Consistent estimation of distribution functions under increasing concave and convex stochastic ordering

A random variable <i>Y</i>₁ is said to be smaller than <i>Y</i>₂ 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 <i>Y</i>₂ in the increasing convex order if E[ψ(Y1)]≤E[ψ(Y2)] for all increasing convex <i>ψ</i>. This article develops nonparametric estimators for the conditional cumulative distribution functions Fx(y)=ℙ(Y≤y|X=x) of a response variable <i>Y</i> given a covariate <i>X</i>, solely under the assumption that the conditional distributions are increasing in <i>x</i> in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the <i>K</i>-sample case X∈{1,…,K} and for continuously distributed <i>X</i>.

若对于所有使得期望存在的递增凹函数$oldsymbol{phi}$，均满足$mathbb{E}[phi(Y_1)] leq mathbb{E}[phi(Y_2)]$，则称随机变量（random variable）$Y_1$在递增凹随机序（increasing concave stochastic order）下小于$Y_2$；若对于所有递增凸函数$oldsymbol{psi}$，均满足$mathbb{E}[psi(Y_1)] leq mathbb{E}[psi(Y_2)]$，则称$Y_1$在递增凸随机序（increasing convex order）下小于$Y_2$。
本文针对给定协变量（covariate）$X$的响应变量（response variable）$Y$，仅在条件分布关于$x$满足递增凹或递增凸随机序的假设下，推导了其条件累积分布函数（conditional cumulative distribution function）$F_X(y) = mathbb{P}(Y leq y mid X=x)$的非参数估计量（nonparametric estimators）。
本文同时针对协变量$X in {1,dots,K}$的K样本情形，以及协变量$X$为连续型分布的情形，建立了估计量的一致相合性（uniform consistency）与收敛速度（rates of convergence）。