2007/10/01 by Franz Merkl, Leila Mohammadi
Engineering · Mathematics · #Advanced Statistical Methods and Models #Applied mathematics #Control Systems and Identification #Cube root #Entropy (arrow of time) #Estimator #Intersection (aeronautics) #Mathematical analysis #Mathematics #Probabilistic logic #Sample size determination #Statistical Methods and Inference #Statistics #Upper and lower bounds #math.ST #msc:62G05 #msc:62G20 #stat.TH
paper · pdf · doi:10.1214/009053607000000325
published as Annals of Statistics 2007, Vol. 35, No. 5, 2193-2218 · Published in at http://dx.doi.org/10.1214/009053607000000325 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2007/10/01 · arxiv created 2007/12/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
This paper is concerned with estimating the intersection point of two densities, given a sample of both of the densities. This problem arises in classification theory. The main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size. As corollaries, we obtain probabilistic bounds for the prediction error in a classification problem. The key to the proof is an entropy estimate. The lower bounds are based on bounds for general estimators, which are applicable in other contexts as well. Furthermore, we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds.