15 am

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