A comparison of approximation methods for the estimation of probability distributions on parameters

Abstract

In this paper, we compare two computationally efficient approximation methods for the estimation of growth rate distributions in size-structured population models. After summarizing the underlying theoretical framework, we present several numerical examples as validation of the theory. Furthermore, we compare the results from a spline based approximation method and a delta function based approximation method for the inverse problem involving the estimation of the distributions of growth rates in size-structured mosquitofish populations. Convergence as well as sensitivity of the estimates with respect to noise in the data are discussed for both approximation methods. 1 Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

Asymptotic Properties of Probability Measure Estimators in a Nonparametric Model

“…[4,8,9,10,15]). However, it was demonstrated in [5] that if the sought-after probability measure is absolutely continuous, then the spline-based approximation methods converge much faster than do the Dirac measure approximation methods (in terms of the value of M). In addition, it was observed in [5] that the spline-based approximation methods also provide convergence for the associated probability density functions while the Dirac measure approximation methods do not do this.…”

A Division-Dependent Compartmental Model for Computing Cell Numbers in CFSE-based Lymphocyte Proliferation Assays

“…Cyton models and branching process models have been formulated to account for various levels of correlation [34,45,73], and these models may be incorporated into the compartmental model framework as described above. Alternatively, it may be possible (given any reasonable, identifiable parameterizations of cell division and death) to place probability distributions on these parameters (e.g., on the functions α i (t) and β i (t)) [6,9,12] in the manner described in Section 3.3.…”

Estimation in time-delay modeling of insecticide-induced mortality

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“…An alternative approach, "optimize in a function space setting and then discretize for computation," would require as a first step the computation of sensitivity (or gradients) of the original delay system (1) solutions with respect to its function space parameters (especially the probability kernel m). To compute these one might develop a set of sensitivity equations which could then be a basis for an (as yet undeveloped ) infinite dimensional nonlinear asymptotic statistical framework (see [8,9] for some initial efforts in this direction) for the computation of functional "confidence bands" for the estimated functions. This of course would involve infinite dimensional function space versions of the sensitivity and covariance matrices that are fundamental in the finite dimensional asymptotic theory for sampling distributions.…”