Bruce M. Hill

Posterior Distribution of Percentiles: Bayes' Theorem for Sampling From a Population

A Bayesian approach to inference about the percentiles and other characteristics of a finite population is proposed. The approach does not depend upon, though it need not exclude, the use of parametric models.Some related questions concerning the existence of exchangeable distributions are considered. It is shown that there are no countably additive exchangeable distributions on the space of observations which give ties probability 0 and for which a next observation is conditionally equally likely to fall in any of the open intervals between successive order statistics of a given sample.

The Three-Parameter Lognormal Distribution and Bayesian Analysis of a Point-Source Epidemic

Abstract Some unusual features of the likelihood function of the three-parameter lognormal distribution ln(t – γ)∼N(μ, σ2) are explored. In particular, it is shown that there exist paths along which the likelihood function of any sample t 1, …, tn tends to ∞ as (γ, μ, σ2) approaches (t (1), – ∞, + ∞), where t (1) is the smallest of the ti , and hence that in a meaningful sense this is the maximum-likelihood estimate. Estimation is then considered from a Bayesian point of view, and some natural posterior distributions are explored. A statistical model for a point-source epidemic is presented, and the theory developed is used in estimating the time of onset and other parameters.

Inference about Variance Components in the One-Way Model

Abstract The estimation of variance components in the one-way model is considered from a subjective Bayesian point of view. The situation in which the classical unbiased estimate of the between variance component is negative is explored in some detail. Exact and approximate posterior distributions are obtained in both the balanced and unbalanced case. Common sense aspects of the problem are emphasized, and some contrasts with other approaches. For example, Bayesianly speaking, a large negative unbiased estimate of the between variance component indicates an uninformative experiment in which the effective likelihood for that variance component is extremely flat, instead of strong evidence that the variance component is nearly zero.

The Rank-Frequency Form of Zipf's Law

Abstract Suppose that there are K regions in a country, with Ni people and Mi cities in the ith region. Let Ni be large, Mi random, given Ni , and such that the distribution of M i N i –1, given Ni , converges to a limiting distribution F, with F(x) ∼ Cx γ as x → 0, γ > 0. Let L (r) be the size of the rth largest city in the country. If, given Mi and Ni , there is a Bose-Einstein allocation of the Ni people to the Mi cities in region i, independently for the various regions, then a plot of L (r) against r will be approximately proportional to r –(1+α), for 1 + α = γ–1.

Zipf's Law and Prior Distributions for the Composition of a Population

Abstract The limiting distribution of frequencies of frequencies as the population size becomes large is obtained under Bose-Einstein and Maxwell-Boltzmann forms of the classical occupancy problem, when the number of cells is random and has a prior distribution. A rich variety of limiting distributions is obtained, including weak forms of Zipfs Law and the Fisher logarithmic series distribution. Lizards are compared to the weak form of Zipfs Law, and some speculations are made in regard to this law.

Information for Estimating the Proportions in Mixtures of Exponential and Normal Distributions

Abstract The Fisher information I(p; f 1, f 2) for estimating the proportion p in a mixture λ(x) = pf 1(x) +(1 − p)f 2(x) of two densities is investigated. A general power series expansion is obtained, which is then explored in detail for the case of two exponential densities, and for the case of two normal densities with equal scale. Simple approximations are obtained, for example when (μ1 − μ2/σ) is near zero in a mixture of two normal distributions with means μ1 and μ2 and common variance σ2, and when α/β is near unity in a mixture of two exponential distributions with mean lives (α)−1 and (β)−1, α < β. Brief tables based on the various approximations are presented, giving an overall picture of the information. The main qualitative conclusion is that extremely large, and often impractical, sample sizes are required to obtain even moderate precision in estimating p unless the mixed distributions are very well separated.

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.

Stronger Forms of Zipf's Law

Abstract It is shown that convergence in probability to Zipfs Law follows from a many family modification of a Bose-Einstein form of the classical occupancy model with a random number of cells.

Posterior Moments of the Number of Species in a Finite Population and the Posterior Probability of Finding a New Species

Abstract By using a model of Hill for sampling from a finite population, the posterior moments of the number of species in the population and the posterior probability of finding a new species are obtained. Results are based on a negative binomial prior distribution on the number of species in the population and are appropriate for sampling both with and without replacement.

Correlated Errors in the Random Model

Abstract The usual one-way random model for the analysis of variance is broadened to allow for negative correlation between true residuals in the same row or cluster. This gives rise to an essential unidentifiability of parameters. Nonetheless, the likelihood function is typically such that the joint posterior distribution of all parameters is quite informative and there is enormous evidence for negative correlation whenever the sum of squares within is sufficiently large. Indeed, as this sum of squares goes to infinity, the joint posterior distribution converges to a limiting distribution which is very nearly the same as that in which it is known a priori that residuals are negatively correlated in the most extreme way possible.