Why the Parameter Set Matters
Admissibility is a statement about risk comparisons at the parameter values under consideration. If a procedure is strictly worse at even one parameter value in the specified space, another procedure may dominate it by matching its risk elsewhere and improving at that point. Remove that parameter value, and the same pair of procedures may no longer satisfy the definition of domination. Their risks have not changed; the set of comparisons that counts has.
This tutorial keeps the statistical model, action space, loss function, and class of decision rules fixed. Only the parameter set changes. This distinction matters: restricting the parameter set is not the same operation as restricting the available actions or the allowed procedures.
The weak inequality must hold throughout the specified parameter set. A lower risk at one parameter value does not establish domination if the competing rule has higher risk at another included value. The strict inequality must also occur within the set: improvement only at parameters that have been removed does not count.
What Restriction Does to a Risk Comparison
For any two rules, define their full-space strict-improvement set by \(S=\{\theta\in\Theta:R(\theta,\delta_1)<R(\theta,\delta_0)\}\), assuming \(R(\theta,\delta_1)\le R(\theta,\delta_0)\) throughout \(\Theta\). Their risk functions are unchanged by restriction; only the points at which the inequalities are tested change.
Proof. The assumed inequality \(R(\theta,\delta_1)\le R(\theta,\delta_0)\) holds at every \(\theta\in\Theta\), so it also holds at every \(\theta\in\Theta_0\). By definition, domination relative to \(\Theta_0\) additionally requires at least one \(\theta\in\Theta_0\) where the inequality is strict. That condition is exactly \(S\cap\Theta_0\ne\varnothing\). If the intersection is empty, there is no strict improvement in the restricted set, so the definition of domination is not met. \(\square\)
Proof. Suppose \(\delta_0\) satisfies the stated pointwise inequalities. For any \(\delta_1\in\mathcal D\), they give \(R(\theta,\delta_0)\le R(\theta,\delta_1)\) for every \(\theta\in\Theta_0\). Thus \(\delta_1\) cannot have risk no greater than \(\delta_0\) everywhere on \(\Theta_0\) and strictly lower risk anywhere in it. No rule in \(\mathcal D\) dominates \(\delta_0\), which is the definition of admissibility. \(\square\)
Worked Examples
Worked Example: Exact Observation and a Removed Improvement
Let \(\Theta=\{0,1,2\}\). The observation is \(X=\theta\) with probability one, the action is a real-valued estimate, and the loss is squared error, \(L(\theta,a)=(a-\theta)^2\). Compare the rules \(\delta_0(x)=x\) for \(x=0,1\), \(\delta_0(2)=1\), and \(\delta_1(x)=x\) for all three possible observations.
Since \(X=\theta\) with probability one, each rule’s risk is its squared error at the observed value:
Thus \(\delta_1\) weakly improves on \(\delta_0\) throughout \(\Theta\) and strictly improves at \(\theta=2\); it dominates \(\delta_0\) on the full space. Now restrict to \(\Theta_0=\{0,1\}\). Both rules have risk zero at both retained parameter values, so \(\delta_1\) does not strictly improve anywhere in \(\Theta_0\).
In fact, \(\delta_0\) is admissible relative to \(\Theta_0\) among all rules with real-valued actions, including randomized rules. Squared-error loss is nonnegative, and \(\delta_0\) already has risk zero at both parameter values. No rule can have lower risk at either point, so none can dominate it on \(\Theta_0\). The strict improvement that established full-space domination was entirely removed.
Worked Example: Nonzero Risk and a Randomized Rule Class
Suppose \(X\) is constant and the available actions are \(A,B,C\). The loss table is therefore also the risk table for the corresponding deterministic rules:
| Action | \(\theta_1\) | \(\theta_2\) | \(\theta_3\) |
|---|---|---|---|
| A | 1 | 1 | 3 |
| B | 1 | 1 | 2 |
| C | 2 | 3 | 0 |
On the full space \(\Theta=\{\theta_1,\theta_2,\theta_3\}\), rule \(B\) has risks \((1,1,2)\) and rule \(A\) has risks \((1,1,3)\). At \(\theta_1\) and \(\theta_2\), their risks are equal; at \(\theta_3\), rule \(B\) has risk \(2<3\). Therefore \(B\) dominates \(A\) on \(\Theta\).
Now take \(\Theta_0=\{\theta_1,\theta_2\}\). Rule \(A\) has risk 1 at each retained parameter value. Each available action has loss at least 1 at both values. A randomized rule chooses a probability distribution over these actions, so its risk at either retained value is an average of losses that are all at least 1. That average is also at least 1. No deterministic or randomized rule can strictly improve on \(A\) at either point without increasing risk somewhere in \(\Theta_0\). Hence \(A\) is admissible relative to \(\Theta_0\), despite being dominated on the full space.
The calculation illustrates why the rule class must be held fixed when making the comparison. Here the conclusion remains valid even when randomization is allowed, because the risk lower bound applies to every mixture of actions.
Worked Example: Restriction Can Create a New Dominance Relation
Again let the observation be constant, and consider two rules \(A\) and \(B\) with the following risks:
| Rule | \(\theta_1\) | \(\theta_2\) | \(\theta_3\) |
|---|---|---|---|
| A | 0 | 2 | 2 |
| B | 1 | 1 | 0 |
On the full space, \(A\) has lower risk at \(\theta_1\), since \(0<1\). But \(B\) has lower risk at \(\theta_2\) and \(\theta_3\), since \(1<2\) and \(0<2\). Neither rule dominates the other: each is worse at a parameter value in \(\Theta\).
Restrict the parameter space to \(\Theta_0=\{\theta_2,\theta_3\}\). At \(\theta_2\), the risks are 2 for \(A\) and 1 for \(B\); at \(\theta_3\), they are 2 for \(A\) and 0 for \(B\). Rule \(B\) now has strictly lower risk at every retained point, so it dominates \(A\) relative to \(\Theta_0\). The restriction has not changed either risk function. It has removed the point where \(A\) was better.
How to Check Admissibility After Restriction
When the parameter set changes, start the risk comparison again on the new set rather than carrying over a full-space conclusion. A practical audit is:
Record the model, loss, action space, and class of rules. Change only the parameter set if that is the question being asked.
For each proposed competitor, compare risks at every \(\theta\in\Theta_0\). Do not use values outside \(\Theta_0\) to establish or refute domination relative to it.
Verify weak improvement at every retained point and strict improvement at at least one retained point. If either condition fails, that competitor does not dominate.
To establish admissibility, rule out every permitted competitor, including randomized rules if they belong to the class. A risk lower bound applying to all rules can make this step immediate.
Common Mistakes
- Carrying a full-space domination conclusion over automatically. A full-space dominator may have had its only strict improvement removed. Check whether the strict-improvement set intersects the restricted parameter set.
- Counting an improvement outside the restricted set. Risk at an excluded parameter value cannot supply the required strict inequality for domination relative to \(\Theta_0\).
- Assuming admissibility is preserved in either direction. The examples show that a full-space dominated rule can be admissible on a subset, while a pair of rules incomparable on the full space can become ordered after restriction.
- Changing more than the parameter set. If the available rules or actions also change, the comparison answers a different question. State the rule class when making an admissibility claim.
Check Your Understanding
For each question, focus on the risk comparisons within the parameter set named.
- A rule weakly improves on another throughout \(\Theta\), with strict improvement only at \(\theta_3\). What happens to their domination relation if the parameter set is restricted to \(\{\theta_1,\theta_2\}\)?
- Why does zero risk at every point of \(\Theta_0\) establish admissibility there under nonnegative loss, provided the rule class is fixed?
- Can restricting the parameter set turn two rules that were incomparable on \(\Theta\) into a dominance pair? Explain using the risk comparisons that must change or disappear.
- In the second worked example, why can randomization over actions \(A,B,C\) not reduce risk below 1 at \(\theta_1\) or \(\theta_2\)?
- Does admissibility relative to \(\Theta_0\) imply admissibility relative to the full space \(\Theta\)? Give the logical reason for your answer.