I have a series of numbers, which are some survival probabilities that form a decaying curve. I would like to fit them with a "Pareto" distribution. I expect to have a smooth fitted curve, thus I can extrapolate it and have a rough prediction.
My data are like 1, 0.987, 0.972, 0.965, ....
How can I achieve my goal?
The survival function for the Pareto distribution (under one of its many parametrizations) is: $$ S(x) = \left(\frac{\theta}{x+\theta}\right)^\alpha $$ where $\alpha$ is the shape and $\theta$ is the scale of the distribution. So a very simple way to fit would be to solve for $\alpha, \theta$ which minimizes the squared difference between the observed vector (which is between 1 and 0 and is taking the place of an empirical survival function) and the $S_{\hat{\alpha}, \hat{\theta}}(x)$ calculated survival values for the fitted distribution.
