EmpCDF

EmpCDF stands for Empirical Cumulative Distribution Function. It is a step function that approximates the cumulative distribution function of a random variable based on a sample of observed data. The EmpCDF assigns a probability to each observed value, indicating the proportion of data points that are less than or equal to that value. For a given set of data points $x_1, x_2, ..., x_n$, the EmpCDF, denoted as $F_n(x)$, is defined as the number of observations less than or equal to $x$ divided by the total number of observations, $n$. Mathematically, $F_n(x) = \frac{1}{n} \sum_{i=1}^n I(x_i \le x)$, where $I(\cdot)$ is the indicator function.

The EmpCDF is a non-decreasing function that ranges from 0 to 1. It is commonly used in non-parametric statistics to estimate the underlying distribution of a population without making assumptions about its specific form. It is particularly useful for visualizing and comparing distributions, and for hypothesis testing, such as the Kolmogorov-Smirnov test. The EmpCDF converges to the true cumulative distribution function as the sample size increases, a property known as Glivenko-Cantelli theorem.