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, 2010 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, 2010 at 17:30
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    $\begingroup$ Gamma integrals can be "controlled analytically," so what distinction are you making? $\endgroup$
    – whuber
    Nov 23, 2010 at 20:44

1 Answer 1


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, 2010 at 17:32
  • $\begingroup$ @mkolar sorry about that, now corrected $\endgroup$ Nov 23, 2010 at 17:47

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