### Using Confidence Bands to Quantify Uncertainty in the Estimation of Probability Distributions

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** Jimena Davis,
North Carolina State University **

We will present and compare computational and statistical results of some approximation methods for a general class of inverse problems in which the underlying dynamics are described by partial differential equations and the unknown quantity of interest is a probability distribution. We are interested in determining the distribution P* from a given family of probability distributions P(Q) that gives the best fit of the underlying model to some given data. In general, this optimization problem involves both an infinite dimensional state space and an infinite dimensional parameter space. Therefore, computationally efficient approximation methods are desired. In choosing these methods, we want the finite dimensional sets P^M(Q) to converge to P(Q) in some sense, and in our efforts, we use the Prohorov metric of weak star convergence of measures for these results. We will also demonstrate how to construct “functional” confidence bands for estimated probability distributions by using the standard asymptotic theory for ordinary least squares (OLS) estimators for finite dimensional parameter estimation problems. The approximation methods and techniques that we will present are applicable to a variety of inverse problems. We will present results for the estimation of growth rate distributions in size-structured mosquitofish populations.

**Abstract Author(s):** Jimena L. Davis and H.T. Banks