Original question and givens
Use one common finite model with parameter indices=Fin(2)={0,…,1}, outcomes=Fin(3)={0,…,2}, complete candidate laws=[[1/9, 1/9, 7/9], [1/2, 1/3, 1/6]], and the single total statistic T=[0, 6, 12] in outcome order. Each law must be nonnegative and sum to 1 before model qualification. T depends on the observation, not the unknown parameter. An observed value is an estimate, while T is the estimator function; no prior or posterior is supplied. Qualified-model parameter index=1. Give (parameter, expectation, statistic).
G001 — Use one common finite model with parameter indices=Fin(2)={0,…,1}, outcomes=Fin(3)={0,…,2}, complete candidate laws=[[1/9, 1/9, 7/9], [1/2, 1/3, 1/6]], and the single total statistic T=[0, 6, 12] in outcome order. Each law must be nonnegative and sum to 1 before model qualification. T depends on the observation, not the unknown parameter. An observed value is an estimate, while T is the estimator function; no prior or posterior is supplied. Qualified-model parameter index=1. Give (parameter, expectation, statistic).
Step-by-step derivation
S001 · implies
Before
source operation=estimator-expectation-at-parameter;original snapshot and mathematics retained
After
explicit ordered source output fields=[parameter, expectation, statistic]
Conditions: none
Keep the historical interface marker in the original snapshot while giving ordered fields here,without JavaScript or an external table.Labels are not mathematical evidence,and each No marker retains the operation's claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
estimator parameter0 complete law sum := sum(1/9, 1/9, 7/9)
After
estimator parameter0 complete law sum = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · implies
Before
original parameter=0;SAME outcome carrier Fin(3);aligned reduced candidate law=[1/9, 1/9, 7/9]
After
sum=1;nonnegative=Yes;law qualified=Yes
Conditions: none
Every model row must be nonnegative and total1.Invalid candidate rows retain false flags;a total statistic does not repair a probability model.
Step identifiers and dependencies
Rule: estimator_complete_laws_separate_from_total_statistic · Uses: G001, S002
S004 · calculation
Before
estimator parameter1 complete law sum := sum(1/2, 1/3, 1/6)
After
estimator parameter1 complete law sum = 1
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
original parameter=1;SAME outcome carrier Fin(3);aligned reduced candidate law=[1/2, 1/3, 1/6]
After
sum=1;nonnegative=Yes;law qualified=Yes
Conditions: none
Every model row must be nonnegative and total1.Invalid candidate rows retain false flags;a total statistic does not repair a probability model.
Step identifiers and dependencies
Rule: estimator_complete_laws_separate_from_total_statistic · Uses: G001, S004
S006 · implies
Before
original parameter carrier Fin(2);common outcome carrier Fin(3);ONE original statistic array=[0, 6, 12]
After
all model laws qualified=Yes;one value per outcome=Yes;SAME statistic map T(o) at EVERY parameter
Conditions: none
The statistic definition takes an observation index,not the unknown parameter.This is functional parameter-independence,not a claim that the law or expectation of T(X) is parameter-free.No estimand,prior or posterior has been supplied.
Step identifiers and dependencies
Rule: estimator_definition_parameter_independence_not_distribution_free · Uses: G001, S005
S007 · calculation
Before
estimator selected law weighted outcome0 := mul(1/2, 0)
After
estimator selected law weighted outcome0 = 0
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · calculation
Before
estimator selected law weighted outcome1 := mul(1/3, 6)
After
estimator selected law weighted outcome1 = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · calculation
Before
estimator selected law weighted outcome2 := mul(1/6, 12)
After
estimator selected law weighted outcome2 = 2
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S008
S010 · calculation
Before
estimator selected parameter expectation := sum(0, 2, 2)
After
estimator selected parameter expectation = 4
Conditions: none
Compute exactly from original roles and established intermediates,retaining indices,denominators,signs and the relevant carrier.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
FIXED original parameter=1;complete law=[1/2, 1/3, 1/6];SAME statistic=[0, 6, 12];weighted terms=[0, 2, 2]
After
E_parameter[T]=4;no unbiasedness or posterior conclusion
Conditions: none
Weight the same statistic by the complete law for one fixed parameter.Without a declared estimand,comparing this expectation with the parameter index does not establish bias or unbiasedness;no prior supplies a posterior.
Step identifiers and dependencies
Rule: estimator_expectation_fixed_law_no_undeclared_estimand · Uses: G001, S010