4 research outputs found
A law of the iterated logarithm for Grenander's estimator
In this note we prove the following law of the iterated logarithm for the
Grenander estimator of a monotone decreasing density: If , , and is continuous in a neighborhood of , then \begin{eqnarray*}
\limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} (
\widehat{f}_n (t_0 ) - f(t_0) ) = \left| f(t_0) f'(t_0)/2 \right|^{1/3} 2M
\end{eqnarray*} almost surely where and T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} ; here is the two-sided Strassen limit set on . The proof relies on laws of the
iterated logarithm for local empirical processes, Groeneboom's switching
relation, and properties of Strassen's limit set analogous to distributional
properties of Brownian motion.Comment: 11 pages, 3 figure
Confidence bands for a log-concave density
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 . 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 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 rate when the log density is -affine.Comment: Added more experiments, other minor change