Comparison of Statistical Experiments by Erik Torgersen

By Erik Torgersen

The current paintings examines some of the tools of evaluating statistical experiments. It starts off by way of introducing statistical experiments and convex research. This bankruptcy is by way of others on video game concept, choice thought, and vector lattices, that are a ordinary framework for learning statistical difficulties. The thought of deficiency, which measures the adaptation in details among experiments, is then brought. The relation among it and different thoughts, comparable to sufficiency, randomization, distance, ordering, equivalence, completeness and convergence also are explored. the rest of the e-book is dedicated to purposes of the speculation to linear types, neighborhood comparability of differentiable experiments, majorization, and discussions of issues complementary to the most textual content.

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Hence, since any ~-measurable bounded function is the uniform limit of ~-measurable step functions, F(Y) = lim; Fn,(Y) exists for all bounded ~­ measurable functions Y. Finally consider a general bounded measurable function Y. By the RadonNikodym theorem there is a ~-measurable bounded function E(YI~) such that IB Y = IBE(YI~) when B E~. Then I XY = I XE(YI~) when X is integrable and measurable. In particular Ft(Y) = F,(E(YI~)); t = 1,2, ... Hence F(Y) = lim; Fn,(Y) exists for all Y E Loo. We have thus shown that the subnet constructed in the first part of the proof actually may be chosen as a sub-sequence when the given net is a sequence.

3 Derived experiments There is a large variety of ways of obtaining 'new' experiments from 'old' ones. Thus experiments may be mixed by first drawing an experiment according to a known chance mechanism and then performing the drawn experiment. e. they may be shown to be equivalent to experiments obtained by mixing with known mixing distributions. Thus in chapter 7 we shall see that experiments, however complicated, may be regarded as mixtures of experiments admitting complete and sufficient statistics.

D) is a measurable space and Jl is a non-negative (i-finite measure on d. e. Jl' for extended real valued measurable functions is complete. In fact a family (Xt : t E T) of extended real valued measurable functions on (~d) possesses a least upper bound for this ordering of the form sup{ X t : t E C} where C is a countable subset of T. Remark. 10, this result is an immediate consequence of the order completeness of L-spaces. Proof If Jl(PI) = 00 then we may, since Jl is (i-finite, decompose PI as PI = Ui=1 Ai where A 1,A 2 ,· ..

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