Transcription of Chapter 3 Total variation distance between measures
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Chapter 3 Total variation distance between measures
Chapter 3 Total variation distance betweenmeasures1. Why bother with different distances?When we work with a family of probability measures ,{P : }, indexedby a metric space , there would seem to be an obvious way to calculatethe distance between measures : use the metric on . For many problems ofestimation, the obvious is what we want. We ask how close (in the metric)we can come to guessing 0, based on an observation fromP 0; we compareestimators based on rates of convergence, or based on expected values of lossfunctions involving the distance from the parametrization is reasonable (whatever that means), distancesmeasured by the metric are reasonable.
4 Chapter 3: Total variation distance between measures If λ is a dominating (nonnegative measure) for which dµ/dλ = m and dν/dλ = n then d(µ∨ν) dλ = max(m,n) and d(µ∧ν) dλ = min(m,n) a.e. [λ]. In particular, the nonnegative measures defined by dµ +/dλ:= m and dµ−/dλ:= m− are the smallest measures for whichµ+A ≥ µA ≥−µ−A for all A ∈ A. Remark. Note that the ...
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