Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density

• DOI: 10.1214/09-ejs505
• Publication Date: 2010-02-17T14:27:54Z
• Authors: Madeleine Cule; Richard Samworth
• Keywords: Hellinger distance; Divergence (linguistics); Total variation; Sequence (biology); Kullback–Leibler divergence

We present theoretical properties of the log-concave maximum likelihood estimator a density based on an independent and identically distributed sample in ℝd. Our study covers both case where true underlying is log-concave, this model misspecified. begin by showing that for sequence densities, convergence distribution implies much stronger types – particular, it Hellinger distance even certain exponentially weighted total variation norms. In our main result, we prove existence uniqueness minimises Kullback–Leibler divergence from over class all also show converges almost surely these norms to minimiser. correctly specified model, demonstrates strong type consistency estimator; misspecified shows closest sense density.