In 2012, at the ILSVRC, an international contest for object recognition accuracy, the University of Toronto team showed an unparalleled recognition rate with an error rate of just under 17% using deep learning, while other teams had an error rate of around 26%. It was also in 2012 that the Obama administration announced its strategic plan for big data research and development (commonly known as the Big Data Initiative). Therefore, we can say that 2012 was the starting point of the third AI boom. Today, only 10 years have passed since then. In particular, the remarkable development of data analysis methods over the past few years has had the biggest impact on me, as I have been involved in statistical data analysis since I was a student. In this paper, I will review the development of statistics over the past 40 years from my point of view, as I have been involved in research in statistics and related fields since the 1980s. I hope that this paper will be of some help in considering future trends in statistics.

We consider hybrid estimation for stochastic differential equations (SDEs) from discrete observations and adaptive estimation of a stochastic partial differential equation (SPDE) based on sample data. In order to obtain the maximum likelihood (ML) type estimators of discretely observed ergodic SDEs, we need a suitable initial value for optimization of the quasi log likelihood (QLL) function. The Bayes type estimators derived from both the reduced data and the thinned data are used as an initial estimator for optimization of the QLL function, and the ML type estimators are obtained by using the initial Bayes estimators. The ML type estimators with the initial Bayes type estimators are called the hybrid type estimators. The asymptotic properties of both the initial Bayes type and the hybrid type estimators are shown and the asymptotic behaviors of both the initial Bayes type and the hybrid type estimators are given through simulations. Next, we treat parametric estimation for a parabolic linear second-order SPDE with a small dispersion parameter based on high frequency data which are observed in time and space. The adaptive estimators of unknown parameters of the SPDE are obtained, and their asymptotic properties are investigated. Furthermore, we give the asymptotic behaviors of the adaptive estimators by simulations.

When the Akaike Information Criterion (AIC) is derived based on its original definition,there is a setting in which the penalty term deviates significantly from twice the number of parameters: propensity score analysis, which is the basis of causal inference. Although a semi-parametric approach based on propensity scores is considered, the formal use of AIC for the problem of selecting the marginal structure in a marginal structure model leads to a large over-fitting. In recent years, a semi-parametric approach that has been widely used is called doubly robust estimation, which is robust against model mis-specification. In this paper, we adopt the idea of covariate balancing for the doubly robust estimation, and change the loss function from ordinary log-likelihood to one that is robust against outliers. Then, we derive a penalty term while maintaining the robustness, and propose a triply robust criterion as an information criterion with validity. In numerical experiments, we first show that the triply robust criterion clearly outperforms the formal criterion with a penalty term twice the number of parameters in terms of predictive performance in the case without model mis-specification or outliers. Next,we deal with the cases of model mis-specification or outliers, and confirm that the triply robust criterion is less sensitive to them.

In Big Data era, large-scale data is being utilized in a variety of fields. On the other hand, as data becomes larger and larger, there are more and more situations where heterogeneous groups are mixed together, and simple statistical modeling based on the conventional “one-model-fits-the-whole-population approach” is not sufficient to perform appropriate statistical analysis. Although various methodologies have already been developed for this situation, there is still a lack of methodologies that can perform flexible statistical modeling at a realistic computational cost. In this paper, we describe methodologies that can simultaneously perform grouping of data (discovery of heterogeneous groups) and estimation of statistical models for each group (discovery of the unique structure of each group) in the context of clustered data and spatial data.