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Register a new userNonparametric Estimation of Band-limited Probability Density Functions
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Added by
Rahul Agarwal
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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In this paper, a nonparametric maximum likelihood (ML) estimator for band-limited (BL) probability density functions (pdfs) is proposed. The BLML estimator is consistent and computationally efficient. To compute the BLML estimator, three approximate algorithms are presented: a binary quadratic programming (BQP) algorithm for medium scale problems, a Trivial algorithm for large-scale problems that yields a consistent estimate if the underlying pdf is strictly positive and BL, and a fast implementation of the Trivial algorithm that exploits the band-limited assumption and the Nyquist sampling theorem (BLMLQuick). All three BLML estimators outperform kernel density estimation (KDE) algorithms (adaptive and higher order KDEs) with respect to the mean integrated squared error for data generated from both BL and infinite-band pdfs. Further, the BLMLQuick estimate is remarkably faster than the KD algorithms. Finally, the BLML method is applied to estimate the conditional intensity function of a neuronal spike train (point process) recorded from a rats entorhinal cortex grid cell, for which it outperforms state-of-the-art estimators used in neuroscience.
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We consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimisation problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package Sshaped.
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Classical two-sample permutation tests for equality of distributions have exact size in finite samples, but they fail to control size for testing equality of parameters that summarize each distribution. This paper proposes permutation tests for equality of parameters that are estimated at root-n or slower rates. Our general framework applies to both parametric and nonparametric models, with two samples or one sample split into two subsamples. Our tests have correct size asymptotically while preserving exact size in finite samples when distributions are equal. They have no loss in local-asymptotic power compared to tests that use asymptotic critical values. We propose confidence sets with correct coverage in large samples that also have exact coverage in finite samples if distributions are equal up to a transformation. We apply our theory to four commonly-used hypothesis tests of nonparametric functions evaluated at a point. Lastly, simulations show good finite sample properties of our tests.
Let ${mathcal S}_m$ be the set of all $mtimes m$ density matrices (Hermitian positively semi-definite matrices of unit trace). Consider a problem of estimation of an unknown density matrix $rhoin {mathcal S}_m$ based on outcomes of $n$ measurements of observables $X_1,dots, X_nin {mathbb H}_m$ (${mathbb H}_m$ being the space of $mtimes m$ Hermitian matrices) for a quantum system identically prepared $n$ times in state $rho.$ Outcomes $Y_1,dots, Y_n$ of such measurements could be described by a trace regression model in which ${mathbb E}_{rho}(Y_j|X_j)={rm tr}(rho X_j), j=1,dots, n.$ The design variables $X_1,dots, X_n$ are often sampled at random from the uniform distribution in an orthonormal basis ${E_1,dots, E_{m^2}}$ of ${mathbb H}_m$ (such as Pauli basis). The goal is to estimate the unknown density matrix $rho$ based on the data $(X_1,Y_1), dots, (X_n,Y_n).$ Let $$ hat Z:=frac{m^2}{n}sum_{j=1}^n Y_j X_j $$ and let $check rho$ be the projection of $hat Z$ onto the convex set ${mathcal S}_m$ of density matrices. It is shown that for estimator $check rho$ the minimax lower bounds in classes of low rank density matrices (established earlier) are attained up logarithmic factors for all Schatten $p$-norm distances, $pin [1,infty]$ and for Bures version of quantum Hellinger distance. Moreover, for a slightly modified version of estimator $check rho$ the same property holds also for quantum relative entropy (Kullback-Leibler) distance between density matrices.
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