An investigation of the inverse Gaussian distribution with an emphasis on Gaussian analogies

The inverse Gaussian (IG) distribution is a versatile and attractive candidate for modelling asymmetric data. The main appeal of IG lies in the fact that many inference methods associated with it are simple and are strikingly analogous to the corresponding normal-theory methods. The rationale underlying the distribution is that in many situations the time to failure of a complex system can be regarded as the time for the accumulation of small random degradations to reach a certain threshold causing the entire system to fail. This process is modelled naturally by the IG distribution and has been effectively applied in a variety of fields ranging from ecology to internet communication.; Despite its wide use and applicability, the IG percentiles are not readily available and few general procedures for ascertaining the appropriateness of IG models currently exist. In this dissertation, we offer many tools to improve the applicability of the distribution and in the process discover several new properties analogous to those of the Gaussian distribution. We construct some new approximations for the percentiles of the distribution that are superior to those existing in the literature. We provide several new tests for the inverse Gaussian goodness-of-fit hypothesis. For example, an independence characterization is used to develop an IG analogue of the Z-test of normality. In the process, we introduce new concepts termed IG-symmetry, IG -skewness and IG-kurtosis. These, in turn, lead to another set of goodness-of-fit tests. We also study inference methods for IG scale parameters and see that, as in the normal-theory case, they are strongly non-robust in the sense that even asymptotically the Type I error and coverage probabilities are not controlled. Hence, we construct robust methods for homogeneity of IG scale parameters when they are unrestricted, and also when they are order-constrained.; The essence of this thesis is that the new procedures and newly discovered analogies with the Gaussian distribution, in addition to the rich collection of existing procedures and analogies, are compelling reasons for advocating the extensive use of the inverse Gaussian distribution for analyzing right-skewed data.