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This paper generalizes the quadratic framework introduced by Le Courtois (2016) and Sumpf (2018), to obtain new credibility premiums in the balanced case, i.e. under the balanced squared error loss function. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.

In the actuarial field, credibility theory is an empirical model used to calculate the premium. One of the crucial tasks of the actuary in the insurance company is to design a tariff structure that will fairly distribute the burden of claims among insureds. In this work, the authors use the weighted balanced loss function (WBLF, henceforth) to obtain new credibility premiums, and WBLF is a generalized loss function introduced by Zellner (1994) (see Gupta and Berger (1994), pp. 371-390) which appears also in Dey et al. (1999) and Farsipour and Asgharzadhe (2004).

The authors declare that there is no conflict of interest and the funding information is not applicable.

This work is motivated by the following: quadratic credibility premium under the balanced loss function is useful for the practitioner who wants to explicitly take into account higher order (cross) moments and new effects such as the clustering effect to finding a premium more credible and more precise, which arranges both parts: the insurer and the insured. Also, it is easy to apply for parametric and non-parametric approaches. In addition, the formulas of the parametric (Poisson–gamma case) and the non-parametric approach are simple in form and may be used to find a more flexible premium in many special cases. On the other hand, this work neglects the semi-parametric approach because it is rarely used by practitioners.

There are several examples of actuarial science (credibility).

In this paper, the authors used the WBLF and a quadratic adjustment to obtain new credibility premiums. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.

This study aims to examine and understand the impact of reverse logistics adoption on firms' digitalization and collaboration activities. Specifically, leveraging the knowledge-based view, this study examines how adopting sustainable logistic practices (reverse logistics) prepares firms to embrace digitalization and encourages them to collaborate with other organizations.

The study used longitudinal survey data from two waves (2017 and 2019) from the Mannheim Centre for European Economic Research. The authors used the negative binomial regression analyses to test the impact of reverse logistics adoption on the digitalization and inter-organizational collaboration dependent count variables.

The study's findings highlight the usefulness of reverse logistics in enabling digitalization and inter-organizational collaboration. The results show that the firms investing in sustainable supply chains will be better positioned to nurture digitalization and inter-organizational collaboration.

For resource-bound managers, this study provides an important insight into prioritizing activities by highlighting how reverse logistics can facilitate digitalization and collaboration. The study demonstrates that the knowledge generated by reverse logistics adoption can be an essential pillar and enabler toward achieving firms' digitalization and collaboration goals.

The study is among the first to examine the effect of reverse logistics adoption on firm activities that are not strictly associated with the circular economy (digitalization and collaboration). Utilizing the knowledge-based view, this study reports on the additional benefits of reverse logistics implementation previously not discussed in the literature.