Exponential Tail Bounds for Chisquared Random Variables

The paper derives some exponential tail bounds for central and non-central chisquared random variables. The bounds are simple and can easily be applied in statistical analysis. Especially relevant are the tail bounds for non-central chisquares, which are different from some of the other exponential bounds available in the literature, for example the one given in [1].


Introduction
The objective of this note is to derive some exponential tail bounds for chisquared random variables. The bounds are non-asymptotic, but they can be used very successfully for asymptotic derivations as well. As a corollary, one can get tail bounds for F-statistics as well. Also, I show how some exact moderate deviation [4] inequalities can be obtained as special cases of these tail bounds.
The chisquared random variables are special cases of sub-exponential random variables. We examine when the bounds obtained here are sharper than the ones that use only the sub-exponentiality of chisquares.
The outline of the next two sections is as follows. Exponential tail bounds for central chisquares are given in Sect. 2. Corresponding bounds for non-central chisquares are given in Sect. 3.

Central Chisquare
We begin with an upper tail bound for central chisquares. The following theorem is proved.
Proof From Markov's inequality in its exponential form (see for example, [2] or [3], one gets Let This proves the theorem. Suppose a = p + c . Then an equivalent way of writing the above result is By the inequality a weaker version of (2) is given by It may be noted that a chisquare random variable is a special case of a sub-exponential random variable. There are several equivalent definitions of sub-exponential random variables. The one we find convenient is given as follows (see [5], p 26). (1) . 8p)) . The inequality given in (3) is sharper than the last one when 0 < c < p . Moreover, as p → ∞ , it follows from (2) that P(X − p > c) ∼ exp(−c 2 ∕(4p)) , while sub-exponentiality continues to yield the same upper bound exp(−c 2 ∕(8p)).
The next inequality is related to the lower tail of a chisquared random variable. The following theorem is proved.
The second inequality is an easy consequence of expansion of a logarithmic function. To prove the first inequality, we begin with Similar as before, let (4), one gets the inequality This proves the theorem.
The exact upper bound given in the rightmost side of Theorem 2 is stronger than the similar sub-exponential bound exp − c 2 8p . Moreover, since p + c > p , it is possible to combine (3) with Theorem 2 to get the inequality Since 2 p ∕p is the average of p iid 2 1 random variables, each with mean 1 and variance 2, the central limit theorem leads to .
It is possible to use Theorems 1 and 2 to obtain some crude tail bounds for the F-statistic as well. To see this, suppose rXx and Y are two independent chisquared random variables with respective degrees of freedom m 1 and m 2 . I write F = It is well-known that asymptotically as m 2 → ∞ , the F-statistic reduces to a chisquare statistic divided by its degrees of freedom. This is also reflected in (6). In particular, we get the inequality 3 Non-Central Chisquare I find in this section, upper and lower tail bounds for non-central chisquare. These upper bounds are not the sharpest ones that one might get, but they are simple enough for potential use in statistics. I begin with the upper bound.
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