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Register a new userRecycled Least Squares Estimation in Nonlinear Regression
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Added by
Ben Boukai
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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We consider a resampling scheme for parameters estimates in nonlinear regression models. We provide an estimation procedure which recycles, via random weighting, the relevant parameters estimates to construct consistent estimates of the sampling distribution of the various estimates. We establish the asymptotic normality of the resampled estimates and demonstrate the applicability of the recycling approach in a small simulation study and via example.
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This paper proposes the capped least squares regression with an adaptive resistance parameter, hence the name, adaptive capped least squares regression. The key observation is, by taking the resistant parameter to be data dependent, the proposed estimator achieves full asymptotic efficiency without losing the resistance property: it achieves the maximum breakdown point asymptotically. Computationally, we formulate the proposed regression problem as a quadratic mixed integer programming problem, which becomes computationally expensive when the sample size gets large. The data-dependent resistant parameter, however, makes the loss function more convex-like for larger-scale problems. This makes a fast randomly initialized gradient descent algorithm possible for global optimization. Numerical examples indicate the superiority of the proposed estimator compared with classical methods. Three data applications to cancer cell lines, stationary background recovery in video surveillance, and blind image inpainting showcase its broad applicability.
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Understanding forest fire spread in any region of Canada is critical to promoting forest health, and protecting human life and infrastructure. Quantifying fire spread from noisy images, where regions of a fire are separated by change-point boundaries, is critical to faithfully estimating fire spread rates. In this research, we develop a statistically consistent smooth estimator that allows us to denoise fire spread imagery from micro-fire experiments. We develop an anisotropic smoothing method for change-point data that uses estimates of the underlying data generating process to inform smoothing. We show that the anisotropic local constant regression estimator is consistent with convergence rate $Oleft(n^{-1/{(q+2)}}right)$. We demonstrate its effectiveness on simulated one- and two-dimensional change-point data and fire spread imagery from micro-fire experiments.
We consider a re-sampling scheme for estimation of the population parameters in the mixed effects nonlinear regression models of the type use for example in clinical pharmacokinetics, say. We provide an estimation procedure which {it recycles}, via random weighting, the relevant two-stage parameters estimates to construct consistent estimates of the sampling distribution of the various estimates. We establish the asymptotic consistency and asymptotic normality of the resampled estimates and demonstrate the applicability of the {it recycling} approach in a small simulation study and via example.
A partial least squares regression is proposed for estimating the function-on-function regression model where a functional response and multiple functional predictors consist of random curves with quadratic and interaction effects. The direct estimation of a function-on-function regression model is usually an ill-posed problem. To overcome this difficulty, in practice, the functional data that belong to the infinite-dimensional space are generally projected into a finite-dimensional space of basis functions. The function-on-function regression model is converted to a multivariate regression model of the basis expansion coefficients. In the estimation phase of the proposed method, the functional variables are approximated by a finite-dimensional basis function expansion method. We show that the partial least squares regression constructed via a functional response, multiple functional predictors, and quadratic/interaction terms of the functional predictors is equivalent to the partial least squares regression constructed using basis expansions of functional variables. From the partial least squares regression of the basis expansions of functional variables, we provide an explicit formula for the partial least squares estimate of the coefficient function of the function-on-function regression model. Because the true forms of the models are generally unspecified, we propose a forward procedure for model selection. The finite sample performance of the proposed method is examined using several Monte Carlo experiments and two empirical data analyses, and the results were found to compare favorably with an existing method.
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