Antithetic variate for Chi-squared distribution?

Question

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?

Your definition of an antithetic variable seems too limited to make the question answerable. But if you follow the definition used at https://stats.stackexchange.com/questions/183453, an answer should be clear. — whuber, Oct 03 '19 at 20:55
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. — Dhruv Mahajan, Oct 03 '19 at 21:30
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. — whuber, Oct 03 '19 at 21:33

score 2 · Accepted Answer · answered Oct 03 '19 at 23:42

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.

Antithetic variate for Chi-squared distribution?

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