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Multiple and single imputation Regression model Resampling Comparison of confidence intervals a b s t r a c tThe problem of imputing missing observations under the linear regression model is considered. It is assumed that observations are missing at random and all the observations on the auxiliary or independent variables are available. Estimates of the regression parameters based on singly and multiply imputed values are given. Jackknife as well as bootstrap estimates of the variance of the singly imputed estimator of the regression parameters are given. These estimators are shown to be consistent estimators. The asymptotic distributions of the imputed estimators are also given to obtain interval estimates of the parameters of interest. These interval estimates are then compared with the interval estimates obtained from multiple imputation. It is shown that singly imputed estimators perform at least as good as multiply imputed estimators. A new nonparametric multiply imputed estimator is proposed and shown to perform as good as a multiply imputed estimator under normality. The singly imputed estimator, however, still remains at least as good as a multiply imputed estimator.

A novel scalable and privacy-preserving distributed parallel optimization that allows the participation of large-scale aggregation of prosumers with residential PV-battery systems in the market for the ancillary service (ASM) is proposed in this paper. To consider both reserve capacity and reserve energy, day-ahead and real-time stages in the ASM are considered. A method, based on hybrid Variable Neighborhood Search (VNS) and distributed parallel optimization is designed for the day ahead and realtime optimization. Different distributed optimization methods are compared and designed and a new distributed optimization method based on Linear Programming (LP) is designed that overcomes previous methods based on integer and Quadratic programming (QP). The proposed LP-based optimization can be easily coded up and implemented on microcontrollers and connected to a designed Internet of Things (IoT) based architecture. As confirmed by simulation results, carried out considering different realistic case studies, both day-ahead and real-time proposed optimization methods, by allocating the computational effort among local resources, are highly scalable and fulfil the privacy of prosumers.

In the context of the energy transition, energy communities are gaining increasing attention all over the world, in recent years. By participating in an energy community, prosumers may take a leading role in the energy transition and improve the self-consumption of renewable energy produced inside the community. Prosumers can carry out energy exchanges inside the energy community and provide ancillary services to the system operators, thus contributing to improve the efficiency and stability of the grid. A novel scalable, privacy-preserving, and real-time distributed parallel optimization is proposed to manage a largescale energy community, considering energy exchanges inside the community according to the model of virtual self-consumption and the provision of ancillary services. The proposed method preserves the privacy of prosumers and allows the assessment of the impact of energy exchanges on the ancillary services provided by an energy community. Simulation results confirmed that the proposed method is superior in terms of privacy if compared with the equivalent centralized optimization and that it has a convergence rate higher than that of the splitting conic solver (SCS).

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