In Section 2.1.3 we have already pointed out that the binwidth
is not the only parameter that governs shape and appearance of the
histogram. Look at Figure 2.5 where four histograms for
the stock returns data have been plotted. We have used the same binwidth
for each histogram but varied the origin
of the bin grid.
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Even though we use the same data and the same binwidth, the
histograms give quite different accounts of some of the key features
of the data:
whereas all histograms indicate that the true pdf is unimodal, only the
upper right histogram suggests a symmetrical pdf. Also, note that the
estimates of differ considerably.
This property of histograms strongly conflicts with the goal of
nonparametric statistics to ``let the data speak for themselves".
Obviously, the same data speak quite differently out of the
different histograms. How can we get rid of the dependency of the
histogram on the choice of the origin of the bin grid? A natural remedy
might be to compute histograms using the same binwidth but different
origins and to average over the different histograms.
We will consider this technique in the next section.