Some New Inequalities for Beta Distributions

This note provides some new inequalities and approximations for beta distributions, including exponential tail inequalities, inequalities of Hoeffding and Bernstein type, and Gaussian tail inequalities and approximations. Some of the results are extended to Gamma distributions.


Introduction
Beta distributions play an important role in statistics and probability theory (Gupta and Nadarajah, 2004). A frequent obstacle in problems involving beta distributions is the lack of analytic expressions for their distribution function, the normalized incomplete beta function. Therefore one often has to resort to inequalities and approximations.
This note provides some new inequalities for the beta distribution Beta(a, b) with parameters a, b > 0, its distribution function B a,b and its density function β a,b on [0, 1], where

Exponential tail inequalites
For p ∈ (0, 1) and x ∈ [0, 1] let In case of a ≥ 1 or b ≥ 1, this inequality can be refined as follows: Lemma 2. Suppose that a ≥ 1. With p r := (a − 1)/(a + b − 1) < p, Suppose that b ≥ 1. With p := a/(a + b − 1) > p, Remark 3. At first glance, the upper bounds in Lemma 2 seem to be weaker than the ones in Proposition 1, at least in the tail regions, because the factor a + b − 1 is strictly smaller than a + b.
But elementary algebra reveals that in case of a ≥ 1, Thus the bounds in Lemma 2 are strictly smaller than the bounds in Proposition 1. This is illustrated in Figure 1 for (a, b) = (4,8).
In case of a = 1, the asserted bounds are sharp, because B(a, b) = 1/b, p r = 0 and B a,b (x) = 1 − (1 − x) b . In case of a > 1, the ratio , so the ratio in question may be rewritten as which is clearly strictly decreasing in x ∈ (0, 1) and decreasing in x ∈ [p r , 1], the latter bound is trivial for x ≤ p r , whereas for x > p r , it becomes minimal for x o = p r .
The inequalities in Lemma 2 imply Hoeffding type exponential inequalities. Indeed, it follows from K(q, q) = 0, This yields the following inequalities: Further tail and concentration inequalities for the Beta distribution have been derived by Marchal and Arbel (2017), Bobkov and Ledoux (2019, Appendix B) and Skorski (2021). Marchal and Arbel (2017) prove that Beta(a, b) is subgaussian with a variance parameter that is the solution of an equation involving hypergeometric functions. An analytic upper bound for the variance parameter is (4(a + b + 1)) −1 , which implies the tail inequalities These bounds are weaker than the one-sided bounds in Corollary 5. For the right tails, the difference Analogously, for the left tails, the difference 10 of Bobkov and Ledoux (2019) states that for X ∼ Beta(a, b) and ε ≥ 0, This inequality is weaker than the result by Marchal and Arbel (2017). Skorski (2021) derives a Bernstein type inequality. With the parameters he shows that The next result shows that our bounds imply a stronger Bernstein type inequality. , .
Proof of Lemma 6. For symmetry reasons, it suffices to derive the upper bound for 1 − B a,b (x).
In view of Lemma 2, it suffices to show that for x ∈ [p, 1], .
To this end we use Lemma S.12 of Dümbgen and Wellner (2022) which implies that .
To simplify notation, we write m := a+b, y := x−p ∈ [0, 1−p] and δ := p−p r = (1−p)/(m−1). Then our lower bound for (a + b − 1)K(p r , x) reads and we want to show that this is greater than or equal to . Furthermore, and in case of 0 < p ≤ 1/2, the right hand side is not larger than p(1 − p) + (2/3)(1 − 2p)y. This proves our assertion in case of 0 < p ≤ 1/2 already, and it remains to treat the case 1/2 < p < 1.
To this end, we have to show that the ratio of (1) and (m + 1)y 2 is not smaller than the ratio of (2) and p(1 − p) + (2/3)(1 − 2p)y. This assertion is equivalent to the inequality With λ := (m − 1) −1 and z := y/(1 − p) ∈ [0, 1], the latter inequality is equivalent to Since the left-hand side is strictly positive and 3p + 2(1 − 2p)z ≥ 3p + 2(1 − 2p) = 2 − p, it suffices to show that If z ≤ 2/3, the second summand on the left-hand side is at least Thus it suffices to consider z ≥ 2/3. It follows from Consequently, it suffices to verify that The second summand on the left-hand side equals an increasing function of λ > 0 for any fixed z ∈ (0, 1]. Consequently, inequality (4) would be a consequence of Since 1/z − 1 = (1 − z)/z ≥ 1 − z, it even suffices to show that The minimiser of the left-hand side, as a function of z ∈ R, is given by If p ≤ 8/9, then z o ≥ 1, so it suffices to verify (5) for z = 1. Indeed, But this inequality is equivalent to (6) (23/4)p + 5/p ≤ 11.

Extensions to gamma distributions
If X ∼ Beta(a, b), then bX has approximatly a gamma distribution Gamma(a) with shape parameter a > 0 and scale parameter one. Denoting the distribution function of Gamma(a) with G a , the bounds in Lemma 2 and Remark 4 lead to the following bounds: Lemma 10. For arbitrary a > 0, G a (x) ≤ x a e −x a a e −a for 0 ≤ x ≤ a.
For a ≥ 1, Instead of the approximation argument, one could verify directly, that G a (x)/[x a e −x ] is monotone increasing in x > 0, and that in case of a ≥ 1, [1−G a (x)]/[x a−1 e −x ] is monotone decreasing in x > 0. Note also that in case of a ≥ 1, the median of Gamma(a) is contained in (a − 1, a), see Groeneveld and Meeden (1977).