Information in additional observations of a non-parametric experiment that is not estimable

Using Le Cam’s notion of deficiency between experiments, one can measure the value of obtaining an additional observation, when one has already been given <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> independent and identically distributed observations. We show that for certain types of non-parametric experiments, the value of an additional observation decreases at a rate of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash StartRoot n EndRoot"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> </mml:mrow> <mml:annotation encoding="application/x-tex">1/\sqrt {n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is distinct from the typical rate of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash n"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1/n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the case of parametric experiments, and from the non-decreasing value in the case of very large experiments. In particular, the rate of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash StartRoot n EndRoot"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> </mml:mrow> <mml:annotation encoding="application/x-tex">1/\sqrt {n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> holds for the experiment given by observing samples from a density about which we know only that it is bounded from below by some fixed positive constant. For such densities, there exists no estimator that is consistent in total variation distance. We have therefore an example of an experiment which is, in this sense, not estimable, but for which the value of additional observations tends to zero.