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


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?
 A: 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.
A: 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.
