Density estimation in the uniform deconvolution model

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data.     

The future of the motorcar: Alternative scenarios

In contrast to conventional forecasts usually based on constant cost projections, the author considers the future of the motorcar as depending on dynamic changes in the pattern of costs and values related to private car ownership and use. The social, environmental, material and technological factors most influencing the future cost of motoring are analysed, and desirable policies are examined. The results are projected into two scenarios, high and low, of the future conditions for private motoring. The analysis is focused on the Netherlands, but is generally applicable to other Western European countries.     

The law of constant travel time and trip-rates

A widely held belief in transport planning is that there is an automatically expanding mobility, based particularly on the increasing use of private motor car use. But the ‘law’ of constant travel time rates states that the average number of daily trips per person and the time budget allocated to transport show stability. This law has raised a number of objections, notably that there has been no satisfactory explanation of it offered—it is merely an incomprehensible black box. This paper aims to rectify that.     

Exponential deconvolution: two asymptotically equivalent estimators

Two isotonic estimators for the distribution function in a specific deconvolution model, the exponential deconvolution model, are considered. The first estimator is a least squares projection of a naive estimator for the distribution function on the set of distribution functions. The second estimator is the well known maximum likelihood estimator. The two estimators are shown to be first order asymptotically equivalent at a fixed point.     

Consistent likelihood-based estimation of a star-shaped distribution

We consider the classical problem of nonparametrically estimating a star-shaped distribution, i.e., a distribution function F on [0,∞) with the property that F(u)/u is nondecreasing on the set {u : F(u) < 1}. This problem is intriguing because of the fact that a well defined maximum likelihood estimator (MLE) exists, but this MLE is inconsistent. In this paper, we argue that the likelihood that is commonly used in this context is somewhat unnatural and propose another, so called ‘smoothed likelihood’. However, also the resulting MLE turns out to be inconsistent. We show that more serious smoothing of the likelihood yields consistent estimators in this model.