## simulating a sum of Uniforms

When considering the distribution of the sum (or average) of N Uniform variates, called either Irwin-Hall for the sum or Bates for the average, simulating the N uniforms then adding them shows a linear cost in N. The density of the resulting variate is well-known,

$f_X(x;N)=\dfrac{1}{2(N-1)!}\sum_{k=0}^N (-1)^k{N \choose k} (x-k)^{N-1}\text{sign}(x-k)$

but similarly is of order N. Furthermore, controlling the terms in the alternating sum may prove delicate, as shown by the R function unifed::dirwin.hall() whose code

for (k in 0:floor(x)) ret1 <- ret1 + (-1)^k * choose(n, k) *
(x - k)^(n - 1)


quickly becomes unreliable (although I managed an easy fix by using logs and a reference value of the magnitude of the terms in the summation). There is however a quick solution provided by [of course!] Devroye (NURVG, Section XIV.3, p.708), using the fact that the characteristic function of the Irwin-Hall distribution [for Uniforms over (-1,1)] is quite straightforward

$\Phi_N(t) = [\sin(t)/t]^N$

which means the density can be bounded from above and results in an algorithm (NURVG, Section XIV.3, p.714) with complexity at most N to the power 5/8, if not clearly spelled out in the book. Obviously, it can be objected that for N large enough, like N=20, the difference between the true distribution and the CLT approximation is quite negligible (reminding me of my early simulating days where generating a Normal was done by averaging a dozen uniforms and properly rescaling!). But this is not an exact approach and the correction proves too costly. As shown by Section XIV.4 on the simulation of sums in NURVG. So… the game is afoot!

### 2 Responses to “simulating a sum of Uniforms”

1. william.cipolli Says:

How did you do the quick fix with the log? I tried the obvious exp(log(.)) of the terms but that seemed to make things worse?

• This is such a long time ago that I cannot remember… Starting from scratch, I would compute the series one term at a time and keep track of the largest so far as reference value.

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