d-Posterior approach in regression
A.A. Zaikin
Kazan Federal University, Kazan, 420008 Russia
Abstract
In this paper, we have used the d-posterior approach in regression. Regression predictions are a sequence of similarly made decisions. Thus, d-risk can be helpful to estimate the quality of such decisions. We have introduced a method to apply the d-posterior approach in regression models. This method is based on posterior predictive distribution of the dependent variable with the given novel input of predictors. In order to make d-risk of the prediction rule meaningful, we have also considered adding probability distribution of the novel input to the model.
The method has been applied to simple regression models. Firstly, linear regression with Gaussian white noise has been considered. For the quadratic loss function, estimates with uniformly minimal d-risks have been constructed. It appears that the parameter estimate in this model is equal to the Bayesian estimate, but the prediction rule is slightly different. Secondly, regression for the binary dependent variable has been investigated. In this case, the d-posterior approach is used for the logit regression model. As for the 0–1 loss function, the estimate with uniformly minimal d-risk does not exist, we suggested a classification rule, which minimizes the maximum of two d-risks. The resulting decision rules for both models are compared to the usual Bayesian decisions and the decisions based on the maximum likelihood principle.
Keywords: Bayesian inference, regression, d-risk
Acknowledgements. This work was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities (project no. 1.7629.2017/8.9) (for Gaussian regression). The study was also supported by the Russian Foundation for Basic Research and the Republic Of Tatarstan according to the research project no. 17-41-160620 (for logit regression).
The work is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.
References
1. Volodin I.N., Simushkin S.V. On 
-posteriori approach to the problem of statistical inference. Proc. 3rd Int. Vilnius Conf. on Probability Theory and Mathematical Statistics, 1981, vol. 1, pp. 100–101.
2. Volodin I.N., Simushkin S.V. Statistical inference with minimal -risk. J. Sov. Math., 1988, vol. 42, no. 1, pp. 1464–1472. doi: 10.1007/BF01098858.
3. Scott J.G., Kelly R.C., Smith M.A., Zhou P., Kass R.E. False discovery rate regression: An application to neural synchrony detection in primary visual cortex. J. Am. Stat. Assoc., 2015, vol. 110, no. 510, pp. 459-471. doi: 10.1080/01621459.2014.990973.
4. Simushkin S.V. Optimal -guarantee procedures for distinguishing two hypothesis. VINITI Acad. Sci. USSR, 1981, no. 55, pp. 47–81. (In Russian)
5. Volodin I.N., Novikov A.A. Asymptotics of the necessary sample size in testing parametric hypotheses: -posterior approach. Math. Methods Stat., 1998, vol. 7, no. 1, pp. 111–121.
Received
October 12, 2017
For citation: Zaikin A.A. d-Posterior approach in regression. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 410–418.
The content is available under the license Creative Commons Attribution 4.0 License.
