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"Machine Learning: A Probabilistic Perspective by Kevin Patrick Murphy" in chapter 1 says

A simple example of a non-parametric classifier is the K nearest neighbor (KNN) classifier. This simply “looks at” the K points in the training set that are nearest to the test input x, counts how many members of each class are in this set, and returns that empirical fraction as the estimate, as illustrated in Figure 1.14. More formally,

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which looks like the empirical distribution function

$${F}_{n}(t)={\frac {{\mbox{number of elements in the sample}}\leq t}{n}}={\frac {1}{n}}\sum _{i=1}^{n}\mathbf {1} _{X_{i}\leq t}$$

I never heard the term empirical fraction, and I cannot find any other reference talked about empirical fraction.

Are "empirical distribution function" and "empirical fraction" the same thing?

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  • $\begingroup$ This is not a "distribution function" in the usual sense. In $d$ dimensions the empirical distribution function tells you, for any vector $x=(x_1,\ldots,x_d),$ the fraction of data all of whose coordinates are less than or equal to the coordinates of $x.$ $\endgroup$ – whuber yesterday
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A fraction is just one thing divided by another thing. Empirical just means you used observed things. So an empirical distribution function is a set of empirical fractions, but not all empirical fractions are (part of an) empirical distribution function.

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  • $\begingroup$ Thanks man! Would you please give an concrete example of empirical fractions being not empirical distribution functions? $\endgroup$ – yaojp Sep 30 '19 at 9:59
  • $\begingroup$ An odds ratio is a ratio (fraction) of two odds. The odds are empirically observed things, so and odds ratio is an empirical fraction, but it is not part of a distribution function. $\endgroup$ – Maarten Buis Sep 30 '19 at 13:12
  • $\begingroup$ Similarly you could look at a ratio of means, e.g. when you say that on average women earn 20% less than men. In that case you have two observed things: mean wage of men and mean wage of women. You compute the ratio, which is a empirical fraction. $\endgroup$ – Maarten Buis Sep 30 '19 at 13:17
  • $\begingroup$ "on average women earn 20% less than men" is a empirical fraction and is being not an empirical distribution functions, right? $\endgroup$ – yaojp Oct 2 '19 at 22:52
  • $\begingroup$ Thanks a lot. Would you please give a concrete example of two odds and their odds ratio? $\endgroup$ – yaojp Oct 2 '19 at 22:52
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Theoretically, an empirical distribution function is an estimate for the cumulative distribution function which converges, with probability 1, to the true underlying distribution per the Glivenko–Cantelli theorem.

However, the difference between an empirical distribution function (which serves as an estimate of the actual underlying distribution function, which can be continuous) and empirical fractions can be practically significant. This is especially true upon the further application transformations to the raw fractions, which can result in modeling bias.

For example, with small n and also the lowest and highest points, a raw empirical fraction can present an issue, for example, with a hazard ratio estimation. See, for example, 'Small Sample Properties of Some Estimators of a Common Hazard Ratio', by Alexander M. Walker available here. Note, the introduction of suggested rules which modify the raw fraction, by adding say a constant, like 0.5, to the numerator and, say 0.5, to the denominator of an observed fraction (which upon transformation, avoids undefined values). See also here which likewise suggests the same 0.5 adjustment as one of the simple alternative corrections.

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