arXiv subjects

Laurie Davies

Publications and source records attributed to Laurie Davies.

13 recordsLinked to original sources

arXiv2025

Connecting model-based and model-free approaches to linear least squares regression

In a regression setting with a response vector and given regressor vectors, a typical question is to what extent the response is related to these regressors, specifically, how well it can be approximated by a linear combination of the latter. Classical methods for this question are based on statistical models for the conditional distribution of the response, given the regressors. In the present paper it is shown that various p-values resulting from this model-based approach have also a purely data-analytic, model-free interpretation. This finding is derived in a rather general context. In addition, we introduce equivalence regions, a reinterpretation of confidence regions in the model-free context.

math.ST↗

arXiv2024

An Approximation Based Theory of Linear Regression

The goal of this paper is to provide a theory linear regression based entirely on approximations. It will be argued that the standard linear regression model based theory whether frequentist or Bayesian has failed and that this failure is due to an 'assumed (revealed?) truth' (John Tukey) attitude to the models. This is reflected in the language of statistical inference which involves a concept of truth, for example efficiency, consistency and hypothesis testing. The motivation behind this paper 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 is intuitive and requires only four simple equations. Given this a valid approximation is one where all the Gaussian P-values are less than a threshold $p0$ specified by the statistician, in this paper with the default value 0.01. This approximations approach is not only much simpler it is overwhelmingly better than the standard model based approach. This will be demonstrated using six real data sets, four from high dimensional regression and two from vector autoregression. Both the simplicity and the superiority of Gaussian P-values derive from their universal exactness and validity. This is in complete contrast to standard F P-values which are valid only for carefully designed simulations. The paper contains excerpts from an unpublished paper by John Tukey entitled `Issues relevant to an honest account of data-based inference partially in the light of Laurie Davies's paper'.

stat.ME↗

arXiv2023

Covariate Selection Based on a Assumpton-free Approach to Linear Regression with Exact Probabilities

In this paper we give a completely new approach to the problem of covariate selection in linear regression. A covariate or a set of covariates is included only if it is better in the sense of least squares than the same number of Gaussian covariates consisting of i.i.d. $N(0,1)$ random variables. The Gaussian P-value is defined as the probability that the Gaussian covariates are better. It is given in terms of the Beta distribution, it is exact and it holds for all data. The covariate selection procedures based on this require only a cut-off value $α$ for the Gaussian P-value: the default value in this paper is $α=0.01$. The resulting procedures are very simple, very fast, do not overfit and require only least squares. In particular there is no regularization parameter, no data splitting, no use of simulations, no shrinkage and no post selection inference is required. The paper includes the results of simulations, applications to real data sets and theorems on the asymptotic behaviour under the standard linear model. Here the stepwise procedure performs overwhelmingly better than any other procedure we are aware of. An R-package {\it gausscov} is available.

stat.ME↗

arXiv2022

Covariate Selection Based on a Model-free Approach to Linear Regression with Exact Probabilities

In this paper we give a completely new approach to the problem of covariate selection in linear regression. A covariate or a set of covariates is included only if it is better in the sense of least squares than the same number of Gaussian covariates consisting of i.i.d. $N(0,1)$ random variables. The Gaussian P-value is defined as the probability that the Gaussian covariates are better. It is given in terms of the Beta distribution, it is exact and it holds for all data making it model-free free. The covariate selection procedures require only a cut-off value $α$ for the Gaussian P-value: the default value in this paper is $α=0.01$. The resulting procedures are very simple, very fast, do not overfit and require only least squares. In particular there is no regularization parameter, no data splitting, no use of simulations, no shrinkage and no post selection inference is required. The paper includes the results of simulations, applications to real data sets and theorems on the asymptotic behaviour under the standard linear model. Here the step-wise procedure performs overwhelmingly better than any other procedure we are aware of. An R-package {\it gausscov} is available.

stat.ME↗

arXiv2022

Linear Regression, Covariate Selection and the Failure of Modelling

It is argued that all model based approaches to the selection of covariates in linear regression have failed. This applies to frequentist approaches based on P-values and to Bayesian approaches although for different reasons. In the first part of the paper 13 model based procedures are compared to the model-free Gaussian covariate procedure in terms of the covariates selected and the time required. The comparison is based on seven data sets and three simulations. There is nothing special about these data sets which are often used as examples in the literature. All the model based procedures failed. In the second part of the paper it is argued that the cause of this failure is the very use of a model. If the model involves all the available covariates standard P-values can be used. The use of P-values in this situation is quite straightforward. As soon as the model specifies only some unknown subset of the covariates the problem being to identify this subset the situation changes radically. There are many P-values, they are dependent and most of them are invalid. The P-value based approach collapses. The Bayesian paradigm also assumes a correct model but although there are no conceptual problems with a large number of covariates there is a considerable overhead causing computational and allocation problems even for moderately sized data sets. The Gaussian covariate procedure is based on P-values which are defined as the probability that a random Gaussian covariate is better than the covariate being considered. These P-values are exact and valid whatever the situation. The allocation requirements and the algorithmic complexity are both linear in the size of the data making the procedure capable of handling large data sets. It outperforms all the other procedures in every respect.

stat.ME↗

arXiv2020

Excess deaths, baselines, Z-scores, P-scores and peaks

The recent Covid-19 epidemic has lead to comparisons of the countries suffering from it. These are based on the number of excess deaths attributed either directly or indirectly to the epidemic. Unfortunately the data on which such comparisons rely are often incomplete and unreliable. This article discusses problems of interpretation of data even when the data is largely accurate and delayed by at most two to three weeks. This applies to the Office of National Statistics in the UK, the Statistisches Bundesamt in Germany and the Belgian statistical office Statbel. The data in the article is taken from these three sources. The number of excess deaths is defined as the number of deaths minus the baseline, the definition of which varies from country to country. In the UK it is the average number of deaths over the last five years, in Germany it is over the last four years and in Belgium over the last 11 years. This means that in all cases the individual baselines depend strongly on the timing and intensity of adverse factors such as past influenza epidemics and heat waves. This makes cross-country comparisons difficult. A baseline defined as the number the number of deaths in the absence of adverse factors can be operationalized by taking say the 10\% quantile of the number of deaths. This varies little over time and European countries within given age groups. It therefore enables more robust and accurate comparisons of different countries. The article criticizes the use of Z-scores which distort the comparison between countries. Finally the problem of describing past epidemics by their timing, that is start and finish and time of the maximum, and by their effect, the height of the maximum and the total number of deaths, is considered.

stat.AP↗

arXiv2019

Lasso, knockoff and Gaussian covariates: a comparison

Given data $\mathbf{y}$ and $k$ covariates $\mathbf{x}_j$ one problem in linear regression is to decide which if any of the covariates to include when regressing the dependent variable $\mathbf{y}$ on the covariates $\mathbf{x}_j$. In this paper three such methods, lasso, knockoff and Gaussian covariates are compared using simulations and real data. The Gaussian covariate method is based on exact probabilities which are valid for all $\mathbf{y}$ and $\mathbf{x}_j$ making it model free. Moreover the probabilities agree with those based on the F-distribution for the standard linear model with i.i.d. Gaussian errors. It is conceptually, mathematically and algorithmically very simple, it is very fast and makes no use of simulations. It outperforms lasso and knockoff in all respects by a considerable margin.

stat.ME↗

arXiv2017

Stepwise Choice of Covariates in High Dimensional Regression

Given data y(n) and p(n)covariates x(n) one problem in linear regression is to decide which if any of the covariates to include. There are many articles on this problem but all are based on a stochastic model for the data. This paper gives what seems to be a new approach which does not require any form of model. It is conceptually and algorithmically simple and consistency results can be proved under appropriate assumptions.

stat.ME↗

arXiv2017

Statistical Analysis of the Ricker Model

The Ricker model was introduced in the context of managing fishing stocks. It is a discrete non-linear iterative model given by $N(t+1)=rN(t)\exp(-N(t))$ where $N(t)$ is the population at time $t$. The model treated in this paper includes a random component $N(t+1)=rN(t)\exp(-N(t)+\varepsilon(t+1))$ and what is observed at time $t$ is a Poisson random variable with parameter $φN(t)$. Such a model has been analysed using `synthetic likelihood' and ABC (Approximate Bayesian Computation). In contrast this paper takes a non-likelihood approach and treats the model in a consistent manner as an approximation. The goal is to specify those parameter values if any which are consistent with the data.

stat.AP↗

arXiv2016

Stylized Facts and Simulating Long Range Financial Data

We propose a new method (implemented in an R-program) to simulate long-range daily stock-price data. The program reproduces various stylized facts much better than various parametric models from the extended GARCH-family. In particular, the empirically observed changes in unconditional variance are truthfully mirrored in the simulated data.

q-fin.ST↗

arXiv2016

On $p$-values

Models are consistently treated as approximations and all procedures are consistent with this. They do not treat the model as being true. In this context $p$-values are one measure of approximation, a small $p$-value indicating a poor approximation. Approximation regions are defined and distinguished from confidence regions.

stat.OT↗

arXiv2016

Functional Choice and Non-significance Regions in Regression

Given data $y$ and $k$ covariates $x$ the problem is to decide which covariates to include when approximating $y$ by a linear function of the covariates. The decision is based on replacing subsets of the covariates by i.i.d. normal random variables and comparing the error with that obtained by retaining the subsets. If the two errors are not significantly different for a particular subset it is concluded that the covariates in this subset are no better than random noise and they are not included in the linear approximation to $y$.

math.ST↗

arXiv2010

Locally adaptive image denoising by a statistical multiresolution criterion

We demonstrate how one can choose the smoothing parameter in image denoising by a statistical multiresolution criterion, both globally and locally. Using inhomogeneous diffusion and total variation regularization as examples for localized regularization schemes, we present an efficient method for locally adaptive image denoising. As expected, the smoothing parameter serves as an edge detector in this framework. Numerical examples illustrate the usefulness of our approach. We also present an application in confocal microscopy.

stat.ME↗