Various New Inequalities for Beta Distributions

This note provides some new inequalities and approximations for beta distributions, including tail inequalities, exponential inequalities of Hoeffding and Bernstein type, Gaussian inequalities and approximations.


Introduction
Beta distributions play an important role in statistics and probability theory (Gupta and Nadarajah, 2004), and they occur in various scientific fields (Skorski, 2023). 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 resorts to inequalities and approximations, as, for example, in the proofs of Dimitriadis  This paper provides some new inequalities for the beta distribution Beta(a, b) with parameters a, b > 0, its distribution function B a,b , survival functionB a,b = 1 − B a,b and density function β a,b on [0, 1]. The latter is given by , and Γ(·) denotes the gamma function. In Section 2, we refine the lower and upper bounds for B a,b andB a,b by Segura (2016) which are particularly accurate in the tails of Beta(a, b). As a by-product we obtain refinements of Segura's (2014) bounds for the gamma distribution and survival functions. In Section 3 we present new exponential inequalities which are stronger than previously known inequalities of Dümbgen (1998), Marchal and Arbel (2017) or Skorski (2023) for a wide range of parameters (a, b). Section 4 presents inequalities for B a,b andB a,b in terms of Gaussian distribution functions.
Finally, Section 5 discusses the approximation of the symmetric distribution β a,a by Gaussian densities with mean 1/2 in the spirit of Dümbgen et al. (2021). Most proofs are deferred to Section 6.
Segura (2016) showed that for x ∈ [0, 1), where c a,b := (a + b)/(a + 1). Numerical examples reveal that these inequalities are rather accurate if x ≪ p. Indeed, both bounds are equal to 1 for x = 0, and the lower bound has derivative −b + c a,b = Q ′ a,b (0) for x = 0. The upper bound is less accurate, because its derivative for x = 0 equals −b + 1/p > Q ′ a,b (0). Our first contribution is an improvement of Segura's bounds.
Then, The next result provides alternative bounds for Q a,b .
Remark 2.5. The lower bounds for Q a,b resulting from Theorem 2.3 are stronger than the ones from Theorem 2.1. Precisely, it is shown in Section 6 that the following inequalities hold true for x ∈ (0, 1]: Moreover, the upper bounds for Q a,b resulting from Theorem 2.3 are stronger than the ones from Theorem 2.1 if b ∈ (0, 3]. Precisely, the following inequalities are derived in Section 6 for x ∈ (0, 1]:    a,b /Q a,b for ℓ = 1 (dark green), ℓ = 2, 3 (blue) and ℓ = 4, 5 (orange, black dashed).

Gamma distributions.
There is a rich literature about inequalities for gamma distribution and survival functions, see, for instance, Qi and Mei (1999), Neuman (2013), Segura (2014) and Pinelis (2020). We just illustrate that our bounds in Theorems 2.1 and 2.3 yield a connection to that literature. It is well-known that for a random variable X a,b ∼ Beta(a, b), the rescaled variable bX a,b converges in distribution to a gamma random variable with shape parameter a and scale parameter 1 as b → ∞. Denoting the corresponding distribution and survival function with G a andḠ a = 1 − G a , respectively, we have G a (x) = lim b→∞ B a,b (x/b), and one can deduce from Theorems 2.1 and 2.3 the following bounds.
Then Γ(a)e xḠ a (x) Some of these bounds are known already or refinements of results in the aforementioned literature, notably Neuman (2013, Theorem 4.1) and Segura (2014, Theorem 10). Note also that for integers a ≥ 1, The bounds in Corollary 2.6 reproduce this equality for a = 1, 2, 3.

Exponential inequalities
Although the upper bounds in Theorems 2.1 and 2.3 are numerically rather accurate in the tails, they can diverge to ∞ at x = p as a, b → ∞. Moreover, it is sometimes desirable to have bounds for log B a,b (x) and logB a,b (x) in terms of simple, maybe rational, functions of x. Numerous exponential tail inequalities for B a,b andB a,b have been derived already. We start with one particular result of Dümbgen (1998, This function K(p, ·) is strictly convex with minimum K(p, p) = 0. For arbitrary x ∈ [0, 1], These inequalities can be improved substantially. The starting point is a rather general inequality.
Combining this proposition with well-known inequalities for the mode and median of beta distributions leads to the following result.
Theorem 3.2. Let p ℓ := (a + 1)/(a + b) and p r := (a − 1)/(a + b). Then, Figure 4 illustrates these bounds for Beta (4,8). Note also that the second upper bound for B a,b (x) andB a,b (x) may be rewritten as follows: This follows essentially from the fact that c(a, b) = a(b − 1)/(a + 1) satisfies If b > 1, all bounds in question can be formulated in terms of a parameter c ∈ (0, b] as follows: in (2).
This implies that the present bound for B a,b (x) is strictly stronger than the bound of Version 8 (Theorem 5), and that the latter bound is strictly stronger than the bound in (2).
The right-hand side is monotone increasing in b > 1 and positive for b ≥ 1.5. Thus, for x ∈ [0, p], the third upper bound for B a,b (x) is smaller than the second one in case of a ≥ b ≥ 1.5.
Analogously, for x ∈ [p, 1], the third upper bound forB a,b (x) is certainly better than the second one in case of b ≥ a ≥ 1.5. (3) and (4) imply Bernstein and Hoeffding type exponential inequalities. It follows from Dümbgen and Wellner (2022, Lemma S.12) and the well-known inequality z(1 − z) ≤ 1/4 for z ∈ R, that

The inequalities
. This leads to the following inequalities: Further tail and concentration inequalities for the Beta distribution have been derived by Marchal and Arbel (2017) and Skorski (2023). 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 3.5. For the left tails, the difference Analogously, for the right tails, the difference

Skorski (2023) derives a Bernstein type inequality. With the parameters
he shows that for X ∼ Beta(a, b) and ε ≥ 0, .
By means of Theorem 3.2 one can derive a similar inequality: Corollary 3.6. For arbitrary a, b ≥ 1 and ε ≥ 0, For p ≥ 1/2, i.e. a ≥ b, the factor a + b + 1 + a/b is at least a + b + 2, and the bound in Corollary 3.6 is stronger than the one of Skorski (2023). For sufficiently small p < 1/2 and sufficiently large ε, Skorski's (2023) bound (6) can be a bit stronger than the one in Corollary 3.6. But the next result shows that our upper bound from Theorem 2.1, combined with (5), implies even stronger inequalities as soon as ε is a sufficiently large multiple of σ a,b = Std(X).

Proofs
Proof of Theorem 2.1. Let Q be a continuous function on some interval [0,

then elementary calculus reveals that
On the one hand, let Q( If we choose c = c a,b , then This proves the bounds in terms of Q and elementary calculations lead to If we choose c = c a,b , then Hence, Q is a lower or upper bound for Q a,b if b ≤ 1 or b ≥ 1, respectively. But this bound can be refined further. Note that for x ∈ [0, x o ), This proves the bounds in terms of Q Proof of Theorem 2.3. Note first that for any x ∈ [0, 1), the integral a 1 0 y a−1 (1 − xy) b−1 dy = Q a,b (x) is well-defined for any a > 0 and b ∈ R. We may also write with Y ∼ Beta(a, 1). Since E(Y ) = a/(a + 1) and it follows from Jensen's inequality that Secondly, it follows from partial integration that Thirdly, note that Since Q a,b (0) = 0, this implies that In particular, Finally, to derive the bounds in terms of Q [4] a,b (x) and Q [5] a,b (x), we use a particular representation of a twice continuously differentiable function f on [0, 1]. Namely, for any y ∈ (0, 1), where ∆ y is a random variable with values in (0, 1) and density function g y (t) := 2 min t y ,
For x ∈ (0, 1] and b ∈ (1, 2], we write so the assertion that Q a,b (x) is equivalent to But the right-hand side is strictly larger than For x ∈ (0, 1] and b ∈ [2, ∞), the asserted inequality Q This inequality is even valid for all b > 1. Indeed, for b = 1, both sides coincide, and the derivatives of the left-hand and right-hand side with respect to b are equal to respectively. The former is strictly decreasing while the latter is constant in b ≥ 1. Thus it suffices to show that Writing 1 − ax/(a + 1) = 1 − x + x/(a + 1), an equivalent claim is that .
Hence the asserted bounds follow from the fact that for b > 4, Q [1] a,b , Q [1] a,b , Q [2] a,b are lower and Q [S,2] a,b , Q [3] a,b , Q [4] a,b are upper bounds for Q a,b .
As toḠ a (x), we writeḠ a (x) = lim b→∞Ba,b (x/b) = lim b→∞ B b,a (1 − x/b) and b,a (1 − x/b), 3 ≤ ℓ ≤ 5, turned out to be useless in this particular context.) One can show that If we can show that J is monotone decreasing on [0, q], then a standard argument for measures with monotone density ratios applies: As to the monotonicity of J, and this is nonpositive indeed.
Proof of Theorem 3.2. Again, for symmetry reasons, it suffices to derive the upper bounds for B a,b . If b ≤ 1, then Proposition 3.1 shows that B a,b (x) ≤ (x/q) a for x ∈ [0, q] and q ∈ (0, 1).
Letting q → 1 reveals that B a,b (x) ≤ x a for x ∈ [0, 1]. If b > 1, the maximizer of q → q a (1 − q) c(a,b) equals a/(a + c(a, b)) = (a + 1)/(a + b) = p ℓ . Thus Proposition 3.1 yields the second bound for B a,b (x), x ∈ [0, p ℓ ]. Moreover, it is well-known that in case of a ≥ b > 1, the median of Beta(a, b) is at least its mean a/(a + b), see Groeneveld and Meeden (1977) or Dharmadhikari and Joag-Dev (1983). Thus B a,b (p) ≤ 2 −1 , whence Proposition 3.1 leads to the third asserted bound for B a,b (y).