Statistical models

Question 1

Parametric statistical model

Answer 1

f₀ ∈ F = {f(.|θ), θ ∈ Θ ⊂ R^p}

Question 2

Sample space and parametric space

Answer 2

y ∈ Y ⊂ Rⁿ
θ = (β_px1, σ²)’ ∈ Θ ⊂ R^p X R⁺

Question 3

Multivariate normal joint distribution

Answer 3

f(y₁, … y_n| μ Σ) = exp[-½(y-μ)’ Σ^-1 (y - μ)] / (2π)^p/2|Σ|^1/2
Y ~ N_n (μ, Σ)

Question 4

Properties of a multivariate normal

Answer 4

• Marginal distributions of order q < n are still multivariate normals and their mean and variance-covariance matrix are obtained by extracting elements from the starting distribution.
• Conditional distribution Y_1a, … Y_ha | Y_1b, … Y_gb ~ N_h.
• If Σ is diagonal <=> corr = 0 <=> Y₁, … Y_n are independent and their joint distribution is the product of univariate N distributions.
• Z = DY+d, D(qxn), d(qx1) => Z ~ N_q(Dμ +d, DΣD’).

Question 5

Standardized gaussian vector

Answer 5

D = Σ^-1/2, d = -Σ^-1/2μ
Z = DY+d ~ N_n(0, In) and Z’Z ~ χ²_n