I saw in some papers that people sometimes use empirical cdf or complementary cdf to quantify the distribution of data.
What is the advantage of using them and what interesting can they tell about data structure?
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As elevendollar mentions the empirical cdf does not require any assumption about functional form. The only parameter(s) you may have to adjust is the bin width if the CDF is determined via histogram, or smoothing width if you are smoothing data, etc..
It is all too often assumed that data fits a commonly used distribution, such as a Gaussian. Parameters are then estimated under the Gaussian assumption, compared, and conclusions drawn, even though there's no a priori reason to believe so.