MDYPL state evolution functions with intercept

Estimation and inference from generalized linear models based on various methods for bias reduction and maximum penalized likelihood with powers of the Jeffreys prior as penalty. The 'brglmFit()' fitting method can achieve reduction of estimation bias by solving either the mean bias-reducing adjusted score equations in Firth (1993) <doi:10.1093/biomet/80.1.27> and Kosmidis and Firth (2009) <doi:10.1093/biomet/asp055>, or the median bias-reducing adjusted score equations in Kenne et al. (2017) <doi:10.1093/biomet/asx046>, or through the direct subtraction of an estimate of the bias of the maximum likelihood estimator from the maximum likelihood estimates as in Cordeiro and McCullagh (1991) <https://www.jstor.org/stable/2345592>. See Kosmidis et al (2020) <doi:10.1007/s11222-019-09860-6> for more details. Estimation in all cases takes place via a quasi Fisher scoring algorithm, and S3 methods for the construction of of confidence intervals for the reduced-bias estimates are provided. In the special case of generalized linear models for binomial and multinomial responses (both ordinal and nominal), the adjusted score approaches to mean and media bias reduction have been found to return estimates with improved frequentist properties, that are also always finite, even in cases where the maximum likelihood estimates are infinite (e.g. complete and quasi-complete separation; see Kosmidis and Firth, 2020 <doi:10.1093/biomet/asaa052>, for a proof for mean bias reduction in logistic regression). The 'mdyplFit()' fitting method fits logistic regression models using maximum Diaconis-Ylvisaker prior penalized likelihood, which also guarantees finite estimates. High-dimensionality corrections under proportional asymptotics can be applied to the resulting objects; see Sterzinger and Kosmidis (2024) <doi:10.48550/arXiv.2311.07419> for details.

Ioannis Kosmidis

brglm2

Bias Reduction in Generalized Linear Models

Federico Boiocchi

Philipp Sterzinger

Kjell Konis

Nicola Sartori

se1 function

<dl><dt>mu</dt>
<dd>aggregate bias parameter.</dd>
<dt>b</dt>
<dd>parameter <code>b</code> in the state evolution functions.</dd>
<dt>sigma</dt>
<dd>square root of the aggregate variance of the MDYPL
estimator.</dd>
<dt>iota</dt>
<dd>limits of the MDYPL estimate for the intercept as the sample size goes to +Inf</dd>
<dt>kappa</dt>
<dd>asymptotic ratio of columns/rows of the design
matrix. <code>kappa</code> should be in <code>(0, 1)</code>.</dd>
<dt>gamma</dt>
<dd>the square root of the limit of the variance of the
linear predictor.</dd>
<dt>alpha</dt>
<dd>the shrinkage parameter of the MDYPL
estimator. <code>alpha</code> should be in <code>(0, 1)</code>.</dd>
<dt>intercept</dt>
<dd>intercept of the logistic regression model</dd>
<dt>gh</dt>
<dd>A list with the Gauss-Hermite quadrature nodes and
weights, as returned from <code>statmod::gauss.quad()</code> with <code>kind = "hermite"</code>. Default is <code>NULL</code>, in which case <code>gh</code> is set to
<code>statmod::gauss.quad(200, kind = "hermite")</code> is used.</dd>
<dt>prox_tol</dt>
<dd>tolerance for the computation of the proximal
operator; default is <code>1e-10</code>. fixed point problem solved via Newton-Raphson</dd></dl>

Arguments

MDYPL state evolution functions with intercept — se1

<dl>

<dt>mu</dt>
<dd>aggregate bias parameter.</dd>


<dt>b</dt>
<dd>parameter <code>b</code> in the state evolution functions.</dd>


<dt>sigma</dt>
<dd>square root of the aggregate variance of the MDYPL
estimator.</dd>


<dt>iota</dt>
<dd>limits of the MDYPL estimate for the intercept as the sample size goes to +Inf</dd>


<dt>kappa</dt>
<dd>asymptotic ratio of columns/rows of the design
matrix. <code>kappa</code> should be in <code>(0, 1)</code>.</dd>


<dt>gamma</dt>
<dd>the square root of the limit of the variance of the
linear predictor.</dd>


<dt>alpha</dt>
<dd>the shrinkage parameter of the MDYPL
estimator. <code>alpha</code> should be in <code>(0, 1)</code>.</dd>


<dt>intercept</dt>
<dd>intercept of the logistic regression model</dd>


<dt>gh</dt>
<dd>A list with the Gauss-Hermite quadrature nodes and
weights, as returned from <code>statmod::gauss.quad()</code> with <code>kind = "hermite"</code>. Default is <code>NULL</code>, in which case <code>gh</code> is set to
<code>statmod::gauss.quad(200, kind = "hermite")</code> is used.</dd>


<dt>prox_tol</dt>
<dd>tolerance for the computation of the proximal
operator; default is <code>1e-10</code>. fixed point problem solved via Newton-Raphson</dd>

</dl>

se1: MDYPL state evolution functions with intercept

Description

Usage

Arguments