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Let's say you go out and you get a sample set of data with some 50,000 observations. You plot a histogram and realize you don't recognize what distribution the data came from. Additionally, you do a normal QQ plot and confirm that the data are definitely not normal. You look through a list of common distributions and also realize that with so many distributions, and to complicate it more, each distribution has a variety of parameters that can change its shape, it seems unlikely you could accurately guess what the underlying population distribution is by using a variety of statistical tests designed for specific distributions.

What is the best way to find out what the underlying population distribution may be assuming you clearly have enough data to make a good conclusion? I'd like to understand what statistical theory has to say about this as well as know how to solve the problem with R code. I'm thinking there should be some sort of best-fit dist R function out there?

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    $\begingroup$ I would warmly recommend changing the direction of your inquiry, because it is predicated on an assumption that often is incorrect: most datasets do not come from any "known" distribution. Moreover, it often is not useful--even detrimental--to attempt to describe data by fitting some such distribution to them. The "best way" you seek depends on what you know and assume about the objects of your study and what your analytical objectives are. Blindly sticking your data into a "best-fit dist" function--such functions do exist--is rarely the solution. $\endgroup$ – whuber Jun 20 '18 at 12:55
  • $\begingroup$ I strongly agree with @whuber. Also, when searching for the best distribution to fit your data, you may want to check the support of your random variable. Once you know this support, and your searching through the adequate subset of distributions, you can consider the ones that gives you the highest values for the log-likelihood. $\endgroup$ – Bruna w Jun 20 '18 at 16:05
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Broadly speaking, there are two kinds of statistical analysis that can be used to analyse data. One kind of analysis is the class of parametric methods where we use a model that assumes some particular group of possible distributions (e.g., a normal distribution, an exponential distribution, etc.). The other kind of analysis is the class of non-parametric methods where we use a model that makes minimal assumptions about the distribution of the data, and includes all common families of distributions, but also pretty much any distribution that is not a common form.

Parametric methods generally involve a model-fitting process based on the mathematical structure of the assumed form of the underlying distribution; once we fit the model we use diagnostic tests to check if the assumed model form is a reasonable representation of the data. These models usually allow the data to be represented by a fixed number of parameters (hence the name 'parametric') and the complexity of the model does not increase as we get more data. OLS linear regression analysis and standard generalised linear models are done in this way. Non-parametric methods generally involve a model-fitting process based on a very broad structure that allows the complexity of the representation of the underlying distribution to increase as we get more data. These methods allow for almost any underlying distributional form (counting out some pathological cases) and they usually model the data using broad representations that add complexity as we get more data. Spline and kernel-based methods are done in this way.

The field of non-parametric statistical methods is absolutely enormous; there are shelves of books on this topic in most university libraries. For that reason, it is not possible to give an account of the field in a stackexchange.CV answer. However, if you would like to learn more about this field, a useful starting point would be to look at a primer on non-parametric density estimation.

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