On a Class of Distributions Closed Under Inversion

This text on inverted distributions and Theory of Inversion relates to the early work of Snedecor (1934), Merington and Thomson (1943). Since 1934 very little attention has been paid to this area, though some work has been done in eighties and nineties. In this book we discuss a theoretical framework for distributions closed under inversion, reviewed inverted distributions and present some concepts of closure and strict closure under inversion (CUI and SCUI). We provide the details of pervious work done by Gumbel and Keeney (1950), Seshadri (1950) and Saunders (1974). We investigate the intrinsic nature of SCUI distributions and present (a) a set of differential equations for generating SCUI distributions, (b) some characterization theorems for SCUI density functions, and (c) a few properties common to all SCUI distributions. We extend a number of concepts relating to SCUI distributions to the CUI case.