A finite set is “hidden” if its elements are not directly enumerable or if its size cannot be ascertained via a deterministic query. In public health, epidemiology, demography, ecology and intelligence analysis, researchers have developed a wide variety of indirect statistical approaches, under different models for sampling and observation, for estimating the size of a hidden set. Some methods make use of random sampling with known or estimable sampling probabilities, and others make structural assumptions about relationships (e.g. ordering or network information) between the elements that comprise the hidden set. In this review, we describe models and methods for learning about the size of a hidden finite set, with special attention to asymptotic properties of estimators. We study the properties of these methods under two asymptotic regimes, “infill” in which the number of fixed-size samples increases, but the population size remains constant, and “outfill” in which the sample size and population size grow together. Statistical properties under these two regimes can be dramatically different.

We present and compare some nonparametric estimation methods (wavelet and/or spline-based) designed to recover a one-dimensional piecewise-smooth regression function in both a fixed equidistant or not equidistant design regression model and a random design model.

Wavelet methods are known to be very competitive in terms of denoising and compression, due to the simultaneous localization property of a function in time and frequency. However, boundary assumptions, such as periodicity or symmetry, generate bias and artificial wiggles which degrade overall accuracy.

Simple methods have been proposed in the literature for reducing the bias at the boundaries. We introduce new ones based on adaptive combinations of two estimators. The underlying idea is to combine a highly accurate method for non-regular functions, e.g., wavelets, with one well behaved at boundaries, e.g., Splines or Local Polynomial. We provide some asymptotic optimal results supporting our approach. All the methods can handle data with a random design. We also sketch some generalization to the multidimensional setting.

To study the performance of the proposed approaches we have conducted an extensive set of simulations on synthetic data. An interesting regression analysis of two real data applications using these procedures unambiguously demonstrates their effectiveness.

Besides the inferential errors that abound in the interpretation of $p$-values, the probabilistic pre-conditions (i.e. random sampling or equivalent) for using them at all are not often met by observational studies in the social sciences. This paper systematizes different sampling designs and discusses the restrictive requirements of data collection that are the indispensable prerequisite for using $p$-values.