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The stochastic restricted ridge estimator in generalized linear models

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Abstract

Many researchers have studied restricted estimation in the context of exact and stochastic restrictions in linear regression. Some ideas in linear regression, where the ridge and restricted estimations are the well known, were carried to the generalized linear models which provide a wide range of models, including logistic regression, Poisson regression, etc. This study considers the estimation of generalized linear models under stochastic restrictions on the parameters. Furthermore, the sampling distribution of the estimators under the stochastic restriction, the compatibility test and choice of the biasing parameter are given. A real data set is analyzed and simulation studies concerning Binomial and Poisson distributions are conducted. The results show that when stochastic restrictions and ridge idea are simultaneously applied to the estimation methods, the new estimator gains efficiency in terms of having smaller variance and mean square error.

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Notes

  1. Here, 2 refers to the table value corresponding to \(95\%\).

  2. If for example we choose \(Var(\phi )=\sigma ^{2}=0.2\), then the probability approximately equals \(P(\left| \beta _{4}-\beta _{5}\right| \le 0.8944)=0.95\).

  3. Same result is also valid for the ridge estimator when we use a biasing parameter that does not depend on \(\sigma ^{2}\) (\(k_{m}\) and \(k_{h}\) depend on \(\sigma ^{2}\) through \({\hat{\beta }}_{c}\)) However, to save space we did not report those results.

  4. Same result is also valid for the ridge estimator when we use a biasing parameter that does not depend on \(\sigma ^{2}\).

  5. This result is true for the ridge estimator when we use a biasing parameter that does not depend on \(\sigma ^{2}\). However, when a biasing parameter that depends on \(\sigma ^{2}\) is used, this result is valid for large sample sizes and \(\gamma ^{2}=0.85,0.90.\)

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Acknowledgements

This work was supported by Research Fund of Çukurova University under Project Number FBI-2017-9024 to M. Revan Özkale, a visiting scholar for one month at Stockholm University, Department of Statistics.

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Authors and Affiliations

  1. Department of Statistics, Faculty Science and Letters, Çukurova University, 01330, Adana, Turkey

    M. Revan Özkale

  2. Department of Statistics, Stockholm University, Stockholm, Sweden

    Hans Nyquist

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  1. M. Revan Özkale
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  2. Hans Nyquist
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Correspondence to M. Revan Özkale.

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Özkale, M.R., Nyquist, H. The stochastic restricted ridge estimator in generalized linear models. Stat Papers 62, 1421–1460 (2021). https://doi.org/10.1007/s00362-019-01142-7

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