We study the problem of testing the null hypothesis that X and Y are conditionally independent given Z, where each of X, Y and Z may be functional random variables. This generalises testing the significance of X in a regression model of scalar response Y on functional regressors X and Z. We show however that even in the idealised setting where additionally (X, Y, Z) has a Gaussian distribution, the power of any test cannot exceed its size. Further modelling assumptions are needed and we argue that a convenient way of specifying these is based on choosing methods for regressing each of X and Y on Z. We propose a test statistic involving inner products of the resulting residuals that is simple to compute and calibrate: type I error is controlled uniformly when the in-sample prediction errors are sufficiently small. We show this requirement is met by ridge regression in functional linear model settings without requiring any eigen-spacing conditions or lower bounds on the eigenvalues of the covariance of the functional regressor. We apply our test in constructing confidence intervals for truncation points in truncated functional linear models and testing for edges in a functional graphical model for EEG data.
翻译:我们研究关于X和Y在条件上独立的无效假设的测试问题, X、 Y和Y在给定Z, 其中每个X、 Y和Z都可能是功能随机变量。 这个概观性测试X在功能递减器X和Z的萨拉回反应Y的回归模型中的重要性。 然而,我们发现,即使在其他(X、Y、Z)具有高尔西亚分布的理想化环境中,任何测试的力量都无法超过其大小,任何测试的力量不能超过高尔西亚分布。还需要进一步的模型假设,并且我们争辩说,说明这些参数的方便方式是基于选择在Z上X、Y和Y的回归方法。我们建议对由此产生的残余物的内部产品进行测试,这种测试将简单进行计算和校准:在模拟预测误差非常小的情况下,对第一类错误进行统一控制。我们指出,即使在功能线性模型环境模型的脊梯回归,无需任何静态条件或更低的界限,因此需要进一步的模拟假设。我们用测试来在功能递增器的磁度测试中,用于在功能性梯度中进行磁极的磁测测。