Minimax Estimation of Discrete Distributions Under <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> Loss
نویسندگان
چکیده
منابع مشابه
Variational Minimax Estimation of Discrete Distributions under KL Loss
We develop a family of upper and lower bounds on the worst-case expected KL loss for estimating a discrete distribution on a finite numberm of points, given N i.i.d. samples. Our upper bounds are approximationtheoretic, similar to recent bounds for estimating discrete entropy; the lower bounds are Bayesian, based on averages of the KL loss under Dirichlet distributions. The upper bounds are con...
متن کاملMinimax Estimation of Discrete Distributions under $\ell_1$ Loss
We consider the problem of discrete distribution estimation under l1 loss. We provide tight upper and lower bounds on the maximum risk of the empirical distribution (the maximum likelihood estimator), and the minimax risk in regimes where the support size S may grow with the number of observations n. We show that among distributions with bounded entropy H , the asymptotic maximum risk for the e...
متن کاملMathML formatting with TEX rules , TEX fonts , and TEX quality ∗
In this paper we investigate the architecture of a MathML formatting engine based on an abstraction of the TEX box primitives. This engine is carefully designed so that the TEX-dependent formatting rules are isolated from the independent ones and is capable of achieving TEX-comparable output quality when used in conjunction with TEX fonts. We show how the formatting rules presented in Appendix ...
متن کاملA Survey of Free Math Fonts for TEX and LATEX
Comparison of Features 17 Creation of this Survey 20 ∗Copyright 2006 Stephen G. Hartke. Permission is granted to distribute verbatim or modified copies of this document provided this notice remains intact. An initial version of this article appeared in The PracTEX Journal, 1, 2006, http://www.tug.org/pracjourn/ 2006-1/hartke/. The permanent home of this article is http://ctan.tug.org/tex-archiv...
متن کاملE-TEX: Guidelines for Future TEX Extensions
With the announcement of TEX 3.0, Don Knuth acknowledged the need of the (ever growing) TEX community for an even better system. But at the same time, he made it clear, that he will not get involved in any further enhancements that would change the TEXbook. TEX started out originally as a system designed to typeset its author’s own publications. In the meantime it serves hundreds of thousands o...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: IEEE Transactions on Information Theory
سال: 2015
ISSN: 0018-9448,1557-9654
DOI: 10.1109/tit.2015.2478816