How to fit a Pareto distribution via Bayesian estimation (with a Pareto prior)?

I don't know Bayesian statistics very well, so I don't know if the question makes sense. Let me give an example.

We assume that the income distribution of a country is a Pareto distribution (the minimum is C, and the shape alpha is 3). Now we want to estimate the income distribution of a specific company (it is inside this country). We assume that it is also a Pareto distribution and we have 100 observations.

With frequentist consideration, we can fit a Pareto distribution with these 100 observations (with MME or MLE). But I am wondering if it is possible to estimate with the prior distribution, which is the income distribution of a whole country.

And we consider that the minimum is the same, but we want to estimate the new shape alpha'.

Is it possible to consider that the prior distribution is the Pareto distribution (of the country, with known C and alpha) and after updating (with the 100 observations), we still get a posterior distribution that is a Pareto (with C, the same minimum and a new alpha)?

• unclear: what is the parameter of interest? Minimum or shape or mean? and which parameter is modelled by your Pareto prior? – Xi'an Feb 14 '16 at 14:44
• Also, company is nested within country? – Wayne Feb 14 '16 at 14:57
• @Xi'an I got the impression from reading the quesiton that the OP doesn't understand what the prior is for and thinks that "Pareto prior" means that you have some vague prior assumption about the shape of the density which is then modified by data to some more complicated result (when of course you'd either have just a Pareto model under both approaches, or something more flexible under both). – Glen_b Feb 15 '16 at 0:27
• @Glen_b Yes, I think that I don't understand bayesian theory very well. So my following consideration means nothing ? I consider that I know perfectly the income distribution of the country (which is a Pareto with known C and alpha). Now I want to calculate the distribution for a company and I assume that C is known (the same), and alpha is different. Is it possible to "update" alpha, with 100 observations from this company. – XR SC Feb 15 '16 at 15:27
• My observation is that you've asked a series of questions about how to do Bayesian analysis, but don't have a strong understanding of the core idea very well. I don't say this to be mean-spirited, but I do think that you could benefit a great deal by picking up a textbook on the subject. This will give you the a level of knowledge that will help you ask more focused, well-informed questions. Gelman's Bayesian Data Analysis is very readable and a very good introduction to the topic. I strongly recommend it. – Sycorax Feb 18 '16 at 18:28

If the model is Pareto, i.e., $x_1,\ldots,x_n\sim\mathcal{P}(c,\alpha)$ and if $c$ has a Pareto prior $c\sim\mathcal{P}(d,\beta)$ then $$c|x_1,\ldots,x_n\sim \pi(c|x_1,\ldots,x_n\sim)\propto \prod_{i=1}^n \mathbb{I}_{c\le x_i} c^{-n\alpha}\times \mathbb{I}_{d\le c} c^{-\beta-1}$$This leads to $$\pi(c|x_1,\ldots,x_n\sim)\propto \mathbb{I}_{d\le c\le \min_i x_i} c^{-n\alpha-\beta-1}$$a truncated Pareto.