m15005m

<p>Angle and photon counts in thin film x-ray refraction.</p>

datasets

The standard linear regression theory whether frequentist or Bayesian is based on an 'assumed (revealed?) truth' (John Tukey) attitude to models. This is reflected in the language of statistical inference which involves a concept of truth, for example confidence intervals, hypothesis testing and consistency. The motivation behind this package was to remove the word true from the theory and practice of linear regression and to replace it by approximation. The approximations considered are the least squares approximations. An approximation is called valid if it contains no irrelevant covariates. This is operationalized using the concept of a Gaussian P-value which is the probability that pure Gaussian noise is better in term of least squares than the covariate. The precise definition given in the paper  "An Approximation Based Theory of Linear Regression".  Only four simple equations are required. Moreover the Gaussian P-values can be simply derived from standard F P-values. Furthermore they are exact and valid whatever the data in contrast F P-values are only valid for specially designed simulations. A valid approximation is one where all the Gaussian P-values are less than a threshold p0 specified by the statistician, in this package with the default value 0.01. This approximations approach is not only much simpler it is overwhelmingly better than the standard model based approach. The will be demonstrated using high dimensional regression and vector autoregression real data sets. The goal is to find valid approximations. The search function is f1st which is a greedy forward selection procedure which results in either just one or no approximations which may however not be valid. If the size is less than than a threshold with default value 21 then an all subset procedure is called which returns the best valid subset. A good default start is f1st(y,x,kmn=15) The best function for returning multiple approximations is f3st which repeatedly calls f1st. For more information see the papers: L. Davies and L. Duembgen, "Covariate Selection Based on a Model-free Approach to Linear Regression with Exact Probabilities", <doi:10.48550/arXiv.2202.01553>, L. Davies, "An Approximation Based Theory of Linear Regression", 2024, <doi:10.48550/arXiv.2402.09858>.

m15005m: m15005m data

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