Many Parameters the Sensitivity Matrix

1 We now consider a family of probability measures {P_θ } on X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq421.tif"/> with densities f(·, θ) relative to a σ-finite measure μ, where θ is a point in R^k ; θ′ = (θ ₁, θ ₂, …, θ_k ). We shall write θ ′ 0   =   ( θ 10 ,   θ 20 ,   … ,   θ k 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq422.tif"/> . As before, we shall write f for f(·, θ), and f ₀ for f(·, θ ₀) where convenient. The partial derivatives with respect to θ_r will be denoted by f ′ r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq423.tif"/> and f ′ r 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq424.tif"/> . L = log f, L ′ r   =   ∂ L / ∂ θ r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq425.tif"/> , L ′ r 0   =   L ′ r   ( ⋅ ,   θ 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq426.tif"/> .