In this book the authors consider so-called ill-posed problems and stability in statistics. Ill-posed problems are certain results where arbitrary small changes in the assumptions lead to unpredictable large changes in the conclusions. In a companion problem published by Nova, the authors explain that ill-posed problems are not a mere curiosity in the field of contemporary probability. The same situation holds in statistics. The objective of the authors of this book is to (1)identify statistical problems of this type, (2) find their stable variant, and (3)propose alternative versions of numerous theorems in mathematical statistics.
The layout of the book is as follows. The authors begin by reviewing the central pre-limit theorem, providing a careful definition and characterization of the limiting distributions. Then, they consider pre-limiting behavior of extreme order statistics and the connection of this theory to survival analysis. A study of statistical applications of the pre-limit theorems follows. Based on these theorems, the authors develop a correct version of the theory of statistical estimation, and show its connection with the problem of the choice of an appropriate loss function. As It turns out,a loss function should not be chosen arbitrarily. As they explain, the availability of certain mathematical conveniences (including the correctness of the formulation of the problem estimation) leads to rigid restrictions on the choice of the loss function. The questions about the correctness of incorrectness of certain statistical problems may be resolved through appropriate choice of the loss function and/or metric on the space of random variables and their characteristics (including distribution functions, characteristic functions, and densities). Some auxiliary results from the theory of generalized functions are provided in an appendix.