OPTIMAL CONFIDENCE REGIONS

19–23.10.1998, Tampere

Lecture 1 : Basic Notions. Location-Scale Families. Two Methods for Estimation. Confidence Regions. Equivariant and Invariant Statistics.

Lecture 2 : Confidence Interval Estimators in Normal and Exponential Cases. Optimal Confidence Regions. Basic Lemma.

Lecture 3 : Optimal Confidence Regions in Normal and Exponential Cases.

Lecture 4 : Optimal Confidence Region for Regression Parameters. Confidence Region for Regression Function at Given Point (and in Given Interval. Optimal Designs.)

Basic Notions

ξ – random variable on the probability space (Ω, F, P), (X , B_{X}) – range space of ξ, X ⊂ R, F_{ξ}(u) = P (ξ ≤ u) – distribution function of ξ, u ∈ R, f_{ξ}(u) = \frac{dF_{ξ}(u)}{du} – density function of ξ, u ∈ R, P_{ξ}(A) = _{A}^{∫} f_{ξ}(u)du – distribution of ξ, A ∈ B_{X} , x = (x_{1}, x_{2}, …, x_{n}) – sample of size n drawn from the distribution P_{ξ}, (X^{n}, B_{X^{n}}) – sample space, X^{n} = X × ⋯ × X ,

P_{ξ}^{(n)}(A) = ∫ … ∫_{A} f_{ξ}(u_{1}) … f_{ξ}(u_{n})du_{1} … du_{n}, A ∈ B_{X^{n}},

(X^{n}, B_{X^{n}}, P_{ξ}^{(n)}) – statistical model (space). ——————————————————————— If P_{ξ} ∈ {P_{θ} : θ ∈ Θ}, Θ ⊂ R^{k}, k ≥ 1, then we call the statistical model parametric and denote it by (X^{n}, B_{X^{n}}, P_{θ}^{(n)}). Saying that we have a parametric model (X^{n}, B_{X^{n}}, P_{θ}^{(n)}) means that we have a sample x of size n drawn from the distribution P_{θ}. If P_{ξ} ∈ {P_{θ} : θ ∈ Θ} then F_{ξ} ∈ {F_{θ} : θ ∈ Θ} and vice versa.