This paper presents minimax rates for density estimation when the data dimension @math is allowed to grow with the number of observations @math rather than remaining fixed as in previous analyses. We prove a non-asymptotic lower bound which gives the worst-case rate over standard classes of smooth densities, and we show that kernel density estimators achieve this rate. We also give oracle choices for the bandwidth and derive the fastest rate @math can grow with @math to maintain estimation consistency.
Unlike in the density estimation setting, there are some related results in the information theory literature which endeavor to address the limits of estimation under the triangular array. Essentially, this work examines the estimation of the joint distribution of a @math -block of random variables observed in sequence from an ergodic process supported on a finite set of points. show that if @math grows like @math , then these joint distributions can be estimated consistently. An extension of these results to the case of a Markov random field embedded in a higher dimension is given by @cite_0 . Our results are slightly slower than these (see cor:d-rate ), but estimating continuous densities rather than finitely supported distributions is more difficult.
@cite_0
2039898347
The paper deals with upper and lower bounds for the quality of (probability) density estimation. Connections are established between these problems and the theory of approximation of functions. Particularly, it is demonstrated how some of Kolmogorov's concepts work