It is often useful to conduct inference for probability densities by constructing “plausible” sets in which the unknown density of given data may lie. Examples of such sets include pointwise intervals, simultaneous bands, or balls in a function space, and they may be frequentist or Bayesian in interpretation. For almost any density estimator, there are multiple approaches to inference available in the literature. Here we review such literature, providing a thorough overview of existing methods for density uncertainty quantification. The literature considered here comprises a spectrum from theoretical to practical ideas, and for some methods there is little commonality between these two extremes. After detailing some of the key concepts of nonparametric inference – the different types of “plausible” sets, and their interpretation and behaviour – we list the most prominent density estimators and the corresponding uncertainty quantification methods for each.

Our interest is whether two binomial parameters differ, which parameter is larger, and by how much. This apparently simple problem was addressed by Fisher in the 1930’s, and has been the subject of many review papers since then. Yet there continues to be new work on this issue and no consensus solution. Previous reviews have focused primarily on testing and the properties of validity and power, or primarily on confidence intervals, their coverage, and expected length. Here we evaluate both. For example, we consider whether a p-value and its matching confidence interval are compatible, meaning that the p-value rejects at level α if and only if the $1-\mathit{\alpha }$ confidence interval excludes all null parameter values. For focus, we only examine non-asymptotic inferences, so that most of the p-values and confidence intervals are valid (i.e., exact) by construction. Within this focus, we review different methods emphasizing many of the properties and interpretational aspects we desire from applied frequentist inference: validity, accuracy, good power, equivariance, compatibility, coherence, and parameterization and direction of effect. We show that no one method can meet all the desirable properties and give recommendations based on which properties are given more importance.

Scan statistics have been a very important and active area of statistical research in the past three decades. Detecting areas with a significant concentration of points is an important task in understanding the underlying phenomena in many fields such as: epidemiology, politics, crime analysis, zoology, etc. This study reviews how scan statistics have developed in the last three decades, the main concerns of researchers in scan statistics, and how researchers have approached these concerns.