The kernel estimate of a probability density function inherits its smoothness properties from the kernel density chosen by the investigator. Nevertheless, for computational (and especially graphical) ...reasons, exact kernel density estimates are often presented in (piecewise constant) discretized or (piecewise linear) interpolated form, using the exact estimate only at some grid of points. The asymptotic integrated mean squared error properties of these modifications are studied. It is seen that discretization may adversely affect the order of magnitude of this risk criterion, so care needs to be taken in practice to limit the degree of discretization employed. This roughness effect essentially disappears when interpolation is used instead; then, it takes a remarkably coarse grid to result in more than a negligible deterioration in the estimate's performance. Data discretization prior to applying the kernel density estimation prescription is also addressed. Such prebinning does not affect the smoothness of the resulting estimate and has a reassuringly minor effect on overall performance, both in conjunction with postbinning schemes and without.
This article attempts to illustrate the utility of isoquant map analysis from the field of production theory in microeconomics for the analysis of criminal justice data. Cross-national comparisons of ...aggregate crime and justice data are used to demonstrate the ability of this technique to reveal important patterns that are often obscured by simple rate comparisons and multivariate treatments such as pooled time-series analysis. For each jurisdiction, aggregate trends in criminal justice processing rates are systematically analyzed as a sequence of two-input production processes: gross imprisonment rates (prison population divided by resident population) can be partitioned in terms of the crime rate and punitiveness (prison population divided by the number of offenses); punitiveness can, in turn, be partitioned in terms of severity and certainty of punishment; certainty of punishment can then be partitioned, seriatim, in terms of the incarceration rate, the conviction rate, and the arrest or clearance rate and the rate at which citizens report crimes. Cross-national data collected by Farrington, Langan, and Wikström are used to illustrate the utility of the method for displaying comparisons of the decomposition of aggregate criminal justice data for the United States, England, and Sweden.
