Abstract
Surveys the different types of probability models for which statistical methods will be later constructed. It describes what situations they are suitable for, and derives some of their key properties, including their behaviour under transformation. It also introduces the exponential family of distributions as a unifying framework for later results to be stated in more generality. The last part of the chapter considers the problem of choosing a particular type of model, whether by first principles or by means of exploratory data analysis, using numerical and graphical summaries.
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Notes
- 1.
The reason for this is that in the natural representation, the parameter appears linearly in the exponent. In the usual representation, the parameter appears nonlinearly, as the image through the function η. This complicates things when we will need to differentiate with respect to the parameter.
- 2.
That is, if we took the line segment x (n) − x (1) and placed equal weights at the points x 1, …, x n , then the point \(\bar{x}\) is where the line segment would balance.
- 3.
In the sense that half the observations must be greater than or equal to the median, and half the observations must be less than or equal to the median.
- 4.
That is, if we took the line segment x (n) − x (1) and placed equal weights at the points x 1, …, x n , then tried to rotate the segment around the point \(\bar{x}\), then the variance is an indicator of how much force we would need to apply. If the observations are spread far from \(\bar{x}\), then we need a lot of force (high sample variance); but if the observations are close to \(\bar{x}\), then our task is easier (low sample variance).
- 5.
To be precise: 25 % of the sample observations are less than or equal to Q 1, and 25 % of the observations are greater than or equal to Q 3.
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Panaretos, V.M. (2016). Regular Probability Models. In: Statistics for Mathematicians. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28341-8_1
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DOI: https://doi.org/10.1007/978-3-319-28341-8_1
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