Reading a Risk Table Coordinate by Coordinate
A risk table turns comparisons between decision rules into comparisons between finite vectors. Each column represents a parameter value, and each entry is the risk of a particular rule at that parameter. The key question is not which row has the smallest average or the most small entries. It is whether one row is no larger in every column and strictly smaller in at least one.
That distinction matters because a rule may have a small risk at some parameter values and a larger risk at others. A single summary across columns can hide those differences. For a finite parameter space, reading the table carefully gives a direct test for whether a listed rule is dominated by another listed rule.
The inequality is always read from the proposed dominator to the rule being compared. Thus, if a row for \(\delta'\) is to dominate a row for \(\delta\), the entries in the \(\delta'\) row must be no larger at any parameter value. Equality in some columns is allowed; the required strict improvement may occur in just one column.
A Coordinate Test for Dominance
For a proposed comparison, subtract the risk of the proposed dominator from the risk of the rule being tested. This fixes the direction of the comparison and makes the check mechanical.
The proposed rule \(\delta'\) dominates \(\delta\) exactly when every \(d_j\) is nonnegative and at least one \(d_j\) is positive. A zero difference means a tie at that parameter value; a negative difference means \(\delta'\) has higher risk there and therefore fails the test.
Proof. If \(\delta'\) dominates \(\delta\), the definition gives \(d_j\geq0\) for every \(j\), with at least one positive. A finite sum of nonnegative numbers, at least one of which is positive, is positive. Conversely, if every \(d_j\geq0\) and their sum is positive, not all \(d_j\) can be zero. So at least one is positive, and the definition of domination is satisfied. \(\square\)
The sum in this result is only a convenient way to test whether at least one coordinate is strictly improved after checking that none is worse. A positive sum alone is not a dominance test: a positive difference at one parameter cannot cancel a negative difference at another.
Worked Example: Comparing Actions in a Finite Decision Problem
Suppose there is no informative observation: \(X=0\) with probability one under each of three parameter values \(\theta_1,\theta_2,\theta_3\). A rule chooses one of four actions, and the losses for those actions are listed below. Since the observation is constant, each rule always chooses its stated action, so its risk at each parameter is the corresponding loss.
| Rule (action) | \(\theta_1\) | \(\theta_2\) | \(\theta_3\) |
|---|---|---|---|
| A | 4 | 2 | 5 |
| B | 5 | 3 | 6 |
| C | 1 | 4 | 3 |
| D | 2 | 2 | 4 |
For example, the risk of rule A at \(\theta_1\) is \(E_{\theta_1}[L(A,\theta_1)]=4\), because the loss is always 4. The same calculation at all three parameter values gives the A row \((4,2,5)\).
Compare D with A, taking D as the proposed dominator. The coordinate differences, risk of A minus risk of D, are \((4-2,\,2-2,\,5-4)=(2,0,1)\). Every difference is nonnegative and at least one is positive, so D dominates A. For B the differences are \((5-2,\,3-2,\,6-4)=(3,1,2)\), so D also dominates B.
Now compare C with D. The differences, risk of C minus risk of D, are \((1-2,\,4-2,\,3-4)=(-1,2,-1)\). The negative entries show that D has higher risk than C at \(\theta_1\) and \(\theta_3\), so D does not dominate C. Reversing the comparison gives differences \((1,-2,1)\), so C does not dominate D either. Among these four listed rules, C and D are undominated; A and B are dominated.
Worked Risks from a Sampling Model
The entries in a risk table come from expectations under the sampling model. For simple procedures, those expectations can be calculated directly before making the coordinate comparisons.
Worked Example: A Binomial Model and Squared-Error Risk
Let \(\Theta=\{0,1,2\}\), and suppose \(X\sim\operatorname{Binomial}(2,\theta/2)\). A rule reports an estimate of \(\theta\), and the loss is squared error, \(L(\theta,a)=(a-\theta)^2\). Compare the data-based rule \(\delta_X(X)=X\) with the constant rules \(\delta_0(X)=0\), \(\delta_1(X)=1\), and \(\delta_2(X)=2\).
At \(\theta=0\), \(X=0\) with probability one. Thus the four risks, in the order \(\delta_X,\delta_0,\delta_1,\delta_2\), are \(0,0,1,4\).
At \(\theta=1\), \(X\sim\operatorname{Binomial}(2,1/2)\), so \(P(X=0)=1/4\), \(P(X=1)=1/2\), and \(P(X=2)=1/4\). Therefore \(R(1,\delta_X)=(1/4)(0-1)^2+(1/2)(1-1)^2+(1/4)(2-1)^2=1/2\). The constant-rule risks are \(R(1,\delta_0)=1\), \(R(1,\delta_1)=0\), and \(R(1,\delta_2)=1\).
At \(\theta=2\), \(X=2\) with probability one. The risks are \(0,4,1,0\), respectively. Collecting these calculations gives:
| Rule | \(\theta=0\) | \(\theta=1\) | \(\theta=2\) |
|---|---|---|---|
| \(\delta_X\) | 0 | 1/2 | 0 |
| \(\delta_0\) | 0 | 1 | 4 |
| \(\delta_1\) | 1 | 0 | 1 |
| \(\delta_2\) | 4 | 1 | 0 |
For \(\delta_X\) versus \(\delta_0\), the differences \(R(\theta,\delta_0)-R(\theta,\delta_X)\) are \((0,1/2,4)\). Hence \(\delta_X\) dominates \(\delta_0\). For \(\delta_X\) versus \(\delta_2\), the differences are \((4,1/2,0)\), so \(\delta_X\) dominates \(\delta_2\) as well.
The comparison with \(\delta_1\) does not establish domination in either direction. The differences for \(\delta_1\) minus \(\delta_X\) are \((1,-1/2,1)\), which includes a negative coordinate. The reversed differences are \((-1,1/2,-1)\), which also include a negative coordinate. Thus, within these four candidates, \(\delta_X\) and \(\delta_1\) are undominated, while \(\delta_0\) and \(\delta_2\) are dominated.
Finite Candidate Classes and Admissibility
A table usually displays only selected rules. Conclusions based on it must therefore be stated relative to the rules it contains. In the binomial example, the calculation shows that \(\delta_0\) and \(\delta_2\) are inadmissible within the four-rule candidate class. It does not, by itself, show that either remaining rule is admissible among every possible decision rule for the model.
Proof. By definition, a rule is admissible within the candidate class if no rule in that class dominates it. The table contains the risk vector for each listed rule at each parameter value. For any pair of listed rules, the coordinate test above is equivalent to the definition of domination. Therefore, checking every other row for a dominating row is both necessary and sufficient for the rule to be undominated within the candidate class. \(\square\)
For a finite candidate class, the checks can also be organized as successive removal of dominated rows. This process is safe because domination is transitive. If \(\delta_1\) dominates \(\delta_2\) and \(\delta_2\) dominates \(\delta_3\), then at every parameter value \(R(\theta,\delta_1)\leq R(\theta,\delta_2)\leq R(\theta,\delta_3)\). At least one of the two comparisons is strict at some coordinate, so the inequality from \(\delta_1\) to \(\delta_3\) is strict at that coordinate. Hence \(\delta_1\) dominates \(\delta_3\).
Because the candidate class is finite, removing one dominated row at a time must stop after finitely many removals. If a removed rule was dominated by another rule that is removed later, transitivity ensures that a rule dominating it remains at the end of that chain of removals. Thus, removal can identify the undominated rows among the candidates. It still cannot certify global admissibility unless the candidate class includes every procedure allowed by the decision problem.
Common Mistakes and Practical Checks
The most frequent error is to replace a coordinatewise comparison with a comparison of averages. For instance, averaging risks across parameter values requires choosing weights; those weights may make one rule look preferable even when it has higher risk at a particular parameter. Such an average is not the dominance criterion.
- Checking only some columns. A proposed dominator must be no worse at every parameter value in the stated parameter space.
- Forgetting the strict coordinate. If all entries tie, the risk vectors are equal; neither rule dominates the other under the definition used here.
- Reversing the subtraction. To test whether \(\delta'\) dominates \(\delta\), check \(R(\theta,\delta)-R(\theta,\delta')\), not the reverse, against the nonnegative condition.
- Claiming global admissibility from a short list. An undominated row is only undominated among the procedures that were checked, unless the table exhausts the full class.
A disciplined table reading has three steps: name the proposed dominator and the rule being tested, compare their risks at every parameter coordinate, and verify both the no-worse condition and at least one strict improvement. Only then should a conclusion about domination or candidate-class admissibility be recorded.
Check Your Understanding
For each question, use coordinatewise risk comparisons rather than averages.
- A proposed dominator has risk differences \(R(\theta_j,\delta)-R(\theta_j,\delta')=(0,2,0)\). Does it dominate \(\delta\)? Explain.
- For differences \((3,-1,2)\), identify exactly which part of the dominance condition fails.
- Why does a row with the smallest sum of risks not necessarily dominate every other row?
- A rule is undominated among five rows in a table. What additional information is needed to conclude it is globally admissible?
- If rule A dominates rule B and rule B dominates rule C, what can be concluded about A and C? State the coordinatewise reason.