Abstract
In this research note, we sketch the idea to use (aspects of) imprecise probability models to handle unobserved heterogeneity in statistical (regression) models. Unobserved heterogeneity (frailty) is a frequent issue in many applications, arising whenever the underlying probability distributions depend on unobservable individual characteristics (like personal attitudes or hidden genetic dispositions). We consider imprecise sampling models where the likelihood contributions depend on individual parameters, varying in an interval (cuboid). Based on this, and a hyperparameter controlling the amount of ambiguity, we directly fit a credal set to the data. We introduce the basic concepts of this credal maximum likelihood approach, sketch first aspects of practical calculation of the resulting estimators by constrained optimization, derive some first general properties and finally discuss some ideas of a data-dependent choice of the hyperparameter.
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- 1.
We consider here directly the set of parametric models; often in the theory of imprecise the whole convex hull of the generating models is used.
- 2.
In the case of multi-dimensional parameters typically only few components of the parameter vector are taken to vary individually. We do not explicitly distinguish the varying and the constant components of the parameter vector notationally.
- 3.
Typically the models are equivalently formulated in terms of a central parameter \(\nu _{\text{ overall }}\) and individual variations \(\nu _i\) around it with zero mean such that \(\vartheta _i=\nu _{\text{ overall }}+\nu _i\).
- 4.
Some ideas how to choose \(\delta \) in a data-dependent way are sketched in Sect. 4.
- 5.
For the sake of conciseness, the formulation is given in terms of \(f_\vartheta (\cdot )\) only; it is immediately adopted to the regression case relying on \(f_\vartheta (\cdot |x_i)\).
- 6.
Throughout this paper vectorial inequalities are understood as inequalities for all components.
- 7.
This corresponds to the determination of the variance of the distribution in the traditional random effects approaches to unobserved heterogeneity, while, from this perspective, in our approach the range of the random effects has to be specified. This analogy should however not hide the major difference: the traditional approach needs a fixed distributional class for the – unobservable (!) – random effect (typically normal distribution), while our approach refrains from any modelling of the hidden process.
Therefore, if one looks at an interpretation of our approach in the context of traditional theory, our approach could be termed ‘nonparametric’.
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Acknowledgement
I am very grateful to the two anonymous referees for very helpful comments and quite stimulating remarks.
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Augustin, T. (2018). Imprecise Sampling Models for Modelling Unobserved Heterogeneity? Basic Ideas of a Credal Likelihood Concept. In: Ciucci, D., Pasi, G., Vantaggi, B. (eds) Scalable Uncertainty Management. SUM 2018. Lecture Notes in Computer Science(), vol 11142. Springer, Cham. https://doi.org/10.1007/978-3-030-00461-3_24
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