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Robust & Non-Robust Models in Statistics (Hardcover)
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Robust & Non-Robust Models in Statistics (Hardcover)
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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 characterisation of the limiting distributions.
Then, they consider pre-limiting behaviour 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 generalised
functions are provided in an appendix.
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