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.
翻译:我们提出了一个新的方法来推断对日对流的分布:我们不使用最大可能性的方法,而是提议将日对流限制纳入对cdf $F的适当的非参数性信任套件中。这种方法的优点是,它自动提供了统计不确定性的量度,从而克服了最大概率估计的明显限制。特别是,我们展示了如何为具有有限样本的密度建立信任带,这种密度保证了信任度。我们在这里采用的对价非参数性信任套件具有有吸引力的计算和统计特性:它能够通过对convex编程的差别,将优化的现代工具用于应对这一问题,并产生最佳的统计推论。我们显示,当日志密度为$k$-afine时,由此产生的信任带的宽度接近对等值$n ⁇ \frac{1 ⁇ 2 ⁇ $。