论文标题
对数凸线密度的置信带
Confidence bands for a log-concave density
论文作者
论文摘要
我们提出了一种新的关于对数符号分布的推理的方法：而不是使用最大似然的方法，而是建议将日志concovity约束纳入适当的非参数置信度，以设置为CDF $ f $。这种方法的优点是它会自动提供统计不确定性的度量，因此克服了最大似然估计的明显限制。特别是，我们展示了如何为具有有限样本保证置信度的密度构建置信带。我们在此处介绍的$ F $设置的非参数置信设置具有有吸引力的计算和统计属性：它允许通过凸面编程的差异带来现代工具从优化来解决此问题，并且可以带来最佳的统计推断。我们表明，当log密度为$ k $ - affine时，所得置信带的宽度几乎以几乎参数$ n^{ - \ frac {1} {2}} $ rate收敛。
We present a new approach for inference about a log-concave distribution: Instead of using the method of maximum likelihood, we propose to incorporate the log-concavity constraint in an appropriate nonparametric confidence set for the cdf $F$. This approach has the advantage that it automatically provides a measure of statistical uncertainty and it thus overcomes a marked limitation of the maximum likelihood estimate. In particular, we show how to construct confidence bands for the density that have a finite sample guaranteed confidence level. The nonparametric confidence set for $F$ which we introduce here has attractive computational and statistical properties: It allows to bring modern tools from optimization to bear on this problem via difference of convex programming, and it results in optimal statistical inference. We show that the width of the resulting confidence bands converges at nearly the parametric $n^{-\frac{1}{2}}$ rate when the log density is $k$-affine.
