From Finite Risk Vectors to General Risk Functions

From Risk Vectors to Risk Functions

When the parameter space is finite, a decision rule can be represented by a finite list of risks, one for each parameter value. For a general parameter space, the same risks form a function. This change in representation is more than a notational convenience: it makes clear that comparing rules means comparing their risks at every parameter value, while convergence of risk functions is a separate question about how values behave across the parameter space.

Let \(\Theta\) be a nonempty parameter space, and let \(\delta\) be a decision rule in a specified class of rules. Assume the loss is measurable and nonnegative, and that the risk is finite at every \(\theta\in\Theta\). Write

Here \(E_\theta\) is expectation under the sampling distribution indexed by \(\theta\). The rule \(\delta\) therefore determines the risk function \(R_\delta:\Theta\to\mathbb R\), given by \(R_\delta(\theta)=R(\theta,\delta)\). A randomized rule can be included in the same notation: its risk is still a function of \(\theta\).

If \(\Theta=\{\theta_1,\ldots,\theta_m\}\), listing \(R_\delta(\theta_i)\) for \(i=1,\ldots,m\) gives the finite risk vector. Its coordinatewise order is precisely the restriction of pointwise order to this finite parameter space. For an infinite \(\Theta\), the function-space order is defined by the same coordinate comparisons, now required at every point of \(\Theta\).

Finite Restrictions and the Function-Space View

For any finite subset \(F=\{\theta_1,\ldots,\theta_m\}\subset\Theta\), define the restriction map

The finite vector records exactly what the risk function does on \(F\), and nothing about parameters outside \(F\). In particular, if \(\pi_F(R_{\delta_1})\leq\pi_F(R_{\delta_2})\) coordinate by coordinate, then \(\delta_1\)'s risk is no greater on \(F\). This statement alone is not global dominance.

Proof. Enumerate \(\Theta=\{\theta_1,\ldots,\theta_m\}\). By definition, weak dominance means \(R_{\delta_1}(\theta)\leq R_{\delta_2}(\theta)\) for every \(\theta\in\Theta\). Applying this statement to each \(\theta_i\) gives the \(m\) coordinate inequalities in the risk vectors. Conversely, those \(m\) inequalities cover every element of \(\Theta\), so they imply weak dominance on \(\Theta\). \(\square\)

The finite result thus extends directly as a definition to general \(\Theta\): use the same inequality at each parameter value. What does not extend automatically is an inference that starts with many separate finite comparisons and concludes that a single rule works on the whole parameter space. The rule giving the best comparison may change as \(F\) changes.

Pointwise Convergence and Finite-Coordinate Information

A basic way to compare risk functions on a general parameter space is through pointwise convergence. A sequence of functions \(r_n:\Theta\to\mathbb R\) converges pointwise to \(r:\Theta\to\mathbb R\) if, for each fixed \(\theta\in\Theta\), the real sequence \(r_n(\theta)\) converges to \(r(\theta)\). The parameter value is held fixed while taking the limit; the definition does not require one bound to work uniformly over all \(\theta\).

The product topology formalizes the idea that a finite set of coordinates can be checked at a time. A basic neighborhood of a function \(r\) requires a candidate function to be close to \(r\) at a specified finite collection of parameter values; it places no restriction on the remaining coordinates. Consequently, approximation on every fixed finite set is the natural local information supplied by this topology. It is not, by itself, one uniform approximation statement over all of \(\Theta\).

Proof. Fix an arbitrary \(\theta\in\Theta\). For every \(n\), \(r_n(\theta)\leq s(\theta)\), where \(s(\theta)\) is a fixed real number. Since \(r_n(\theta)\to r(\theta)\), the limit of this sequence cannot exceed \(s(\theta)\). More explicitly, if \(r(\theta)>s(\theta)\), let \(\varepsilon=(r(\theta)-s(\theta))/2>0\). For all sufficiently large \(n\), convergence gives \(r_n(\theta)>r(\theta)-\varepsilon=(r(\theta)+s(\theta))/2>s(\theta)\), contradicting the assumed inequality. Thus \(r(\theta)\leq s(\theta)\). Since \(\theta\) was arbitrary, the inequality holds throughout \(\Theta\). \(\square\)

The hypothesis uses the same comparator \(s\) for every \(n\), with the inequality holding at every parameter value. It does not say that approximate rules which work on different finite subsets can be assembled into one rule satisfying a global inequality. That is a separate existence question requiring additional structure.

Pointwise Is Not Uniform

When the risk functions are bounded, uniform convergence is a stronger alternative. A sequence \(r_n\) converges uniformly to \(r\) if \(\sup_{\theta\in\Theta}|r_n(\theta)-r(\theta)|\to0\). Pointwise convergence checks each fixed parameter separately; uniform convergence controls the largest discrepancy over the whole parameter space. On an infinite parameter space these conditions can differ, so a finite-coordinate argument should not be described as a uniform risk bound.

Common Mistakes and Why the Distinction Matters

• Promoting a finite restriction to a global comparison. A rule that improves on a finite set \(F\) may depend on \(F\). Global dominance requires one rule whose risk inequality holds for every \(\theta\in\Theta\).
• Confusing weak and strict dominance. A rule weakly dominates itself, because its risk equals itself everywhere. It does not strictly dominate itself, because strict dominance requires a strict inequality somewhere.
• Calling pointwise convergence uniform. Pointwise convergence controls each fixed parameter in turn. It gives no common error bound across an infinite parameter space, as the exponential-risk example demonstrates.
• Assuming a pointwise infimum is an attainable risk. The pointwise best value at each parameter may be supplied by a different rule. An infimum function is not automatically the risk function of any available rule.

The function-space formulation keeps two logical tasks separate. First, compare risk functions by the pointwise order, which is the direct generalization of coordinatewise comparison of finite risk vectors. Second, specify the mode of convergence when limits are involved. If one sequence of risks is pointwise no greater than a fixed comparison function, the inequality survives its pointwise limit. But finite restrictions that use different rules, or pointwise approximation without a common bound, do not alone establish global domination or uniform risk control.