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The Box-Cox quantile regression model using the two stage method introduced by Chamberlain (1994) and Buchinsky (1995) provides an attractive extension of linear quantile regression techniques. However, a major numerical problem exists when implementing this method which has not been addressed so far in the literature. We suggest a simple solution modifying the estimator slightly. This modification is easy to implement. The modified estimator is still [square root] n-consistent and its asymptotic distribution can easily be derived. A simulation study confirms that the modified estimator works well.
Atmospheric aerosols and their effect on clouds are thought to be important for anthropogenic radiative forcing of the climate, yet remain poorly understood1. Globally, around half of cloud condensation nuclei originate from nucleation of atmospheric vapours2. It is thought that sulfuric acid is essential to initiate most particle formation in the atmosphere3,4, and that ions have a relatively minor role5. Some laboratory studies, however, have reported organic particle formation without the intentional addition of sulfuric acid, although contamination could not be excluded6,7. Here we present evidence for the formation of aerosol particles from highly oxidized biogenic vapours in the absence of sulfuric acid in a large chamber under atmospheric conditions. The highly oxygenated molecules (HOMs) are produced by ozonolysis of α-pinene. We find that ions from Galactic cosmic rays increase the nucleation rate by one to two orders of magnitude compared with neutral nucleation. Our experimental findings are supported by quantum chemical calculations of the cluster binding energies of representative HOMs. Ion-induced nucleation of pure organic particles constitutes a potentially widespread source of aerosol particles in terrestrial environments with low sulfuric acid pollution.