Statistical Model

A Statistical Model is a non-deterministic mathematical model that is used in statistical hypothesis testing to estimate population parameters and to describe probability distributions.

• Context:
  • It is usually represented as ([math]\displaystyle{ S, \mathcal{P} }[/math]) with [math]\displaystyle{ \mathcal{P}=\{P_{\theta} : \theta \in \Theta\} }[/math], where [math]\displaystyle{ S }[/math] is the sample space, [math]\displaystyle{ \mathcal{P} }[/math]is the set of probability distribution of the sample space, and [math]\displaystyle{ \Theta }[/math] is the set of all possible values of [math]\displaystyle{ \theta }[/math] (model parameters). If [math]\displaystyle{ \Theta }[/math] has a finite dimension than the statistical model is parametric, while if [math]\displaystyle{ \Theta }[/math] as infinite dimension the statistical model is nonparametric.
  • It can be interpreted as the hypothesized probability distribution for the observation of the random variable X.
  • It can range from being a Parametric Statistical Model to being a Nonparametric Statistical Model.
• Example(s):
  • a Linear Regression Model,
  • a Statistical Language Model.
  • …
• Counter-Example(s):
  • Statistical Hypothesis.
  • Sampling Distribution.
  • Population Parameter.
• See: Statistical Inference, Statistical Hypothesis Testing Task, Sample Space, Probability distribution, Population Parameter.

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References

2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Statistical_model

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

The assumptions embodied by a statistical model describe a set of probability distributions, some of which are assumed to adequately approximate the distribution from which a particular data set is sampled. The probability distributions inherent in statistical models are what distinguishes statistical models from other, non-statistical, mathematical models.

A statistical model is usually specified by mathematical equations that relate one or more random variables and possibly other non-random variables. As such, "a model is a formal representation of a theory" (Herman Adèr quoting Kenneth Bollen).

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

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