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Given $n$ independent and identically distributed observations and measuringthe value of obtaining an additional observation in terms of Le Cam's notion ofdeficiency between experiments, we show for certain types of non-parametricexperiments that the value of an additional observation decreases at a rate of$1/\sqrt{n}$. This is distinct from the known typical decrease at a rate of$1/n$ for parametric experiments and the non-decreasing value in the case ofvery large experiments. In particular, the rate of $1/\sqrt{n}$ holds for theexperiment given by observing samples from a density about which we know onlythat it is bounded from below by some fixed constant. Thus there exists anexperiment where the value of additional observations tends to zero but forwhich no estimator that is consistent (in total variation distance) exists.

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