This book presents new findings on nonregular statistical estimation.
Unlike other books on this topic, its major emphasis is on helping
readers understand the meaning and implications of both regularity and
irregularity through a certain family of distributions. In particular,
it focuses on a truncated exponential family of distributions with a
natural parameter and truncation parameter as a typical nonregular
family. This focus includes the (truncated) Pareto distribution, which
is widely used in various fields such as finance, physics, hydrology,
geology, astronomy, and other disciplines. The family is essential in
that it links both regular and nonregular distributions, as it becomes a
regular exponential family if the truncation parameter is known. The
emphasis is on presenting new results on the maximum likelihood
estimation of a natural parameter or truncation parameter if one of them
is a nuisance parameter. In order to obtain more information on the
truncation, the Bayesian approach is also considered. Further, the
application to some useful truncated distributions is discussed. The
illustrated clarification of the nonregular structure provides
researchers and practitioners with a solid basis for further research
and applications.