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Keywords:

  • Adaptive bandwidth;
  • Bandwidth selection;
  • Bootstrap;
  • Cosmological models;
  • Mass bias parameter;
  • Minimum χ2 estimation;
  • Sloan Digital Sky Survey;
  • Two-point correlation function

Summary.  We focus on selecting optimal bandwidths for non-parametric estimation of the two-point correlation function of a point pattern. We obtain these optimal bandwidths by using a bootstrap approach to select a bandwidth that minimizes the integrated squared error. The variance term is estimated by using a non-parametric spatial bootstrap, whereas the bias term is estimated with a plug-in approach using a pilot estimator of the two-point correlation function based on a parametric model. The choice of parametric model for the pilot estimator is very flexible. Depending on applications, parametric statistical point models, physical models or functional models can be used. We also explore the use of the procedure for selecting adaptive optimal bandwidths. We investigate the performance of the bandwidth selection procedure by using a simulation study. In our data example, we apply our method to a Sloan Digital Sky Survey galaxy cluster catalogue by using a pilot estimator based on the power law functional model in cosmology. The resulting non-parametric two-point correlation function estimate is then used to estimate a cosmological mass bias parameter that describes the relationship between the galaxy mass distribution and the underlying matter distribution.