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Concept
Empirical distribution function
Summary
In statistics, an empirical distribution function (commonly also called 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.
Let (X1, ..., Xn) be independent, identically distributed real random variables with the common cumulative distribution function F(t). Then the empirical distribution function is defined as
where is the indicator of event A. For a fixed t, the indicator is a Bernoulli random variable with parameter p = F(t); hence is a binomial random variable with mean nF(t) and variance nF(t)(1 − F(t)). This implies that is an unbiased estimator for F(t).
However, in some textbooks, the definition is given as
The mean of the empirical distribution is an unbiased estimator of the mean of the population distribution.
which is more commonly denoted
The variance of the empirical distribution times is an unbiased estimator of the variance of the population distribution, for any distribution of X that has a finite variance.
The mean squared error for the empirical distribution is as follows.
Where is an estimator and an unknown parameter
For any real number the notation (read “ceiling of a”) denotes the least integer greater than or equal to . For any real number a, the notation (read “floor of a”) denotes the greatest integer less than or equal to .
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