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[email protected]
Correction to:
On the Chambers–Mallows–Stuck Method
for Simulating Skewed Stable Random Variables.
Rafal Weron
The Hugo Steinhaus Center for Stochastic Methods,
Technical University of Wroclaw,
50–370 Wroclaw, Wyspianskiego 27, Poland.
E–mail: [email protected]
In the paper Weron (1996), I gave a proof to the equality in law of a skewed stable
variable and a nonlinear transformation of two independent uniform and exponential
variables. The Chambers et al. (1976) method of computer generation of a skewed stable
random variable is based on this equality.
The lack of an explicit proof of this formula has led to some inaccuracies in the
literature. Unfortunately an error crept into my calculations as well.
Equation (3.3) of Theorem 3.1 (in Weron (1996)) gives a formula for a stable random
variable X ∼ S1 (σ2 = 1, β, 0) (in representation (2.5)). To obtain formula (3.9), which
is supposed to be S1 (σ = 1, β, 0) (in representation (2.1)), I then have merely multiplied
the expression in (3.3) by π2 , forgetting that for α = 1, this gives a random variable with
σ = 1 and µ = − π2 β log π2 (see the last formula in Weron (1996)). Adding π2 β log π2 back
in to obtain a random variable with σ = 1 and µ = 0 effectively places a π2 in front of the
W inside the log, which is exactly equation (2.4) of Chambers et al. (1976).
In view of the above, formula (3.9) should read as follows:
2 π
( + βV ) tan V − β log
X=
π 2
π
W cos V
2
π
+ βV
2
.
Acknowledgements
I am indebted to J. Huston McCulloch for careful reading of the paper and finding
the error.
References
Chambers, J.M., Mallows, C.L. and Stuck, B.W. (1976), A method for simulating
stable random variables, J. Amer. Statist. Assoc. 71, 340–344.
Weron, R. (1996), On the Chambers-Mallows-Stuck method for simulating skewed
stable random variables, Statist. Probab. Lett. 28, 165-171