I have data on the first to fourth moments of a continuous random variable and I am trying to find what distribution best fits the data. Wikipedia has a list of about 20 distributions that could fit the data and I would like to try each of them to see which is the best. Some of these distributions have one parameter, some have two or three. Is there some test I can use to compare the fit of each distribution? Or at least a test to compare distributions with the same number of parameters?
Try Hahn and Shapiro Statistical Models in Engineering and the discussion of the Pearson Distribution. This is illustrated within Wikipedia and relationships to well-known distributions are shown in section 4 of the same page. You can also find more background information here.
There is an extensive derivation and discussion of the Pearson family in Chapter 5 of our book:
- Rose and Smith, "Mathematical Statistics with Mathematica", Springer, NY
You can download the chapter for free here:
See Section 5.2 D (p.159 on) for a derivation of Pearson Coefficients in terms of the first 4 population moments. [There is also a fun derivation in terms of the first 6 moments, which may well have been a first at the time.] The Pearson system generally works rather nicely if the skewness and kurtosis of the data are not tooooo large ... it is best to check to see where $\beta_1$ and $\beta_2$ lie on the Pearson diagram as a first step. If, however, skewness and/or kurtosis are 'large', and you are working with sample data rather than population data, then the estimates of the moments can become unreliable, and the method is less useful.
P.S. If you would like to add your moment data to the question (specifying if they are raw moments or central moments), I'd be happy to generate the Pearson fit for you. Only takes a few seconds to do. Ideally, please specify the central moments $\mu_2$, $\mu_3$ and $\mu_4$ and the mean ... but if you only have raw moments, that should be fine too.
I would suggest maximum entropy density estimation, but I could not find an off-the-shelf application to do it. However, you may find this paper useful: A Computationally Efficient Multivariate Maximum-Entropy Density Estimation (MEDE) Technique.
I like your question! But I think even when using 20 or more distributions you can fail easily. If your data has e.g. multiple modes, then a Gaussian mix maybe of 2 or even more Gaussians could fit well, but also mixes of other distributions like log-norm or Student T. To decide which is best you will really need a lot of sample points like N>>1000. Consider to exploit also other information, like or all samples >0, or are there sharp bounds (like uniform) or long tails (often leading to non-existing high-order moments!)?? Maybe you know more about your design, like the key variables have Gaussian distribution, but transfer function is a polynomial (or exponential or rational or whatever)?