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Recent research has shown that interval estimators with good coverage properties are achievable for some functions of quantiles, even when sample sizes are not large. Motivated by this, we consider interval estimators for the ratios of independent quantiles and interquantile ranges that will be useful when comparing location and scale for two samples. Simulations show that the intervals have excellent coverage properties for a wide range of distributions, including those that are heavily skewed. Examples are also considered that highlight the usefulness of using these approaches to compare location and scale.

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Skewness measures can be used to measure the level of asymmetry of a distribution. Given the prevalence of statistical methods that assume underlying symmetry, and also the desire for symmetry in order to make meaningful judgements for common summary measures (e.g. the sample mean), reliably quantifying asymmetry is an important problem. There are several measures, among them generalizations of Bowley's well known skewness coefficient, that use sample quartiles and other quantile-based measures. The main drawbacks of many measures is that they are either limited to quartiles and do not take into account more extreme tail behavior, or that they require one to choose other quantiles (i.e. choose a value for p different from 0.25) in place of the quartiles. Our objective is to (i) average the skewness measures over all p and (ii) provide interval estimators for the new measure with good coverage properties. Our simulation results show that the interval estimators perform very well for all distributions considered.