论文标题
单调回归中贝叶斯估计和测试的收敛速率
Convergence Rates for Bayesian Estimation and Testing in Monotone Regression
论文作者
论文摘要
在统计建模中通常会自然出现形状限制，例如对功能的单调性。 我们考虑了一种贝叶斯的方法来估计单调回归函数和单调性测试。我们使用分段常数函数构建先前的分布。为了估计，事先对这些步骤的高度施加单调性是明智的，但是在理论上很难进行后部分析。我们考虑使用``投射 - 后者''方法，其中使用了共轭正常先验，但是单调性约束是通过单调函数空间的投影图对后验样品施加的。我们表明，在$ l_1 $ -metric下，以最佳利率$ n^{ - 1/3} $以最佳速率的后验合同，在经验$ l_p $ -metrics下以$ 0 <p \ le 2 $的经验$ l_p $ -metrics下的几乎最佳利率。投射 - 后者方法在计算上也更方便。我们还使用一组单调函数的缩小邻居的后部概率构建了贝叶斯测试，以假设单调性。我们表明，所得测试具有通用的一致性属性，并获得了分离率，从而确保所得的功率函数接近一个。
Shape restrictions such as monotonicity on functions often arise naturally in statistical modeling. We consider a Bayesian approach to the problem of estimation of a monotone regression function and testing for monotonicity. We construct a prior distribution using piecewise constant functions. For estimation, a prior imposing monotonicity of the heights of these steps is sensible, but the resulting posterior is harder to analyze theoretically. We consider a ``projection-posterior'' approach, where a conjugate normal prior is used, but the monotonicity constraint is imposed on posterior samples by a projection map on the space of monotone functions. We show that the resulting posterior contracts at the optimal rate $n^{-1/3}$ under the $L_1$-metric and at a nearly optimal rate under the empirical $L_p$-metrics for $0<p\le 2$. The projection-posterior approach is also computationally more convenient. We also construct a Bayesian test for the hypothesis of monotonicity using the posterior probability of a shrinking neighborhood of the set of monotone functions. We show that the resulting test has a universal consistency property and obtain the separation rate which ensures that the resulting power function approaches one.
