Guha, Pritha (2012)
Ph.D. thesis, University of Birmingham.
Quantile-quantile plots are in use to compare univariate distributions for a long time, but as there is no ordering in higher dimension, there is no straight forward generalisation of quantiles for the multivariate data and hence there is no visual tool which can be considered as a generalisation of quantile-quantile plots to compare multivariate distributions. In this work we have considered some notions of multivariate ranks, quantiles and data depths. Based on spatial rank, we have constructed central rank regions and some measures of scale. We proposed a scale-scale plot, which can be used to compare multivariate distributions. Under spherical symmetry, our scale curves have some nice closed form formula, however they are not equivariant under affine transformations. We discussed this issue with illustrations and proposed an affine equivariant version based on data-driven transformations. We established some characterisation results for the proposed affine equivariant scale curves under elliptic symmetry and used the fact to propose some visual test of location and scale in the family of elliptically symmetric distributions. Our proposed scale-scale plot is based on volume functionals of central rank region. We gave some asymptotic results regarding the distribution of the volume functional and constructed a test statistic based on the volume functional. We proposed some asymptotic results regarding the distribution of the test statistic and also studied the power of the proposed test of multivariate normality. As further applications to our scale-scale plots, we discuss the behaviour of our proposed scale-scale plots when the distribution is not elliptically symmetric with illustrations and study the power of the test of for skew elliptic and g and h distribution based on the previously defines test statistic. Among other application of the scale-scale plots, we propose a kurtosis plot, which can be used to study the peakedness and tail behaviour of the multivariate distributions, a visual test of location and scale.
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