均值倾斜松弛分位数回归：固定内容区间泛函与广义贝叶斯计算

中文摘要 AI 辅助

概率内容本身并不决定区间的位置。我们研究由松弛分位数回归（RQR）诱导的区间泛函，其残积检查损失无需预先指定端点分位数即可估计两个未标记的原始根。在明确条件下，无约束总体最小化器是唯一的连续内容-c区间，其保留均值等于总体均值。固定均值倾斜在保留内容的同时将该保留均值移至μ+δ。内部可允许倾斜索引所有有限根内部内容-c窗口，边界成员通过合格的单侧极限获得。因此，等尾区间和最短连续区间具有特定于分布的恢复倾斜。我们使用伪非对称拉普拉斯正态-指数增广构造基于损失的广义后验。固定速率均值倾斜采样器在合适高斯先验下覆盖静态回归，普通RQR在零倾斜时精确获得。实现的普通分支还支持使用西村-萨查德增广（RHS-NS）的条件高斯正则化马蹄适配器，且冻结的深度回声状态网络特征矩阵是同一静态扫描的确定性非线性设计特例。动态线性扩展用交替根特定的前向滤波后向采样步骤替换系数块：堆叠状态先验为高斯分布，但联合增广观测核为四次方。当前软件和实证证据关注普通RQR；非零倾斜算法已推导但尚未实现或验证。所有更新涉及损失和先验下的区间根泛函，而非响应似然或后验预测响应。

英文摘要

Intervals with the same probability content can have different endpoint placements and widths. This matters for tolerance inference, where a reported interval must also satisfy a repeated-sampling content-confidence statement. We develop mean-tilted intervals (MTIs), a fixed-content family indexed by retained-mean balance. The zero-tilt member is the mean-preserving interval (MPI) induced by the residual-product criterion of Pouplin et al.; nonzero tilts move through admissible contiguous windows, including distribution-specific central and shortest intervals. For tolerance inference, we introduce TCSP, a tolerance-calibrated shortest-path action. TCSP chooses the retained order-statistic count by distribution-free scan calibration and reports the shortest closed window at that count. This keeps the certified interval action separate from generalized-posterior endpoint summaries. We also study a calibrated MTI-ECM comparator that profiles fitted content and tilt over a prespecified grid and applies an independent Dirichlet-process content-probability check. In iid simulations at tolerance confidence 0.95, we compare TCSP, MTI-ECM, Young-Mathew interpolation, and Wilks intervals across feasible content-sample-size cells and eight continuous distributions. The study emphasizes skewed distributions, where placement matters most, and excludes cells where the sample range cannot support the requested two-sided distribution-free statement.

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22 pages, 5 figures