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Statistical inference for inverse problems. (English) Zbl 1137.62325

Summary: We study statistical inference for certain inverse problems. We go beyond mere estimation purposes and review and develop the construction of confidence intervals and confidence bands in some inverse problems, including deconvolution and the backward heat equation. Further, we discuss the construction of certain hypothesis tests, in particular concerning the number of local maxima of the unknown function. The methods are illustrated in a case study, where we analyze the distribution of heliocentric escape velocities of galaxies in the Centaurus galaxy cluster, and provide statistical evidence for its bimodality.

MSC:

62G07 Density estimation
62G15 Nonparametric tolerance and confidence regions
62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
85A35 Statistical astronomy
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