# Three problems in high-dimensional statistical parameter estimation

## Abstract/Contents

- Abstract
- Statistical parameter estimation is the process of estimating parameters of a system from direct or indirect observations. As noted by Bradly Efron in his recent book [29], at its inception parameter estimation was about simple questions using large datasets, e.g., estimating mean and variance of the age of a population using survey data. Later it evolved into problems involving still simple questions but now small amount of data. Examples included hypothesis testing and estimating parameters of a communication channel using short training sequences. This setting resulted in a fascinating literature and promotion of ideas like statistical eciency and power. In recent years, a new class of parameter estimation has been emerging which is best exemplified by the Netflix challenge. These problems are characterized by vast amount of input data but at the same time complex and large set of parameters to be estimated. In the case of the Netflix challenge, the data consist of about 10^8 known movie ratings and the challenge was to estimate about 10^10 unknown ratings. In this work, we consider three problems of this type, namely, learning and controlling high- dimensional dynamical systems, and statistical signal processing for time of flight mass spectrometry. We present both theoretical and experimental results. First, we consider the problem of learning high-dimensional dynamical systems. A continuous time autonomous dynamical system is described by a stochastic differential equation while a discrete time system is described by a state evolution equation. We consider the problem of learning the drift coefficient of a p-dimensional stochastic differential equation from a sample path of length T. We assume that the drift is parametrized by a high-dimensional vector, and study the support recovery problem in the limit in which both p and T tend to infinity. In particular, we prove a general lower iv bound on the sample-complexity T by using a characterization of mutual information as a time integral of conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic differential equations, the drift coefficient is parametrized by a p-̳>̳

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2013 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Ibrahimi, Morteza | |
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Associated with | Stanford University, Department of Electrical Engineering. | |

Primary advisor | Montanari, Andrea | |

Thesis advisor | Montanari, Andrea | |

Thesis advisor | Pauly, John (John M.) | |

Thesis advisor | Van Roy, Benjamin | |

Advisor | Pauly, John (John M.) | |

Advisor | Van Roy, Benjamin |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Morteza Ibrahimi. |
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Note | Submitted to the Department of Electrical Engineering. |

Thesis | Thesis (Ph.D.)--Stanford University, 2013. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2013 by Morteza Ibrahimi
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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