Comparison among simultaneous confidence regions for nonlinear diffusion models
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Comparison among simultaneous confidence regions for nonlinear diffusion models Claudia Furlan1
· Cinzia Mortarino1
Received: 7 December 2017 / Accepted: 23 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Accuracy measures for parameter estimates represent a tricky issue in nonlinear models. Practitioners often use separate marginal confidence intervals for each parameter in place of a simultaneous confidence region (sCR). However, this can be extremely misleading due to the curvature of the parameter space of the nonlinear model. For low parameter dimensions, routines for evaluating approximate sCRs are available in the most common software programs; however, the degree of accuracy depends on the curvature of the parameter space and the sample size. Exact sCRs are computationally intensive, and for this reason, in the past, they did not receive much attention. In this paper, we perform a comparison among exact, asymptotic exact, approximate sCRs, and marginal confidence intervals. More modern regions based on bootstrap are also examined as an alternative approach (both parametric and nonparametric). Their degree of accuracy is compared with both real data and simulation results. Among the nonlinear models, in this paper, the focus is on two of the most widespread diffusion models of products and technologies, that is, the Bass and Generalized Bass models. Three different empirical studies are analyzed here. Simulation studies are also performed for lifecycles with the same diffusion characteristics as those of the empirical studies. Our results show that, as the parameter dimension increases, overlapping among the alternative sCRs reduces. The approximate sCR shows inadequate values of overlapping with the exact sCR, even for moderate parameter dimension. Bootstrap regions also exhibit good performance in describing the shape of the exact region when curvature is present, but they fail to spread up to its boundary. The coverage probability of each region is assessed with simulations. We observe that the coverage probability of the approximate sCR decreases rapidly, even for moderate parameter dimension, and it is smaller than the nominal level for bootstrap regions.
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Claudia Furlan [email protected] Cinzia Mortarino [email protected]
1
Department of Statistical Sciences, University of Padova, Padova, Italy
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C. Furlan, C. Mortarino
Keywords Estimation accuracy · Bootstrap · Bass model · Cumulative data · Tukey’s depth
1 Introduction Nonlinear models are the natural modeling framework for many real-world phenomena. Unlike linear models, accuracy measures for parameter estimates, such as confidence intervals or confidence regions, may represent a difficult task due to the curvature of the parameter space. A common mistake is relying on marginal confidence intervals, which can be misleading. The topic of constructing a simultaneous confidence region (sCR), both exact and approximate, was mostly developed in the 1960s–1980s, but this research was limited by t
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