For the purpose of this question, please consider me a stats newbie.
I'm working on a (very fun!) research project which involves estimating a pdf of "personal values" -- i.e. how much a certain person values a certain good they can buy.
I wanted to find the functional form of the PDF so that I could generate data according to the distribution I saw in the empirical data I've been working with.
But I received a suggestion to follow Maximum Likelihood Estimation, which isn't something I've done in the past. I've been experimenting but my data doesn't seem to likely fit any typical distribution... it's more of a "bactrian camel" distribution.
So what I'm interested in knowing is:
- Is my current approach correct? (see below)
- How might I approach this with Maximum Likelihood Estimation?
Here's what I've done so far:
- Constructed a CDF of the data I have (taking them as "valuations")
- Literally... found the OLS estimator of the CDF, in functional form, so now I have a polynomial estimation of the CDF
- Took the derivative of that function (which should equal the PDF)
- Generated data according to that derivative.
Overall, this approach has performed fairly well. I don't have any metrics to compare my generated data with the actual data -- I don't know any! -- but from comparing the two datasets, they're fairly close in mean and variance.
-- An enthusiastic college kid
Edit: Adding details:
Here's what my density function looks like. I'd like to approximate a functional form of this, because I need to do two things:
Infer the distribution of data above 150; according to theory, there exist data that is above 150, but it's not in this distribution.
Generate random data (say a sample size of 5,000 -> 10,000)
The distribution I approximated through my take-derivative-of-CDF-estimation looks like this (ignore the improper scaling of y-axis -- this is scaled up)