Non Parametric Probability Distribution

While in some cases one can presume that a random variable follows a specific distribution such as Normal, Weibull or Gamma in many cases one cannot make any assumption concerning the shape of the probability distribution of a given random variable. For those instances we have to rely on Non Parametric distributions.

To illustrate this we show below the distribution of 20 random data generated using a mixture of Normal(100,100) and Lognormal (2, 2) distributions. We use an histogram, a normal, a lognormal and a non parametric (Kernel) distributions. The distribution are fitted using Maximum Likelihood Estimation.

We expect the non parametric distribution to be the best fit.

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The Fitted Probability Densities chart showing Normal series, LogNormal series, Kernel (Non Parametric) series. The Fitted Cumulative Distribution chart showing Normal series, LogNormal series, Kernel (Non Parametric) series, True Cumulative series.


As can be seen on the chart showing the cumulative functions, only the Kernel Non Parametric distribution is systematically close to the True Cumulative - without making any special effort - despite the normal and lognormal distribution parameters being re-estimated using the Maximum Likelihood Estimation technique.