Empirical distribution function

The empirical distribution function assigns probability mass 1/n to each of n observations, counting repeated observations with their multiplicities. It estimates the population distribution from a finite sample.

Cumulative distribution function: A function giving the probability that a random variable is no greater than each real value. The empirical distribution function is the cumulative distribution function of the sample’s discrete probability measure.
Kolmogorov–Smirnov test: A statistical test based on the largest distance between cumulative distribution functions. Its test statistic compares the empirical function with a hypothesized distribution or another sample’s function.
Probability measure: A measure assigning probabilities to events in a sample space. The empirical function summarizes a probability measure that places mass on observed values.
Kernel density estimation: A nonparametric method estimating a continuous density by smoothing observations with kernel functions. Unlike the empirical function’s jumps, kernel estimates smooth sample contributions across nearby values.
Order statistic: An observation’s value when sample observations are arranged in increasing order. Sorted observations determine the locations and sizes of the empirical function’s jumps.
Quantile estimation: The estimation of values that divide a probability distribution at specified cumulative probabilities. Inverting the empirical function yields a direct, data-based quantile estimate.
Random sample: A collection of observations generated according to a sampling process. The empirical distribution function is constructed from the realized sample.
Parametric distribution: A probability distribution drawn from a family specified by a finite set of parameters. A parametric model replaces the empirical function’s unrestricted sample-based shape with a chosen family.
Sample quantile: A quantile estimated from ordered observations in a finite sample. The empirical function’s generalized inverse gives sample quantiles.
Bootstrap method: A resampling method that estimates statistical uncertainty by drawing repeatedly from observed data. The empirical distribution provides the sampling distribution used for nonparametric bootstrap resamples.

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