ABSTRACT

Let Z n = { Z i } i = 1 n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0001.tif"/> be an IID (independent and identically distributed) random sample of n ∈ ℕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0002.tif"/> observations, where Z i ⊤ = ( X i ⊤ , Y i ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0003.tif"/> , X i ∈ X ⊂ ℝ p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0004.tif"/> and Y i ∈ [ q ] = { 1 , … , q } ( i ∈ [ n ] ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0005.tif"/> . Here, p , q ∈ ℕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0006.tif"/> , where ℕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0007.tif"/> is the set of natural numbers (zero exclusive), ℝ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0008.tif"/> is the set of real numbers, and ( ⋅ ) ⊤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429446177/1dffcb2a-0fb0-4190-8367-09301fa51e47/content/C003_Equ_0009.tif"/> is the matrix transposition operator. We call X i and Y i the feature vector and label of the ith observation from the sample Z n, respectively.