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 $f$.

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). 


Why use a parametric statistical test

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 estimating the parameters of an equation (i.e., $$\beta_0$$, $$\beta_1$$, $$\beta_2$$, etc.). Compared to non-parametric approaches, they do not need as many observations, but because they start off with an assumption, they may not fit the true $$f$$.

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- Parametric statistical methods
- Non-parametric statistical methods

Parametric and non-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

Parametric Statistical Methods

https://www.investopedia.com/terms/n/nonparametric-method.asp

Parametric Method and Nonparametric Method

Parametric Statistical Methods have Datasets that match the following Criteria:
-Dependent variable data are continuous, not categorical 
-Data are (more or less) normally distributed 
-Variance (R^2) are similar/homogenous within each predictor group
-Data samples are randomly selected without bias 
-Sample size is adequately large (n>25) 

If these criteria are not met, then a non-parametric statistical test should be applied 

Assumption Criteria for Parametric v. Non-Parametric Method Selection

Nonparametric models are highly flexible statistical descriptions that do not rely on strong, simplifying assumptions about the underlying data generation process. A defining common theme of nonparametric approaches is that their level of complexity grows as the amount of available training data increases. While parametric models are necessary when data is scarce to prevent overfitting, the abundance of data in modern machine learning allows for the use of fully nonparametric models that can better fit complex realities.

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