Summary: The need for new methods to deal with big data is a common theme in most scientific fields, although its definition tends to vary with the context. Statistical ideas are an essential part of this, and as a partial response, a thematic program on statistical inference, learning and models in big data was held in 2015 in Canada, under the general direction of the Canadian Statistical Sciences Institute, with major funding from, and most activities located at, the Fields Institute for Research in Mathematical Sciences. This paper gives an overview of the topics covered, describing challenges and strategies that seem common to many different areas of application and including some examples of applications to make these challenges and strategies more concrete.

The self-similar gravitational collapse solutions to the Einstein-axion–dilaton system have already been discovered. Those solutions become invariants after combining the spacetime dilation with the transformations of internal SL(2, R). We apply nonlinear statistical models to estimate the functions that appear in the physics of Black Holes of the axion–dilaton system in four dimensions. These statistical models include parametric polynomial regression, nonparametric kernel regression and semi-parametric local polynomial regression models. Through various numerical studies, we reached accurate numerical and closed-form continuously differentiable estimates for the functions appearing in the metric and equations of motion.

In many surveys, we often deal with situations where measuring the study variable is expensive; however, there are easy-to-measure characteristics which can be used as ranking information to obtain more representative samples from the population. Ranked set sampling is successfully employed in these cases as an alternative to commonly used simple random sampling. When the data is ordinal categorical, it is common to apply the ordinal logistic regression approach to ranked set sampling data for the estimation of parameters. This technique first depends on the information of training data. Besides, one is not capable of using the ranking information in the estimation process. In this paper, we propose a ranked set sampling scheme in which ranking information from multiple sources can be combined and incorporated efficiently into both data collection and estimation. The ranked set sampling data is used for non-parametric and maximum likelihood estimation of ordinal categorical population. Through extensive simulation studies, the performance of estimators is evaluated. The methods are finally applied to analyze bone disorder data and obesity data.