The mathematics of Benford’s law: a primer

This article provides a concise overview of the main mathematical theory of Benford’s law in a form accessible to scientists and students who have had first courses in calculus and probability. In particular, one of the main objectives here is to aid researchers who are interested in applying Benford’s law, and need to understand general principles clarifying when to expect the appearance of Benford’s law in real-life data and when not to expect it. A second main target audience is students of statistics or mathematics, at all levels, who are curious about the mathematics underlying this surprising and robust phenomenon, and may wish to delve more deeply into the subject. This survey of the fundamental principles behind Benford’s law includes many basic examples and theorems, but does not include the proofs or the most general statements of the theorems; rather it provides precise references where both may be found.

     

The Family Unpredictability Scale: Reliability and Validity

We developed the Family Unpredictability Scale (FUS) to better understand the factors that underlie the presence and consequences of family dysfunction. Familial unpredictability is defined as a lack of consistency in family behaviors and regulatory systems. Family researchers and therapists generated and evaluated items, which were self‐administered by married parents who had children aged 2 to 18 (N= 314 ). Principal‐components analysis yielded four subscales (discipline, nurturance, meals, and money; 22 items total) that were internally consistent; confirmatory factor analysis based on a separate subsample yielded an adequate preliminary representation of this factor structure. The FUS related significantly to known measures of family and individual functioning but not to social desirability. The present study provides preliminary reliability, construct validity, and concurrent validity for this new multidimensional, self‐report assessment of family unpredictability. Research and clinical implications are discussed.     

Short- and long-term effects of divorce and separation on housing tenure in England and Wales

This paper investigates the effects of marital and non-marital separation on individuals’ housing tenure in England and Wales. We apply competing risks event history models to data from the British Household Panel Survey and the UK Household Longitudinal Study to analyse the risk of a residential move to different tenure types, for single, married, cohabiting, and separated men and women. Separated individuals are more likely to move and experience a tenure change than those who are single or in a relationship. Among separated people, private renting is the most common outcome of a move; however, women are also likely to move to social renting, whereas men tend to move to homeownership. This pattern persists when we account for time since separation and order of move, indicating a potential long-term effect of separation on housing tenure. This long-term effect is especially pertinent to separated women who cannot afford homeownership.     

APPROPRIABLE RENTS FROM YELLOWSTONE PARK: A CASE OF INCOMPLETE CONTRACTING

The history of Yellowstone Park is best understood in the context of the theory of the firm as developed by Armen Alchian. In the late nineteenth century, the Northern Pacific Railroad recognized the potential amenity rents from Yellowstone and captured them by vertically integrating its monopoly on transportation to the region with the supply of tourist facilities in the park. This gave the railroad de facto ownership of the park that lasted until automobiles were allowed to enter in 1915. Thereafter the tragedy of the commons ensued and rents were dissipated instead of maximized.     

Infant‐Directed Speech Facilitates Word Segmentation

There are reasons to believe that infant‐directed (ID) speech may make language acquisition easier for infants. However, the effects of ID speech on infants' learning remain poorly understood. The experiments reported here assess whether ID speech facilitates word segmentation from fluent speech. One group of infants heard a set of nonsense sentences spoken with intonation contours characteristic of adult‐directed (AD) speech, and the other group heard the same sentences spoken with intonation contours characteristic of ID speech. In both cases, the only cue to word boundaries was the statistical structure of the speech. Infants were able to distinguish words from syllable sequences spanning word boundaries after exposure to ID speech but not after hearing AD speech. These results suggest that ID speech facilitates word segmentation and may be useful for other aspects of language acquisition as well. Issues of direction of preference in preferential listening paradigms are also considered.     

Some subjective bayesian considerations in the selection of models

Some considerations relating to the post-data selection of models are discussed. These include some difficulties with orthodox theory, implementation of the likelihood principle, and Bayesian tests of hypotheses.