A note on the null distribution of the local spatial heteroscedasticity (LOSH) statistic

Recently, Ord and Getis (Ann Reg Sci 48:529–539, 2012) developed a local statistic $$H_i$$Hi, called local spatial heteroscedasticity statistic, to identify boundaries of clusters and to describe the nature of heteroscedasticity within clusters. Furthermore, in order to implement the hypothesis testing, Ord and Getis suggested a chi-square approximation method to approximate the null distribution of $$H_i$$Hi, but they said that the validity of the chi-square approximation remains to be investigated and some other approximation methods are still worthy of being developed. Motivated by this suggestion, we propose in this paper a bootstrap procedure to approximate the null distribution of $$H_i$$Hi and conduct some simulation to empirically assess the validity of the bootstrap and chi-square methods. The results demonstrate that the bootstrap method can provide a more accurate approximation than the chi-square method at the cost of more computation time. Moreover, the power of $$H_i$$Hi in identifying boundaries of clusters is empirically examined using the proposed bootstrap method to compute $$p$$p values of the tests, and the multiple comparison issue is also discussed.