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Let x𝑖∈ R𝑑 and 𝑦𝑖∈ R be continuously distributed random variables supported on X=[0, 1] 𝑑 and Y=[0, 1], respectively. We are interested in estimating the conditional distribution function and its derivatives: 𝜃𝜇, 𝝂 (𝑦| x)=

𝜕 𝜇

𝜕 𝑦𝜇

𝜕 𝝂

𝜕 x 𝝂 𝐹 (𝑦| x), where 𝜇∈ N, and 𝝂∈ N𝑑 representing multi-indices.(In the main paper we only consider the estimation of conditional density and derivatives thereof, that is, we set 𝝂= 0 and 𝜇= 𝜗+ 1≥ 1.) To present our estimation strategy, we start from 𝜃0, 𝝂, the conditional distribution function and its derivatives with respect to the conditioning variable, and apply the local polynomial method: