Abstract
We study the problem of nonparametric estimation of probability density functions (pdf) with a product form on the domain $\triangle =\{(x_{1},\ldots ,x_{d})\in{\mathbb{R}} ^{d},0\leq x_{1}\leq \dots\leq x_{d}\leq 1\}$. Such pdf’s appear in the random truncation model as the joint pdf of the observations. They are also obtained as maximum entropy distributions of order statistics with given marginals. We propose an estimation method based on the approximation of the logarithm of the density by a carefully chosen family of basis functions. We show that the method achieves a fast convergence rate in probability with respect to the Kullback-Leibler divergence for pdf’s whose logarithm belong to a Sobolev function class with known regularity. In the case when the regularity is unknown, we propose an estimation procedure using convex aggregation of the log-densities to obtain adaptability. The performance of this method is illustrated in a simulation study.
Highlights
We estimate densities with product form on the simplex = {(x1, . . . , xd) ∈ Rd, 0 ≤ x1 ≤ · · · ≤ xd ≤ 1} by a nonparametric approach given a sample of n independent observations Xn = (X1, . . . , Xn)
We restrict our attention to densities which can be written in the form, for x = (x1, . . . , xd) ∈ Rd: d
Let us denote the support of a probability density g by supp (g) = {x ∈ Rd, g(x) > 0}
Summary
Densities of this form arise, in particular, as solutions for the maximum entropy problem for the distribution of order statistics with given marginals, or in the case of the random truncation model. Adaptive methods for function estimation based on a random sample include Lepski’s method, model selection, wavelet thresholding and aggregation of estimators. The convex aggregate estimator fλ takes the form: fλ = exp d mi λm θi,kφi,k(xi) − ψλ 1 , m∈Mn i=1 k=1 with λ ∈ Λ+ = {λ = (λm, m ∈ Mn), λm ≥ 0 and m∈Mn λm = 1} and normalizing constant ψλ given by: ψλ = log exp d mi λm θi,k φi,k (xi ). We select λ∗n by maximizing a penalized version of the log-likelihood function We show that this method gives a sequence of estimators fλ∗n, free of the smoothness parameters r1, .
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