The probability density function (pdf) is the first derivative of the cumulative distribution (cdf) for a continuous random variable. I take it that this only applies to well-defined distributions like the Gaussian, t-distribution, Johnson SU, etc, though.
If given real data that we know does not conform to some prior distribution (perfectly), does that mean that (it would be safe to assume that) the real data's cdf cannot be differentiated, and therefore has no pdf, making us resort to histogram, or kernel density, or log-spline approximations, of the continuous data's pdf?
just trying to rationalize the whole model-fitting craze (Gaussian, t-, Cauchy) that is always encountered in statistics, and why it always overrides approximation approaches (histogram, kernel density).
In other words, rather than use an estimator on the empirical data (histogram, kernel density), we are trained to look for a best match model (Gaussian, t-, Cauchy) instead, even though we know the real data's pdf diverges from that model.
What makes the "modeling" approach better than "approximation"? Is it, and how is it, more right?