2021/06/30 by Constantinos Daskalakis, Vasilis Kontonis, Daskalakis, Constantinos +5 · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2106.15908
openalex publication_date 2021/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show a statistical version of Taylor's theorem and apply this result to non-parametric density estimation from truncated samples, which is a classical challenge in Statistics \citewoodroofe1985estimating, stute1993almost. The single-dimensional version of our theorem has the following implication: "For any distribution P on [0, 1] with a smooth log-density function, given samples from the conditional distribution of P on [a, a + ε] ⊂ [0, 1], we can efficiently identify an approximation to P over the whole interval [0, 1], with quality of approximation that improves with the smoothness of P." To the best of knowledge, our result is the first in the area of non-parametric density estimation from truncated samples, which works under the hard truncation model, where the samples outside some survival set S are never observed, and applies to multiple dimensions. In contrast, previous works assume single dimensional data where each sample has a different survival set S so that samples from the whole support will ultimately be collected.