I have $30$ unequal samples ($n_i >1000$) from a supposedly same population. How should I go about identifying the underlying distribution of the population using these $30$ samples simultaneously? Distribution tests (particularly, Individual Distribution Identification feature of Minitab) which returns Anderson-Darling statistic are meant for identifying the distribution of the population from a single sample and not more than one samples.

Practical problem: I am looking at marks obtained at each rank of selected candidates list in a competitive examination for government jobs in India for the past $30$ years, hence $30$ unequal samples (as the number of vacancies varies by year).

  • $\begingroup$ With sample sizes that large why do you feel the need to fit a probability distribution with a minimal number of parameters? Why not fit a nonparametric density for each year and plot all on top of each other? (density in R, SmoothKernelDistribution in Mathematica, etc.). $\endgroup$ – JimB May 22 '18 at 14:44
  • $\begingroup$ What is the point of detecting/fitting the distribution? (@JimB I imagine that a density function with parameters that depend on the year would be useful when one wishes to extrapolate the fit to future years) $\endgroup$ – Sextus Empiricus Aug 20 '18 at 8:50
  • $\begingroup$ @martijnweterings. Agreed IF the objective is to predict rather than to describe and IF a very small number ( like 1 or 2) parameters adequately describe the distribution. $\endgroup$ – JimB Aug 20 '18 at 16:04
  • $\begingroup$ @JimB also when the objective is to describe to gain insight rather than to describe to interpolate or display, a kernel smoother is not so useful. $\endgroup$ – Sextus Empiricus Aug 20 '18 at 17:08
  • $\begingroup$ @MartijnWeterings Again, no disagreement. It's just my experience that someone with a lot of data and apparently no underlying model based on experience nor a particular objective other than looking at interesting data needs to start with an exploratory analysis. Extreme clarity with nice fitting standard distributions just doesn't happen too often (although it can happen on occasion). That's why I would suggest nonparametric densities to get an idea of the shape (bimodal? trimodal? skewed? etc.). In other words data analysis needs more than just data. $\endgroup$ – JimB Aug 21 '18 at 0:43

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