Statistical Model
From GM-RKB
A Statistical Model is a non-deterministic mathematical model that used in statistical hypothesis testing to estimate population parameters and to describe probability distributions.
- Context:
- It is usually represented as ([math]S, \mathcal{P}[/math]) with [math]\mathcal{P}=\{P_{\theta} : \theta \in \Theta\}[/math], where [math]S[/math] is the sample space, [math]\mathcal{P}[/math]is the set of probability distribution of the sample space, and [math]\Theta[/math] is the set of all possible values of [math]\theta[/math] (model parameters). If [math]\Theta[/math] has a finite dimension than the statistical model is parametric, while if [math]\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):
- Counter-Example(s):
- See: Statistical Inference, Statistical Hypothesis Testing Task, Sample Space, Probability distribution, Population Parameter.
References
2016
- 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).^{[1]}
- 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.