Consistent Estimation of Distribution Functions under Increasing Concave and Convex Stochastic Ordering
在仅假设条件分布满足递增凹或递增凸随机序的条件下，提出了条件累积分布函数的非参数估计量，并证明了其一致性和收敛速度，适用于K样本和连续协变量情形。
A random variable Y1 is said to be smaller than Y2 in the increasing concave stochastic order if E[ϕ(Y1)]≤E[ϕ(Y2)] for all increasing concave functions ϕ for which the expected values exist, and smaller than Y2 in the increasing convex order if E[ψ(Y1)]≤E[ψ(Y2)] for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions Fx(y)=ℙ(Y≤y|X=x) of a response variable Y given a covariate X, solely under the assumption that the conditional distributions are increasing in x in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the K-sample case X∈{1, …, K} and for continuously distributed X.