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On Semi-supervised Estimation of Discrete Distributions under f-divergences
- Publication Year :
- 2024
-
Abstract
- We study the problem of estimating the joint probability mass function (pmf) over two random variables. In particular, the estimation is based on the observation of $m$ samples containing both variables and $n$ samples missing one fixed variable. We adopt the minimax framework with $l^p_p$ loss functions. Recent work established that univariate minimax estimator combinations achieve minimax risk with the optimal first-order constant for $p \ge 2$ in the regime $m = o(n)$, questions remained for $p \le 2$ and various $f$-divergences. In our study, we affirm that these composite estimators are indeed minimax optimal for $l^p_p$ loss functions, specifically for the range $1 \le p \le 2$, including the critical $l_1$ loss. Additionally, we ascertain their optimality for a suite of $f$-divergences, such as KL, $\chi^2$, Squared Hellinger, and Le Cam divergences.<br />Comment: Full version. Presented in ISIT-24. arXiv admin note: text overlap with arXiv:2305.07955
- Subjects :
- Mathematics - Statistics Theory
Computer Science - Information Theory
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2405.09523
- Document Type :
- Working Paper