The t-statistic is a widely-used scale-invariant statistic for testing the null hypothesis that the mean is zero. Martingale methods enable sequential testing with the t-statistic at every sample size, while controlling the probability of falsely rejecting the null. For one-sided sequential tests, which reject when the t-statistic is too positive, a natural question is whether they also control false rejection when the true mean is negative. We prove that this is the case using monotone likelihood ratios and sufficient statistics. We develop applications to the scale-invariant t-test, the location-invariant $\chi^2$-test and sequential linear regression with nuisance covariates.

翻译：t统计量是一种广泛使用的尺度不变统计量，用于检验均值为零的原假设。鞅方法使得t统计量能够在每个样本量下进行序贯检验，同时控制错误拒绝原假设的概率。对于单边序贯检验（当t统计量过于正向时拒绝原假设），一个自然的问题是当真实均值为负时，这些检验是否也能控制错误拒绝。我们利用单调似然比和充分统计量证明了情况确实如此。我们进一步将结果应用于尺度不变的t检验、位置不变的χ²检验以及带有冗余协变量的序贯线性回归。