In this paper, we derive power guarantees of some sequential tests for bounded mean under general alternatives. We focus on testing procedures using nonnegative supermartingales which are anytime valid and consider alternatives which coincide asymptotically with the null (e.g. vanishing mean) while still allowing to reject in finite time. Introducing variance constraints, we show that the alternative can be broaden while keeping power guarantees for certain second-order testing procedures. We also compare different test procedures in multidimensional setting using characteristics of the rejection times. Finally, we extend our analysis to other functionals as well as testing and comparing forecasters. Our results are illustrated with numerical simulations including bounded mean testing and comparison of forecasters.

翻译：本文推导了在一般备择假设下，关于有界均值若干序贯检验的功效保证。我们聚焦于使用非负上鞅的检验方法，这类方法具有任意时间有效性，并考虑与零假设渐近重合（例如趋于零的均值）但仍能在有限时间内拒绝零假设的备择假设。通过引入方差约束，我们证明在保持某些二阶检验程序功效保证的同时，可以扩展备择假设的范围。此外，我们利用拒绝时间的特征在多维场景下比较了不同的检验程序。最后，我们将分析拓展至其他泛函形式以及预测模型的检验与比较。通过包含有界均值检验和预测模型比较的数值模拟，我们对研究结果进行了例证说明。