... smoothest¹

A smooth curve here is one with no rapid changes or oscillations, so the smoothest is the one with the least change in slope over its length.

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... linear²

Write a linear segment as $y_i = mx_i + c$, so summing several linear segments, at the point $(x_i, y_i)$ gives $y_i = \Sigma_{j=1}^n ({\mbox{\boldmath $m$}}_j x_i + {\mbox{\boldmath $c$}}_j)... ...igma_{j=1}^n {\mbox{\boldmath $m$}}_j + \Sigma_{j=1}^n {\mbox{\boldmath $c$}}_j$ which is obviously a linear segment.

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... averaged³

A shifted histogram being one with the same data and bin width but with a different starting point on the x axis.

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... function⁴

A probability density function is a function f defined on an interval (a, b) and having the following properties.

1. $f(x) \ge 0$ for every $x$
2. $\int_a^b f(x) dx = 1$

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... estimator⁵

For nearest neighbour the density is taken to be inversely proportional to the distance between the data item and the next nearest, by some measure, data item.

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... sphere⁶

The volume of the dimensional sphere is given by

see McDonald (2003) for derivation and evaluations of . This gives values for as follows

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