Statisticians distinguish between three levels of modeling assumptions;
Whatever level of assumption is made, correctly calibrated inference, in general, requires these assumptions to be correct; i.e. that the data-generating mechanisms really have been correctly specified.
Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.
However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, such randomized studies are uneconomical or unethical.
Different schools of statistical inference have become established. These schools—or "paradigms"—are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms.
This paradigm calibrates the plausibility of propositions by considering (notional) repeated sampling of a population distribution to produce datasets similar to the one at hand. By considering the dataset's characteristics under repeated sampling, the frequentist properties of a statistical proposition can be quantified—although in practice this quantification may be challenging.
The topics below are usually included in the area of statistical inference.