MLE of the Pareto Distribution

While not as well-known as the bell-shaped Normal (Gaussian) distribution, the Pareto distribution is a powerful tool for modeling a variety of real-life phenomena. It is named after the Italian economist Vilfredo Pareto (1848-1923), who developed the distribution in the 1890s as a way to describe the allocation of wealth in society. He famously observed that 80% of society’s wealth was controlled by 20% of its population, a concept now known as the “Pareto Principle” or the “80-20 Rule”. You can read more about it

Analytical estimate

Consider the Pareto distribution:

\[ f(x|x_0, \theta) = \theta x_0^{\theta}x^{-\theta - 1}, x \geq x_0 \geq 0 , \theta > 1 \]

Assuming that \(x_1, x_2, · · · , x_n\) are an i.i.d. samples, calculate the MLE of \(\theta\) analytically.

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SGD

Write the SGD algorithm that will estimate \(\theta\). Experiment with the learning rate and the number of iterations and mini-batch size.

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