We consider kernel desnity estimation using the stagewise minimization algorithm with its dictionary having various bandwidths. Let X_i~T =(X_(i1),X_(i2),…，X_(id))， i =1,2，...，m + n， be d-dimensional i.i.d. sample generated from the d-dimensional density finction f. For this i.i.d. sample, we define X*_i≡X_i =1，2，...，m， and use them for the dictionary. The rest of the sample, X_i, i = m +1,m + 2,...,m + n, are utilized for the algorithm of density estimation. Denoting B to be a set of d x dimensional bandwidth matrices H_j, j =1,2，...,|B|, we define the dictionary, D={ΦH_j(·-X*_i)|H_j ∈ B,i = 1，2,...，m，j =1,2,..., |B|},|D| = m ×|B|, where ΦH_j(·-X*_j) is a density function with its mean X*_j and variance-covariance matrix H_j. Each word in D is denoted to be Φ_s(x|X*),s =1，2,...,m ×|B|, where each index number s corresponds to a combination of X*_i and H_j one-to-one.
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机译：我们考虑使用具有各种带宽的分支最小化算法考虑内核Desnity估计。让x_i〜t =（x_（i1），x_（i2），...，x_（id）），i = 1,2，...，m + n，为d维i.i.d.d.从D尺寸密度的样品产生的样品。为此，i.D.样本，我们定义x *_i≡x_i= 1,2，...，m，并为字典使用它们。样品的其余部分X_I，I = M + 1，M + 2，...，M + N用于密度估计算法。表示B为一组DX尺寸带宽矩阵H_J，J = 1,2，...，| B |，我们定义字典，D = {φH_J（·-X * _I）| H_J∈B，i = 1,2，...，m，j = 1,2，...，| b |}，| d | = m×| b |，其中φh_j（·-x * _j）是具有其平均x * _j和方差协方差矩阵h_j的密度函数。 d中的每个单词都表示为φ_s（x | x *），s = 1,2，...，m×| b |，其中每个索引号s对应于x * _i和h_j的组合一体化-一。
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