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. (What else could I say?) Howeverit is not hard to concoct examples where the metric is misleading.<1> , denote the joint distribution fornindependent obser-vations from theN( ,1)distribution, with R.

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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