Every linear statistical model is parametric because it assumes a linear functional form for $f$ to simplify the problem to only estimating $\beta_0, \beta_1, \beta_2, \dots, \beta_p$ coefficients, instead of estimating the complete form of the $p$-dimensional $f$.

University of Michigan - Ann Arbor

Parametric Statistical Methods start off with an assumption for the shape of f. As a result, when training the model, parametric methods simplify the process of finding f by reducing the entire problem to just figuring out the parameters of an equation (i.e., $\beta_0$, $\beta_1$, $\beta_2$, etc). Compared to non-parametric approaches, it does not need as many observations, but because it starts off with an assumption, it may not fit the actual f.

Parametric Statistical Methods

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

Examples of parametric statistical methods

Parametric statistics methods are applied to populations with other known distribution types. Parametric methods are often those for which we know that the population is approximately normal, or we can approximate using a normal distribution after we invoke the central limit theorem. Well-known statistical methods include :
- ANOVA
- Pearson's corrlation
- t-test

Examples of parametric statistics methods

1. Parametric tests in general are more powerful, meaning require a smaller sample size than non parametric tests if the data is truly normally distributed. 
2. They can be better at distinguishing two arms of the distribution or they are more able to lead to a rejection of H0.
3. Parametric test results are easier to interpret as they use actual data (mean) instead of rankings of the values in data (eg. median). 


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