A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process.

A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is a formal representation of a theory (Herman Adèr quoting Kenneth Bollen).

All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference.

Could you explain each sentence in simple terms?
Here are some simplified explanations and paraphrases of the sentences in the definition of a statistical model:

1) A statistical model represents a theory or idea of how data is generated. It is a mathematical equation that assumes certain things about how the data came to be.

2) The model specifies relationships between random variables (variables that can take on a range of values by chance) and non-random variables (variables that are determined or fixed). The model shows how the variables influence or relate to each other.

3) Statistical hypothesis tests and estimators are tools used to analyze data and draw inferences. They are derived from or based on the assumptions and relationships specified in a statistical model.

4) More broadly, statistical models are fundamental to statistics and statistical inference. They underlie the methods and techniques used to study and draw insights from data.

The key purpose of a statistical model is to represent a data-generating process. It allows statisticians to make inferences about a larger group based on a smaller sample. The model makes assumptions and specifies variables/relationships that can then be used as a tool for analyzing data and results. Does this help explain the definition and role of statistical models? Let me know if you have any other questions!