For hypothesis testing of functional parameters, given a functional statistic $T_n$ and a functional depth $D$ with respect to the distribution $P_n$ of $T_n$, we propose the depth value $DT_n \equiv D(T_n;P_n)$ as a test statistic, which we refer to as a depth statistic. In practice, its sampling distribution is approximated by a resampling method such as bootstrap. While achieving accurate sizes, a test based on the proposed depth statistic produces stronger power, as it remains sensitive even to subtle variations arising from complex functional patterns in the alternatives. Moreover, it is broadly applicable to a broad range of inference problems for functional parameters, including two-sample tests, analysis of variance, regression, etc. We provide its theoretical guarantee under mild assumptions along with examples of bootstrap methods and functional depths that satisfy these conditions. Its effectiveness is thoroughly investigated through numerical studies under two popular frameworks: (i) two-sample functional mean tests and (ii) mean response inference for function-on-function regression. The proposed depth statistic is illustrated with two data examples: Canadian weather and German electricity prices datasets.

翻译：针对函数型参数的假设检验问题，给定函数型统计量 $T_n$ 及其分布 $P_n$ 对应的函数型深度 $D$，我们提出以深度值 $DT_n \equiv D(T_n;P_n)$ 作为检验统计量，并将其称为深度统计量。在实际应用中，其抽样分布可通过自助法（bootstrap）等重抽样方法进行近似。基于所提出的深度统计量的检验在保持精确检验水平的同时，能够产生更强的检验功效，因为即使备择假设中由复杂函数模式引起的细微变异，该统计量仍能保持敏感性。此外，该方法广泛适用于函数型参数的各类推断问题，包括双样本检验、方差分析、回归分析等。我们在温和假设条件下给出了该方法的理论保证，并提供了满足这些条件的自助法与函数型深度实例。通过两种主流框架下的数值研究，我们对其有效性进行了全面考察：（i）函数型均值双样本检验；（ii）函数对函数回归中的均值响应推断。本文通过两个数据实例——加拿大气候数据集与德国电价数据集——对所提出的深度统计量进行了演示。