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Full-Text Articles in Physical Sciences and Mathematics
Statistical Topology Via Morse Theory, Persistence And Nonparametric Estimation, Peter Bubenik, Gunnar Carlsson, Peter T. Kim, Zhiming Luo
Statistical Topology Via Morse Theory, Persistence And Nonparametric Estimation, Peter Bubenik, Gunnar Carlsson, Peter T. Kim, Zhiming Luo
Mathematics and Statistics Faculty Publications
In this paper we examine the use of topological methods for multivariate statistics. Using persistent homology from computational algebraic topology, a random sample is used to construct estimators of persistent homology. This estimation procedure can then be evaluated using the bottleneck distance between the estimated persistent homology and the true persistent homology. The connection to statistics comes from the fact that when viewed as a nonparametric regression problem, the bottleneck distance is bounded by the sup-norm loss. Consequently, a sharp asymptotic minimax bound is determined under the sup–norm risk over H¨older classes of functions for the nonparametric regression problem on …
The Cross-Validated Adaptive Epsilon-Net Estimator, Mark J. Van Der Laan, Sandrine Dudoit, Aad W. Van Der Vaart
The Cross-Validated Adaptive Epsilon-Net Estimator, Mark J. Van Der Laan, Sandrine Dudoit, Aad W. Van Der Vaart
U.C. Berkeley Division of Biostatistics Working Paper Series
Suppose that we observe a sample of independent and identically distributed realizations of a random variable. Assume that the parameter of interest can be defined as the minimizer, over a suitably defined parameter space, of the expectation (with respect to the distribution of the random variable) of a particular (loss) function of a candidate parameter value and the random variable. Examples of commonly used loss functions are the squared error loss function in regression and the negative log-density loss function in density estimation. Minimizing the empirical risk (i.e., the empirical mean of the loss function) over the entire parameter space …
Existence And Uniqueness For A Variational Hyperbolic System Without Resonance, Peter W. Bates, Alfonso Castro
Existence And Uniqueness For A Variational Hyperbolic System Without Resonance, Peter W. Bates, Alfonso Castro
All HMC Faculty Publications and Research
In this paper, we study the existence of weak solutions of the problem
□u + ∇G(u) = f(t,x) ; (t,x) є Ω ≡ (0,π)x(0,π)
u(t,x) = 0 ; (t,x) є ∂Ω
where □ is the wave operator ∂2/∂t2 - ∂2/∂x2, G: Rn→R is a function of class C2 such that ∇G(0) = 0 and f:Ώ→R^n is a continuous function having first derivative with respect to t in (L2,(Ω))n and satisfying
f(0,x) = f(π,x) = 0
for all x є [0,π].