Sensitivity of a Family of Probability Measures with Respect to a Parameter

1 We shall consider a set {P_θ ; θ ∈ Θ} of probability measures on the space X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq46.tif"/> with densities f(·, θ) relative to a σ-finite measure μ, where Θ is a real interval. To simplify notation we shall write f for f(·, θ), and f_r for f(·, θ_r ) where convenient. The derivatives with respect to θ, where they exist, will be denoted by f ′ ,   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/eq47.tif"/> . For the present we denote by ρ, = ρ(P_θ , P _{θ 0} ), the distance of P_θ from a fixed probability measure P _{θ 0} . ρ 2   =   ∫ ( √ f   −   √ f 0 ) 2   d μ   = 2   −   2   ∫ √ ( f f 0 )   d μ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351076777/1531b062-750f-426f-9035-7dcbd680db0f/content/eq48.tif"/>