ABSTRACT
Assume that https://www.w3.org/1998/Math/MathML"> X 1 , … , X n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8065.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is an observation from a collection of distributions https://www.w3.org/1998/Math/MathML"> P θ , θ ∈ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8066.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> Θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8067.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ) that depends on a 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/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8068.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> ranging over a set https://www.w3.org/1998/Math/MathML"> Θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8069.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . A popular method for finding an estimator https://www.w3.org/1998/Math/MathML"> θ ˆ n = θ ˆ n X 1 , … , X n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8070.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is to maximize a criterion of the type https://www.w3.org/1998/Math/MathML"> θ ↦ M n ( θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8071.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> over the parameter set https://www.w3.org/1998/Math/MathML"> Θ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8072.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . For notational simplicity, the dependence of https://www.w3.org/1998/Math/MathML"> M n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8073.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> in the observations is implicit. Such an estimator is often called an M-estimator. When https://www.w3.org/1998/Math/MathML"> X 1 , … , X n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8074.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are i.i.d., the criterion https://www.w3.org/1998/Math/MathML"> θ ↦ M n ( θ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8075.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is often chosen to be the sample average of some known functions https://www.w3.org/1998/Math/MathML"> m θ : X → R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8076.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , () https://www.w3.org/1998/Math/MathML"> M n ( θ ) = n - 1 ∑ t = 1 n m θ X t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq8077.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
