Weighted quantile-based estimation for a class of transformation distributions

Quantile-based methods appear in both statistical inference and exploratory data analysis. Inferential methods based on order statistics generally have extensive theoretical bases, while exploratory data analysis tends to emphasize graphical methods and often uses selected sets of quantiles such as the "letter-values" of Tukey (Exploratory Data Analysis, Addison-Wesley, Reading, MA, 1977b). Since transformations of random variables give rise to families of distributions defined through their quantile functions, quantile-based methods could be considered a natural approach when using such families. This paper considers quantile-based methods for fitting two such families of distributions (formed by transforming the standard normal), the g-and-k and the adapted g-and-h distributions, which have been developed to take advantage of certain shape functionals.The effects of different quantiles are taken into account by considering weighted sums of estimates based on quantiles within the data, these sets of estimates arising from matching shape, location and scale functionals. The methods considered correspond to different criteria for the weighted sums. These iteratively reweighted methods use approximations to means and variances of the functionals, and so not only produce parameter estimates, but also approximations of the mean and variance for these estimates, and weights which indicate which functionals of the quantiles of the data are found to be most important. A simulation study is included, and the procedures, distributions and approximations are also illustrated by fitting two air pollution datasets.Comparisons are made with a quick method that uses the median of the set of estimates, and with numerical maximum likelihood estimation which tends to be not efficient for these families until very large sample sizes are available (Rayner and MacGillivray, 2002). The results indicate that the weighted methods perform better in a number of ways than numerical maximum likelihood estimation for smaller and moderately-sized samples. MATLAB software to carry out the weighted method is available on request.

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