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Boundary estimation in the presence of measurement error with unknown variance

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Boundary estimation appears naturally in economics in the context of productivity analysis. The performance of a firm is measured by the distance between its achieved output level (quantity of goods produced) and an optimal production frontier which is the locus of the maximal achievable output given the level of the inputs (labor, energy, capital, etc.). Frontier estimation becomes difficult if the outputs are measured with noise and most approaches rely on restrictive parametric assumptions. This paper contributes to the direction of nonparametric approaches. A slightly simplified version of the general problem can be written as Y=X Z, where Y is the observable output, X is the unobserved variable of interest with support [0,τ] and density f, and Z is the noise. Suppose that f(τ)>0, and that Z is independent of X and is log-normally distributed with log Z~N(0,σ2) for some unknown variance σ 2. The novelty of our approach consists in proposing a method for simultaneous estimation of τ and σ. The asymptotic consistency and the rate of convergence of the estimators are established, and simulations are carried out to verify the performance of the estimators for small samples. We also describe how the approach could be extended to the problem of estimating a frontier function. (This is joint work with Alois Kneip and Leopold Simar.)

This talk is part of the Statistics series.

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