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I am using antithetic sampling for variance reduction. I know for standard normal $z$, it's antithetic variate is $-z$ ($1-U$ for uniform etc). But I cannot figure out what would be antithetic variate for samples from chi-squared distribution since I don't find symmetry.

Also for $t$ distribution, would it be $-t$? Since it is symmetric around the origin?

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    $\begingroup$ Your definition of an antithetic variable seems too limited to make the question answerable. But if you follow the definition used at stats.stackexchange.com/questions/183453, an answer should be clear. $\endgroup$
    – whuber ♦
    Commented Oct 3, 2019 at 20:55
  • $\begingroup$ I understand that I can get a general antithetic variable for any distribution D following the steps : Step1) Generate u from U(0,1) Step2) Get Value from inverse cdf Dinv(u) Step3) Get value from inverse cdf Dinv(1-u) and then take average. But i wanted to know if there is any definite closed form answer for chi-squared like there is for uniform and normal. $\endgroup$
    – Dhruv Mahajan
    Commented Oct 3, 2019 at 21:30
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    $\begingroup$ Yes: it's called the incomplete gamma function. Statisticians know it better as the chi-squared quantile function or inverse chi-squared cumulative distribution. Any general-purpose statistical software can compute it. $\endgroup$
    – whuber ♦
    Commented Oct 3, 2019 at 21:33

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Using the definition of antithetic variable from Calculating integral with antithetic variables, that is, $x_i^*$ is the antithetic variable for $x_i$ when $F(x_i^*) + F(x_i)=1$, $F$ the cdf, the answer should be clear. For simulation of the chi-squared random variable $X$ use the inversion method, see How does the inverse transform method work?.

So the algorithm is, with $F$ the chi-squared cdf:

  • $U\sim \mathcal{U}(0,1)$
  • $V=1-U$
  • $X=F^{-1}(U)$
  • $X^* = F^{-1}(V)$ is the antithetic variable.
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