Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion

Many different methods have been proposed to construct nonparametric estimates of a smooth regression function, including local polynomial, (convolution) kernel and smoothing spline estimators. Each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set. In this paper an improved version of a criterion based on the Akaike information criterion (AIC), termed AIC_C, is derived and examined as a way to choose the smoothing parameter. Unlike plug-in methods, AIC_C can be used to choose smoothing parameters for any linear smoother, including local quadratic and smoothing spline estimators. The use of AIC_C avoids the large variability and tendency to undersmooth (compared with the actual minimizer of average squared error) seen when other 'classical' approaches (such as generalized cross-validation (GCV) or the AIC) are used to choose the smoothing parameter. Monte Carlo simulations demonstrate that the AIC_C-based smoothing parameter is competitive with a plug-in method (assuming that one exists) when the plug-in method works well but also performs well when the plug-in approach fails or is unavailable.     

Estimation of the number of studies with positive trends when studies with negative trends are present

A two-point estimator is proposed for the proportion of studies with positive trends among a collection of studies, some of which may demonstrate negative trends. The proposed estimator is the y-intercept of the secant line joining the points (a, F?(a)) and (b, F?(b)), where F?(p) is the empirical distribution function of p-values from one-tailed tests for positive trend derived from the individual studies. Although this estimator is negatively biased for any choice of the points 0 ≤ a < b ≤ 1, the bias is less than that of the previously proposed one-point estimator defined by setting b = 1. The bias of the two-point estimator is smallest when a and b approach the inflection point of the true distribution function, E [F?(p)]. The utility of the two-point estimator is demonstrated by using it to estimate the number of male-mouse liver carcinogens among carcinogenicity studies conducted by the National Toxicology Program.     

A new class of measurement-error models,with applications to dietary data

Measurement-error modelling occurs when one cannot observe a covariate, but instead has possibly replicated surrogate versions of this covariate measured with error. The vast majority of the literature in measurement-error modelling assumes (typically with good reason) that given the value of the true but unobserved (latent) covariate, the replicated surrogates are unbiased for latent covariate and conditionally independent. In the area of nutritional epidemiology, there is some evidence from biomarker studies that this simple conditional independence model may break down due to two causes: (a) systematic biases depending on a person's body mass index, and (b) an additional random component of bias, so that the error structure is the same as a one-way random-effects model. We investigate this problem in the context of (1) estimating distribution of usual nutrient intake, (2) estimating the correlation between a nutrient instrument and usual nutrient intake, and (3) estimating the true relative risk from an estimated relative risk using the error-prone covariate. While systematic bias due to body mass index appears to have little effect, the additional random effect in the variance structure is shown to have a potentially important effect on overall results, both on corrections for relative risk estimates and in estimating the distribution of usual nutrient intake. However, the effect of dietary measurement error on both factors is shown via examples to depend strongly on the data set being used. Indeed, one of our data sets suggests that dietary measurement error may be masking a strong risk of fat on breast cancer, while for a second data set this masking is not so clear. Until further understanding of dietary measurement is available, measurement-error corrections must be done on a study-specific basis, sensitivity analyses should be conducted, and even then results of nutritional epidemiology studies relating diet to disease risk should be interpreted cautiously.     

OPTIMAL PUNISHMENT SCHEMES WITH STATE-DEPENDENT PREFERENCES

The optimal tradeoff between the certainty and severity of punishment is studied in a setting in which individuals are more sensitive to changes in certainty than severity and have state-dependent preferences—being convicted of a crime reduces utility at every wealth level. In this setting it is not optimal to set the probability of punishment at one even if it is costless to convict criminals, because reducing the certainty of punishment also reduces the probability that offenders' utility functions shift downward. The optimal level of certainty is therefore lower than that implied by prior studies based on state-independent preferences. (JEL K42, D81)     

WORK‐RELATED ACCIDENTS AND THE LEVEL OF MARKET COMPETITION: AN ANALYSIS OF WORKER INJURY RATES AT U.S. STEEL CORPORATION, 1907–1939

We investigate the relationship between accident rates and industry structure for the United States’ steel industry during the first four decades of the 20th century. We develop a dominant firm theoretical model linking accident rates to number of competitors, showing a positive correlation between accident avoidance and the number of fringe competitors. We then test this theory empirically and, when controlling for other influences, find that reductions in the dominant firm’s market share reduce worker injury rates substantially. (JEL L13, L61, N62, N82)     

A class of local likelihood methods and near-parametric asymptotics

The local maximum likelihood estimate θ^ _t of a parameter in a statistical model f ( x , θ) is defined by maximizing a weighted version of the likelihood function which gives more weight to observations in the neighbourhood of t . The paper studies the sense in which f ( t , θ^ _t ) is closer to the true distribution g ( t ) than the usual estimate f ( t , θ^) is. Asymptotic results are presented for the case in which the model misspecification becomes vanishingly small as the sample size tends to ∞. In this setting, the relative entropy risk of the local method is better than that of maximum likelihood. The form of optimum weights for the local likelihood is obtained and illustrated for the normal distribution.     

Bayesian analysis of human movement curves

Summary.  We consider the Bayesian analysis of human movement data, where the subjects perform various reaching tasks. A set of markers is placed on each subject and a system of cameras records the three-dimensional Cartesian co-ordinates of the markers during the reaching movement. It is of interest to describe the mean and variability of the curves that are traced by the markers during one reaching movement, and to identify any differences due to covariates. We propose a methodology based on a hierarchical Bayesian model for the curves. An important part of the method is to obtain identifiable features of the movement so that different curves can be compared after temporal warping. We consider four landmarks and a set of equally spaced pseudolandmarks are located in between. We demonstrate that the algorithm works well in locating the landmarks, and shape analysis techniques are used to describe the posterior distribution of the mean curve. A feature of this type of data is that some parts of the movement data may be missing—the Bayesian methodology is easily adapted to cope with this situation.