We give a finite-sample analysis of predictive inference procedures after model selection in regression with random design. The analysis is focused on a statistically challenging scenario where the number of potentially important explanatory variables can be infinite, where no regularity conditions are imposed on unknown parameters, where the number of explanatory variables in a "good" model can be of the same order as sample size and where the number of candidate models can be of larger order than sample size. The performance of inference procedures is evaluated conditional on the training sample. Under weak conditions on only the number of candidate models and on their complexity, and uniformly over all data-generating processes under consideration, we show that a certain prediction interval is approximately valid and short with high probability in finite samples, in the sense that its actual coverage probability is close to the nominal one and in the sense that its length is close to the length of an infeasible interval that is constructed by actually knowing the "best" candidate model. Similar results are shown to hold for predictive inference procedures other than prediction intervals like, for example, tests of whether a future response will lie above or below a given threshold.
Conditional predictive inference for stable algorithms
我们调查一般适用和直观吸引力的预测区间的基础上$k$倍交叉验证。我们专注于建议的区间的条件覆盖概率，给定训练样本中的观察结果（因此，训练条件有效性），并表明它在适当的意义上接近标称水平，前提是用于计算点预测的基础算法在省略特征-响应对时足够稳定。我们的研究结果是基于有限样本分析的经验分布函数的$k$倍交叉验证残差，并保持在非参数设置只有最小的假设上的误差分布。为了说明我们的结果，我们还将其应用于高维线性预测，在那里我们获得一致的渐近训练条件有效性，因为样本量和维度都以相同的速率趋于无穷大，并且一致的参数估计通常失败。这些结果表明，尽管存在着严重的问题，restling程序推断的未知参数（参见。Bickel和Freedman，1983; El Karoui和Purdom，2018; Mammen，1996），交叉验证方法可以成功地应用于获得可靠的预测推理，即使在高维和有条件的训练数据。
Inference for Multivariate Regression Model based on synthetic data generated under Fixed-Posterior Predictive Sampling: comparison with Plug-in Sampling
