Most of the time in statistics, the modelled data are intrinsically attached to a notion of "scale" in a physical sense. Think for instance of a discrete sampling of a stochastic process in finance: at microscopic scales - up to the level of seconds of less - the path of a price process very much looks like a marked point process, whereas at macroscopic scales -- for daily sampling say -- it will rather be modelled by a continuous diffusion. Of course, the choice of this scale heavily affects the statistical properties of the resulting model. We will explore through various examples what happens as far as statistical inference is concerned if we add up an extraneous -- nuisance -- scale parameter, and ask whether one can adapt to scales: in particular, how does a procedure designed to work at a certain scale behaves if the data are more suited to another scale? Some examples of what can be proved (and also what we cannot prove) will be presented informally, ranging from Poisson and Hawkes point process models to continuous diffusions. We will also discuss the consequences for some financial applications in high frequency trading.
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