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We develop a statistical estimator to infer the redshift probabilitydistribution of a photometric sample of galaxies from its angularcross-correlation in redshift bins with an overlapping spectroscopic sample.This estimator is a minimum variance weighted quadratic function of the data: aquadratic estimator. This extends and modifies the methodology presented byMcQuinn & White (2013). The derived source redshift distribution is degeneratewith the source galaxy bias, which must be constrained via additionalassumptions. We apply this estimator to constrain source galaxy redshiftdistributions in the Kilo-Degree imaging survey through cross-correlation withthe spectroscopic 2-degree Field Lensing Survey, presenting results first as abinned step-wise distribution in the range z < 0.8, and then building acontinuous distribution using a Gaussian process model. We demonstrate therobustness of our methodology using mock catalogues constructed from N-bodysimulations, and comparisons with other techniques for inferring the redshiftdistribution.

2dFLenS and KiDS: Determining source redshift distributions with cross-correlations