ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for computing the Empirical Cumulative Distribution Function (ECDF) of an observational sample and the associated the various properties. More...
Data Types | |
interface | setECDF |
Compute and return the Empirical Cumulative Distribution Function (ECDF) of a univariate (optionally weighted) sample of size size(ecdf) . More... | |
Variables | |
character(*, SK), parameter | MODULE_NAME = "@pm_sampleECDF" |
This module contains classes and procedures for computing the Empirical Cumulative Distribution Function (ECDF) of an observational sample and the associated the various properties.
An empirical Cumulative Distribution Function (eCDF) is the distribution function associated with the empirical measure of a sample.
This cumulative distribution function is a step function that jumps up by \(1 / N\) at each of the \(N\) data points.
Its value at any specified value of the measured variable is the fraction of observations of the measured variable that are less than or equal to the specified value.
The empirical distribution function is an estimate of the cumulative distribution function that generated the points in the sample.
It converges with probability 1 to that underlying distribution, according to the Glivenko–Cantelli theorem.
A number of results exist to quantify the rate of convergence of the empirical distribution function to the underlying cumulative distribution function.
Definition
Let \((X_1, \ldots, X_N)\) be independent, identically distributed real random variables with the common cumulative distribution function \(F(t)\).
Then the empirical distribution function is defined as,
\begin{equation} {\widehat{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} ~, \end{equation}
where \({\mathbf{1}}_{{A}}\) is the indicator of event \(A\).
For a fixed \(t\), the indicator \(\mathbf{1}_{X_{i}\leq t}\) is a Bernoulli random variable with parameter \(p = F(t)\).
Hence, \(N{\widehat{F}}_{N}(t)\) is a binomial random variable with mean \(N\times F(t)\) and variance \(N\times F(t)(1 − F(t))\).
This implies that \({\widehat{F}}_{N}(t)\) is an unbiased estimator for \(F(t)\).
Final Remarks ⛓
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For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
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character(*, SK), parameter pm_sampleECDF::MODULE_NAME = "@pm_sampleECDF" |
Definition at line 75 of file pm_sampleECDF.F90.