On the definition of sharp bounds for errors measurements

Question

Questions on the definition of sharpness in the context of bounds and inequalities have already been asked in this site, for example here and here. However I am interested in the specific case where sharp bounds are derived for some error measurement.

Let us consider some function $r:\mathbb{R}\rightarrow\mathbb{R}$ which can be interpreted as the size of some error which we are trying to bound; moreover there exists $x^\star\in\mathbb{R}$ such that $r(x^\star)=0$ that is the error is zero at $x^\star$. This is the case for example when approximating a function $f(h)$ around zero by using its Taylor expansion $f(0)+f^\prime(0)h$: when the shift $h$ becomes null then the value coincides with the approximation.

I don't have any reference at hand but when reading papers where an error is measured, I often find theorems or propositions where the author(s) have established a "sharp bound" (also referred to as "sharp estimate") for the error; this has the form of some inequality. So let us assume we have found a function $g:\mathbb{R}\rightarrow\mathbb{R}$ such that: \begin{align} & r(x)\leq g(x), \quad \forall\ x\in\mathbb{R} \tag{1a}\\ & r(x^\star)=g(x^\star) \tag{1b} \end{align} According to the definitions on sharpness (see for example this answer), the function $g$ provides a sharp upper bound on the error. Yet because we have assumed $r(x^\star)=0$, we can always find a constant $C$ such that $Cg(x)$ also satisfies $(1)$ because $Cg(x^\star)=C\times0=r(x^\star)$. Therefore the "sharpness" of the bound can be arbitrarily widened or tightened.

Would the above be still considered a sharp bound (in the context of error measurement)? Does a sharp bound for error estimates mean something else?

Answer 1

If we don't have $0<C<1$ where $r(x^*) \leq Cg(x^*)$, yet $r(x^*) \leq g(x^*)$, then the only solution is to have $r(x^*) = g(x^*)$.

What I'm trying to illustrate is that the problem with picking and choosing which $x$ you inspect and expecting the inequality to be airtight, is that you effectively force the LHS to be equal to RHS for every $x$. Which is no longer a bound, it is an equality.

A bound means that for every $x$ (or $x_1,x_2,...$ in the case of multiple parameters), the inequality holds, and a bound being tight means for SOME $x$ (or $x_1,x_2,...$), we have equality.

For your specific question, no, we cannot simply make $g(x)$ tighter by a factor of $C$ since $g(x)$ may be non-zero at points other than $x^*$. For example:

Answer 2

Regardless of what the value is, notice that $\not \exists f: \mathbb{R} \longrightarrow \mathbb{R}$ such that $r(x) \le f(x) < g(x),\forall x \in \mathbb{R}$—otherwise, $r(x^\star) < g(x^\star)$, a contradiction.