Inference for the Crossing Point of Two Continuous CDF's

Abstract

Let $\mathscr{F}$ denote the set of cdf's on $\mathbb{R}$ with density everywhere positive. Let $C_A = \{(F,G) \in \mathscr{F} \times \mathscr{F}$: there exists a unique $x^\ast \in \mathbb{R}$ such that $F(x) > G(x) \text{for} x << x^\ast \text{and} F(x) < G(x) \text{for} x > x^\ast\}, C_B - \{(F,G) \in \mathscr{F} \times \mathscr{F}: (G,F) \in C_A\}$. Based on independent random samples from $F$ and $G$ (assumed unknown), we give distribution-free tests of $H_0: F = G$ versus the alternatives that $(F,G) \in C_A, (F,G) \in C_B \text{or} (F,G) \in C_A \cup C_B$. Next, assuming that $(F,G) \in C_A$ (or in $C_B$), a point estimate of the crossing point $x^\ast$ is obtained and is shown to be strongly consistent and asymptotically normal. Finally, an asymptotically distribution-free confidence interval for $x^\ast$ is obtained. All inferences are based on a special criterion functional of $F$ and $G$, which yields $x^\ast$ when maximized (minimized) if $(F,G) \in C_A \lbrack(F,G) \in C_B\rbrack$.