Information and complexity in statistical modeling
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With some other examples, though, the calculation can be difficult, or even impractical e. For an assumption to constitute a statistical model, such difficulty is acceptable: doing the calculation does not need to be practicable, just theoretically possible. The intuition behind this definition is as follows. It is assumed that there is a "true" probability distribution induced by the process that generates the observed data.
A parameterization is generally required to have distinct parameter values give rise to distinct distributions, i. A parameterization that meets the requirement is said to be identifiable. Suppose that we have a population of school children, with the ages of the children distributed uniformly , in the population. The height of a child will be stochastically related to the age: e. This implies that height is predicted by age, with some error. An admissible model must be consistent with all the data points. Gaussian, with zero mean. In this instance, the model would have 3 parameters: b 0 , b 1 , and the variance of the Gaussian distribution.
The parameterization is identifiable, and this is easy to check. There are two assumptions: that height can be approximated by a linear function of age; that errors in the approximation are distributed as i. A statistical model is a special class of mathematical model. What distinguishes a statistical model from other mathematical models is that a statistical model is non- deterministic. Thus, in a statistical model specified via mathematical equations, some of the variables do not have specific values, but instead have probability distributions; i.
Statistical models are often used even when the data-generating process being modeled is deterministic. For instance, coin tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic via a Bernoulli process. Choosing an appropriate statistical model to represent a given data-generating process is sometimes extremely difficult, and may require knowledge of both the process and relevant statistical analyses.
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Relatedly, the statistician Sir David Cox has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis". Here, k is called the dimension of the model. As an example, if we assume that data arise from a univariate Gaussian distribution , then we are assuming that.
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In this example, the dimension, k , equals 2. As another example, suppose that the data consists of points x , y that we assume are distributed according to a straight line with i.
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Gaussian residuals with zero mean : this leads to the same statistical model as was used in the example with children's heights. The dimension of the statistical model is 3: the intercept of the line, the slope of the line, and the variance of the distribution of the residuals. Note that in geometry, a straight line has dimension 1. A statistical model is semiparametric if it has both finite-dimensional and infinite-dimensional parameters.
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Parametric models are by far the most commonly used statistical models. Regarding semiparametric and nonparametric models, Sir David Cox has said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies".
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Two statistical models are nested if the first model can be transformed into the second model by imposing constraints on the parameters of the first model. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions.
Information and Complexity in Statistical Modeling by Jorma Rissanen
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