## Abstract

Consider the problem of nonparametric estimation of an unknown β -Hölder smooth density pXY at a given point, where X and Y are both d dimensional. An infinite sequence of i.i.d. samples (Xi,Yi) are generated according to this distribution, and two terminals observe (Xi) and (Yi) , respectively. They are allowed to exchange k bits either in oneway or interactively in order for Bob to estimate the unknown density. We show that the minimax mean square risk is order (k k )-2β d+2β for one-way protocols and k-2β d+2β for interactive protocols. The logarithmic improvement is nonexistent in the parametric counterparts, and therefore can be regarded as a consequence of nonparametric nature of the problem. Moreover, a few rounds of interactions achieve the interactive minimax rate: the number of rounds can grow as slowly as the super-logarithm (i.e., inverse tetration) of k. The proof of the upper bound is based on a novel multi-round scheme for estimating the joint distribution of a pair of biased Bernoulli variables, and the lower bound is built on a sharp estimate of a symmetric strong data processing constant for biased Bernoulli variables.

Original language | English (US) |
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Pages (from-to) | 7867-7886 |

Number of pages | 20 |

Journal | IEEE Transactions on Information Theory |

Volume | 69 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2023 |

Externally published | Yes |

## Keywords

- Density estimation
- communication complexity
- learning with system constraints
- nonparametric statistics
- strong data processing constant

## ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Library and Information Sciences