Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1301.5722

Asymptotically Optimal Detection of Changes in Stochastic Models with Switching Regimes

Boris Brodsky·Boris Darkhovsky
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper deals with the problem of asymptotically optimal detection of changes in regime-switching stochastic models. We need to divide the whole obtained sample of data into several sub-samples with observations belonging to different states of a stochastic models with switching regimes. For this purpose, the idea of reduction to a corresponding change-point detection problem is used. Both univariate and multivariate switching models are considered. For the univariate case, we begin with the study of binary mixtures of probabilistic distributions. In theorems 1 and 2 we prove that type 1 and type 2 errors of the proposed method converge to zero exponentially as the sample size tends to infinity. In theorem 3 we prove that the proposed method is asymptotically optimal by the rate of this convergence in the sense that the lower bound in the a priori informational inequality is attained for our method. Several generalizations to the case of multiple univariate mixtures of probabilistic distributions are considered. For the multivariate case, we first study the general problem of classification of the whole array of data into several sub-arrays of observations from different regimes of a multivariate stochastic model with switching states. Then we consider regression models with abnormal observations and switching sets of regression coefficients. Results of a detailed Monte Carlo study of the proposed method for different stochastic models with switching regimes are presented.

Explore related subjects

math.STstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Boris Brodsky, Boris Darkhovsky. 2013-01-24. Asymptotically Optimal Detection of Changes in Stochastic Models with Switching Regimes. https://arxiv.org/abs/1301.5722

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Functional independent component analysis by choice of norm: a framework for near-perfect classification

We develop a theory for functional independent component analysis in an infinite-dimensional framework using Sobolev spaces that accommodate smoother functions. The notion of penalized kurtosis is introduced motivated by Silverman's method for smoothing principal components. This approach allows for a classical definition of independent components obtained via projection onto the eigenfunctions of a smoothed kurtosis operator mapping a whitened functional random variable. We discuss the theoretical properties of this operator in relation to a generalized Fisher discriminant function and the relationship it entails with the Feldman-Hájek dichotomy for Gaussian measures, both of which are critical to the principles of functional classification. The proposed estimators are a particularly competitive alternative in binary classification of functional data and can eventually achieve the so-called near-perfect classification, which is a genuine phenomenon of high-dimensional data. Our methods are illustrated through simulations, various real datasets, and used to model electroencephalographic biomarkers for the diagnosis of depressive disorder.

math.ST↗
arXiv2026

Trace-Class Results for MCMC Algorithms for Student-$t$ Regression Models

In this paper, we consider MCMC algorithms for Student-$t$ regression models. In three cases, we investigate the efficiency of Markov chains based on the algorithms in terms of whether trace-class results hold or not. First, we consider the case where the parameters follow a matrix-normal-inverse-Wishart distribution and show that the Markov operator associated with a standard data augmentation algorithm is trace-class. Second, we consider the case of an improper prior and univariate outcomes. In this case, the standard Markov operator is not trace-class but the Markov operator associated with a collapsed Gibbs algorithm is trace-class. Third, we consider the case of an improper prior and multivariate outcomes. We obtain a trace-class result for a parameter expanded data augmentation algorithm which is based on a univariate working parameter. Finally, we consider the problem of numerially estimating a convergence rate of the trace-class Markov operator in the second case.

math.ST↗
arXiv2026

The Manifold Hypothesis under Unknown Gaussian Noise:Conditional Certificates and Consistent Dimension Estimation

We study what noisy data can establish about the Manifold Hypothesis under explicit identification and regularity conditions. A population residual certificate combines independent-view localization, Gaussian concentration, membership uncertainty, and population transfer. Existing rectifiability criteria then yield a covered-scale consequence. For a local smooth manifold with positive Hölder density, the actual-ball covariance limit identifies the spectral crossing with geometric dimension. We prove almost-sure eventual recovery under repeated observations. Reusing accurate localization averages improves the sufficient point-sample condition from $Nr^{d+4}\gg\log N$ to $Nr^d\gg\log N$, with replication $kr^2\gg\log N$. A two-mass certificate controls incorrect geometric-dimension emissions under declared class bounds. For single observations with unknown Gaussian noise, affine-support or known coordinate-bound restrictions provide noise intervals and consistent Gaussian correlation-dimension estimators. Ahlfors regularity identifies this exponent with Hausdorff dimension and with the geometric dimension of a homogeneous smooth class. Exact Cantor calculations delineate the limits of integer spectral counts and adjacent-radius slopes. We credit established local PCA, rectifiability, concentration, binomial inference, and deconvolution results before specifying our constructions. Reproducible experiments distinguish point estimation, finite-scale coverage, and certificate emission.

math.ST↗