Numerical methods

Estimates of population parameters based on samples are not exact: there is always some error involved. In principle, one can estimate a population parameter with any estimator, but some will be better than others.

Maximum likelihood estimates of a distribution Maximum likelihood estimation (MLE) is a method to estimate the parameters of a random population given a sample. I described what this population means and its relationship to the sample in a previous post.

tl;dr: Parametric bootstrap methods can be used to test hypothesis and calculate p values while assuming any particular population distribution we may want. Non-parametric bootstrapping methods can be used to test hypotheses and calculate p values without having to assume any particular population as long as the sample can be assumed to be representative of the population and one can transform the data adequately to take into account the null hypothesis.

tl;dr: P-values are tail probabilities calculated from the sampling distribution of a sample-based statistic. This sampling distribution will depend on the size of the sample, the statistic being calculated and assumptions about the random population from which the data could have been sampled.

tl;dr: Given any model that maps inputs to outputs, Dual Numbers can be used to calculate the exact derivative of any output with respect to any input without requiring the user to calculate these derivatives.