MMP.13:4.3 - Construct an estimator and its frequentist uncertainty when needed

An estimator is a rule q_hat=T(Y), chosen before its repeated-use properties are assessed. Derive it from the target and observation law: for example, invert an observable expectation, solve an estimating equation, or maximize a likelihood and obtain the required function of the fitted values. State the selected property, such as a controlled error probability or mean squared error. Likelihood maximization alone supplies no interval.

One general confidence construction chooses, for each admitted theta, an acceptance region A_theta of possible records such that

P_theta(Y in A_theta) >= 1-alpha.

After observing y, retain those theta for which y lies in A_theta, and map them through g. The resulting set C(y) has coverage

P_theta(g(theta) in C(Y)) >= 1-alpha

under the stated law. A target-specific statistic can make this much cheaper than constructing a set for every nuisance parameter. The binomial inversion in :5.1 is a small example.

Coverage describes the procedure across repetitions allowed by the model, at fixed theta. It does not assign posterior probability to the parameter in the one observed interval. Nor is a confidence set the set of all logically possible values: a value outside it may still assign a small positive probability to these records.

Use an exact distribution or justified pivot when available. An asymptotic approximation needs its sample-size, regularity and boundary conditions. A bootstrap needs a resampling unit and mechanism that represent the dependence and the fitted procedure; resampling individual readings cannot reproduce an omitted common calibration error. Retain approximate coverage as approximate unless a stronger result is available.

The observation plan belongs to the guarantee. An interval justified at a fixed sample size need not preserve its coverage when repeatedly inspected until it looks favorable. Use a method valid for the actual stopping plan. For countably many looks, one conservative construction allocates error probabilities alpha_j with sum at most alpha to valid per-look intervals; the union bound gives simultaneous coverage. An applicable confidence-sequence method can provide a less conservative construction. This is a choice of inferential guarantee, not a requirement to collect more data.