E = our approximated quantile using
p1 = .136 + .235q + q^2 + .0066min(n,10) - .05max(a,1).
S = simulated quantile from 1M simulations each of n sums.
R = the ratio E/S,
And as always b=1.
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The approximation (E) of the q-th quantile (e.g. q=.05) is made as follows:
(1) p1 is computed as above.
(2) p2 = q/p1.
(3) y = b / (1-p2^(1/n))^(1/a)
(4) mu = (a*b^a*y^(1-a)-a*b)/(1-a)/(1-b^a*y^(-a))
(5) s = sqrt((a*b^a*y^(2-a)-a*b^2)/(2-a)/(1-b^a*y^(-a))-mu^2) 
(6) E = s*qnorm(p1)*sqrt(n) + n*mu

where qnorm(p1) means the p1-th quantile of the standard normal.
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Summary of the results for this choice of p1:
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I chose a (i.e. alpha) = 0.5, 0.7, 0.9, 0.99, 1.01, 1.1, 1.3, 1.5, 1.7, 1.9, 1.99;  
q = 0.001, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.08, 0.1, 0.2;
n = 2, 5, 10, 20, 30, 50.
So, a total of 11*10*6 = 660 different combinations.
For most, the approximation (E) was within 1% of the simulated quantile (S).
In fact, whenever n=10 or more, the approximation did well, with just a few 
exceptions.
Also, whenever q was 0.05 or less, the approximation did well, again with just
a few exceptions and these all were when n=2.
For every value of a, the approximation started to have problems for n=2 when 
q=0.05 or so, and as q increased this problem increased. By the time q=0.1,
the problem was very severe for n=2 and moderately severe for n=5. 
However, even when q=.2 the approximation worked well for n=10 or more.
There were no special problems for values of a such as 0.99, 1.01, or 1.99. 
However, there were some problems for a=.5 (the smallest value I tried): for 
a=.5 and q=.1 or q=.2, there were problems for all n, and for a=.5 and n=2 or 5, 
there were big problems for most choices of q.