Distribution-Free Confidence Intervals for Functions of Quantiles

Submission note: A thesis submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Engineering and Mathematical Sciences, College of Science, Health and Engineering, La Trobe University, Victoria, Australia.

The objective of this thesis is to provide new distribution free point and interval estimators for measures of spread, relative spread and skewness involving quantiles. The main advantages of these new quantile-based measures are that they are comparatively efficient to compute with minimal distributional assumptions. In the two-samples case, the location and scale of two independent samples can be compared using ratios of linear combinations of quantiles. In the single-sample case, distribution-free inferences can be made using quantile versions of the coefficient of variation and measures of skewness. The new estimators can be directly applied to many areas, such as economics, medical statistics, bio-statistics and social sciences. This thesis consists of four papers, either published or submitted at the time of thesis submission, and also includes some introductory material. In paper I, the distribution-free point and interval estimators were introduced for ratios of independent quantiles to compare the location of two independent samples. The interquantile range, which is the most natural quantile-based estimator of scale, was considered to compare the scale of two independent samples. The distribution-free point and interval estimators were introduced to the squared ratios of interquantile ranges. The best choice of the probability to achieve the minimum asymptotic variance for the squared ratio of interquantile ranges was proposed. Robustness properties of the estimators were investigated using partial influence functions. The simulation results reveal that all the intervals provide excellent coverage probabilities even for small sample sizes and for a wide range of distributions. An R shiny web application was developed and is publicly available to readers to run the simulations as they desire. Some real-world data examples were considered, and the results suggest that new estimators perform really well compared to the classical parametric tests such as t-test and F-test. In paper II, the median absolute deviation which is the most robust estimator of scale with respect to the breakdown point was considered. The distribution-free point and interval estimators were introduced to the median absolute deviation and the difference and squared ratio of median absolute deviations to make inferences on spread of a single sample and xiii to compare the spread of two populations respectively. Robustness properties of the new estimators were investigated using an influence function and partial influence functions. Simulations were conducted to check the performance of the new estimators and the results suggest that the coverage probabilities are very close to the nominal coverage even with small sample sizes and for a variety of distributions. A real-world data example was considered, and the results suggest that the new estimators are more robust to the outliers when compared to the F-test. In paper III, two robust versions of the coefficient of variation based on linear combinations of quantiles were considered to make distribution-free inferences of relative dispersion for a single sample. The first measure was the interquartile range divided by the median and the second measure was the median absolute deviation divided by the median. The distribution-free point and interval estimators were constructed to the two robust versions of the coefficient of variations and the robustness properties were investigated using influence functions. The performance of the new estimators was compared with several existing estimators via simulations and the results suggest that the new methods perform well for a wide variety of distributions. The R shiny web application was developed and is publicly available to compare the performance of the new estimators of spread with some existing estimators. The interval estimators were introduced to the ratios of the robust coefficient of variations as an extension to compare the relative dispersion between two independent samples. The examples reveal that different conclusions can be made based on robust and non-robust versions of the coefficient of variation. In paper IV, some integrated versions were constructed to existing measures of skewness based on ratios of linear combinations of quantiles. These existing skewness measures are some generalizations of Bowley’s well-known skewness coefficient. The validity of the properties that any measures of skewness should satisfy was tested for new measures of skewness. The distribution-free point and interval estimators were introduced for new measures of skewness to make inferences about the skewness of a single sample. A simulation study was conducted to compare the performance of the new estimators with the existing estimators and the results suggest that the new measures perform well for wide range of distributions. The R shiny web application was developed and is publicly available to compare the performance of new skewness estimators. The interval estimators were introduced to the difference of the measures of skewness to compare the skewness between two independent samples. Some real-world examples were used and different conclusions were observed based on different methods.