ABSTRACT
Suppose that we have formulated a statistical model https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1475.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for the phenomenon under consideration: https://www.w3.org/1998/Math/MathML"> X ∼ P θ ∈ P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1476.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . We want to draw conclusions about the distribution https://www.w3.org/1998/Math/MathML"> P θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1477.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> on the basis of the data https://www.w3.org/1998/Math/MathML"> x ∈ X . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1478.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> If https://www.w3.org/1998/Math/MathML"> P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1479.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a parametric model, one is mostly interested in statements about the finite-dimensional parameter https://www.w3.org/1998/Math/MathML"> θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1480.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . If the underlying model is not parameterized by a finite-dimensional parameter, the quantity of interest can be the distribution function, some other function or a finite-dimensional parameter depending on the unknown distribution https://www.w3.org/1998/Math/MathML"> P θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1481.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . To formalize this approach, we introduce a function https://www.w3.org/1998/Math/MathML"> g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1482.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> defined on the parameter space https://www.w3.org/1998/Math/MathML"> Θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1483.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and take values in a set https://www.w3.org/1998/Math/MathML"> Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1484.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , i.e., https://www.w3.org/1998/Math/MathML"> g : Θ → Γ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1485.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Our aim is to derive conclusions about the value of https://www.w3.org/1998/Math/MathML"> g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1486.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> at https://www.w3.org/1998/Math/MathML"> θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1487.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , i.e., about the quantity https://www.w3.org/1998/Math/MathML"> γ = g ( θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429098789/719b1b93-f3d4-48ae-9170-7b872bf41df1/content/eq1488.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
