1,909 research outputs found
A change-point problem and inference for segment signals
We address the problem of detection and estimation of one or two
change-points in the mean of a series of random variables. We use the formalism
of set estimation in regression: To each point of a design is attached a binary
label that indicates whether that point belongs to an unknown segment and this
label is contaminated with noise. The endpoints of the unknown segment are the
change-points. We study the minimal size of the segment which allows
statistical detection in different scenarios, including when the endpoints are
separated from the boundary of the domain of the design, or when they are
separated from one another. We compare this minimal size with the minimax rates
of convergence for estimation of the segment under the same scenarios. The aim
of this extensive study of a simple yet fundamental version of the change-point
problem is twofold: Understanding the impact of the location and the separation
of the change points on detection and estimation and bringing insights about
the estimation and detection of convex bodies in higher dimensions.Comment: arXiv admin note: substantial text overlap with arXiv:1404.622
Near-Optimal Recovery of Linear and N-Convex Functions on Unions of Convex Sets
In this paper we build provably near-optimal, in the minimax sense, estimates
of linear forms and, more generally, "-convex functionals" (the simplest
example being the maximum of several fractional-linear functions) of unknown
"signal" known to belong to the union of finitely many convex compact sets from
indirect noisy observations of the signal. Our main assumption is that the
observation scheme in question is good in the sense of A. Goldenshluger, A.
Juditsky, A. Nemirovski, Electr. J. Stat. 9(2) (2015), arXiv:1311.6765, the
simplest example being the Gaussian scheme where the observation is the sum of
linear image of the signal and the standard Gaussian noise. The proposed
estimates, same as upper bounds on their worst-case risks, stem from solutions
to explicit convex optimization problems, making the estimates
"computation-friendly.
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