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The authors offer a response to the commentaries made on their article on statistical estimation and privacy. They discuss the issues raised by the discussants including the advantages and disadvantages of local notions of privacy as compared to classical differential and approximate differential privacy, the issue of interactivity, and the importance of local differential privacy. The goal of statistical inference is also mentioned.

Electricity load forecasts are an integral part of many decision-making processes in the electricity market. However, most literature on electricity load forecasting concentrates on deterministic forecasts, neglecting possibly important information about uncertainty. A more complete picture of future demand can be obtained by using distributional forecasts, allowing for more efficient decision-making. A predictive density can be fully characterized by tail measures such as quantiles and expectiles. Furthermore, interest often lies in the accurate estimation of tail events rather than in the mean or median. We propose a new methodology to obtain probabilistic forecasts of electricity load that is based on functional data analysis of generalized quantile curves. The core of the methodology is dimension reduction based on functional principal components of tail curves with dependence structure. The approach has several advantages, such as flexible inclusion of explanatory variables like meteorological forecasts and no distributional assumptions. The methodology is applied to load data from a transmission system operator (TSO) and a balancing unit in Germany. Our forecast method is evaluated against other models including the TSO forecast model. It outperforms them in terms of mean absolute percentage error and mean squared error. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

Weighting methods that adjust for observed covariates, such as inverse probability weighting, are widely used for causal inference and estimation with incomplete outcome data. Part of the appeal of such methods is that one set of weights can be used to estimate a range of treatment effects based on different outcomes, or a variety of population means for several variables. However, this appeal can be diminished in practice by the instability of the estimated weights and by the difficulty of adequately adjusting for observed covariates in some settings. To address these limitations, this article presents a new weighting method that finds the weights of minimum variance that adjust or balance the empirical distribution of the observed covariates up to levels prespecified by the researcher. This method allows the researcher to balance very precisely the means of the observed covariates and other features of their marginal and joint distributions, such as variances and correlations and also, for example, the quantiles of interactions of pairs and triples of observed covariates, thus, balancing entire two- and three-way marginals. Since the weighting method is based on a well-defined convex optimization problem, duality theory provides insight into the behavior of the variance of the optimal weights in relation to the level of covariate balance adjustment, answering the question, how much does tightening a balance constraint increases the variance of the weights? Also, the weighting method runs in polynomial time so relatively large datasets can be handled quickly. An implementation of the method is provided in the new package sbw for R. This article shows some theoretical properties of the resulting weights and illustrates their use by analyzing both a dataset from the 2010 Chilean earthquake and a simulated example. [ABSTRACT FROM PUBLISHER]