antithetic variable

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A variable used in a variance reduction technique for simulation. It refers to the use of simulated values that are not independent but are negatively correlated.

Suppose, for example, that we wish to estimate the expectation of the random variable X, and that we have a simulation procedure using the function g of the pseudo-random numbers u1, u2,…, un (0≤uj≤1, for all j):

x=g(u1, u2,…, un).

The antithetic variable procedure makes use of the antitheses of the random numbers, namely (1−u1), (1−u2),…, (1−un), to form x′ given by

x′=g(1−u1, 1−u2,…, 1−un).

Write X′′ as the corresponding random variable. It can be shown that the variance of ½(X+X′) is never greater than half the variance of X. There is therefore a gain in precision as well as an increase in efficiency, since each pseudo-random number is used twice.

Subjects: Probability and Statistics.

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