Let $X \sim \chi^2_k$ be a chi-squared distributed random variable with $k$ degrees of freedom. What are the sharpest known bounds for the following probabilities

$$ \mathbb{P}[X > t] \leq 1 - \delta_1(t, k) $$


$$ \mathbb{P}[X < z] \leq 1 - \delta_2(z, k) $$

where $\delta_1$ and $\delta_2$ are some functions. Pointers to relevant papers would be appreciated.

  • 3
    $\begingroup$ If you define the deltas to be complementary incomplete gamma functions, you obtain exact equalities. Obviously these are the sharpest possible bounds! I guess the point of this question is that your calculator doesn't compute incomplete gammas and you're looking for an approximation, but that still omits essential information: how can we answer this question until we know just what your calculator can compute? $\endgroup$
    – whuber
    Nov 23 '10 at 16:04
  • $\begingroup$ I am not interested in computing an upper bound, but obtaining something that I can control analytically. The answer that robin has provided is exactly what I was looking for. The question is, are there more precise bounds than those provided by Massart and Laurent? $\endgroup$
    – mkolar
    Nov 23 '10 at 17:30
  • 2
    $\begingroup$ Gamma integrals can be "controlled analytically," so what distinction are you making? $\endgroup$
    – whuber
    Nov 23 '10 at 20:44

The Sharpest bound I know is that of Massart and Laurent Lemma 1 p1325.

A corollary of their bound is:

$$P(X-k\geq 2\sqrt{kx}+2x)\leq \exp (-x) $$

$$P(k-X\geq 2\sqrt{kx})\leq \exp (-x) $$

  • 1
    $\begingroup$ the second inequality seems to be incorrect or am I missing something? $\endgroup$
    – mkolar
    Nov 23 '10 at 17:32
  • $\begingroup$ @mkolar sorry about that, now corrected $\endgroup$ Nov 23 '10 at 17:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.