Adaptive modelling of dose–response relationships using smoothing splines

We consider the use of smoothing splines for the adaptive modelling of dose–response relationships. A smoothing spline is a nonparametric estimator of a function that is a compromise between the fit to the data and the degree of smoothness and thus provides a flexible way of modelling dose–response data. In conjunction with decision rules for which doses to continue with after an interim analysis, it can be used to give an adaptive way of modelling the relationship between dose and response. We fit smoothing splines using the generalized cross‐validation criterion for deciding on the degree of smoothness and we use estimated bootstrap percentiles of the predicted values for each dose to decide upon which doses to continue with after an interim analysis. We compare this approach with a corresponding adaptive analysis of variance approach based upon new simulations of the scenarios previously used by the PhRMA Working Group on Adaptive Dose‐Ranging Studies. The results obtained for the adaptive modelling of dose–response data using smoothing splines are mostly comparable with those previously obtained by the PhRMA Working Group for the Bayesian Normal Dynamic Linear model (GADA) procedure. These methods may be useful for carrying out adaptations, detecting dose–response relationships and identifying clinically relevant doses. Copyright © 2009 John Wiley & Sons, Ltd.     

基于赔付期望的财产保险业产出模型研究

本文从《中国保险年鉴》和中国保监局广东省监管局网站获取广东省（不含深圳）财产保险业统计数据,应用移动平均值法和适应性预期法对赔付率进行平滑;鉴于赔付率数据能够较好拟合差分自回归移动平均模型,采用小波降噪去除白噪声的方法,平滑赔付率数据。三种方法中,小波降噪平滑后的数据与原数据误差最小,平滑效果最好。三种平滑方法均能够抵消巨额未决赔款拉低总产出的影响,较好地平滑了财产保险业的总产出。     

Statistical inference on partial linear additive models with distortion measurement errors

We consider statistical inference for partial linear additive models (PLAMs) when the linear covariates are measured with errors and distorted by unknown functions of commonly observable confounding variables. A semiparametric profile least squares estimation procedure is proposed to estimate unknown parameter under unrestricted and restricted conditions. Asymptotic properties for the estimators are established. To test a hypothesis on the parametric components, a test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we further show that its limiting distribution is a weighted sum of independent standard chi-squared distributions. A bootstrap procedure is further proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.     

Adaptive likelihood estimator of conditional variance function

Modelling volatility in the form of conditional variance function has been a popular method mainly due to its application in financial risk management. Among others, we distinguish the parametric GARCH models and the nonparametric local polynomial approximation using weighted least squares or gaussian likelihood function. We introduce an alternative likelihood estimate of conditional variance and we show that substitution of the error density with its estimate yields similar asymptotic properties, that is, the proposed estimate is adaptive to the error distribution. Theoretical comparison with existing estimates reveals substantial gains in efficiency, especially if error distribution has fatter tails than Gaussian distribution. Simulated data confirm the theoretical findings while an empirical example demonstrates the gains of the proposed estimate.     

Accurate and efficient construction of bootstrap likelihoods

The simplest construction of bootstrap likelihoods involves two levels of bootstrapping, kernel density estimation, and non-parametric curve-smoothing. We describe more accurate and efficient constructions, based on smoothing at the first level of nested bootstraps and saddlepoint approximation to remove second-level bootstrap variation. Detailed illustrations are given.     

Variation in height acceleration in the Fels growth data

The Fels growth data record at half-yearly intervals the heights of children from birth to adulthood, and are the basis for pediatricians' growth charts used throughout North America. Aspects of human growth are the subject of a large medical and statistical literature. This paper uses smoothing splines to study the variation in height acceleration. By use of a functional version of principal-components analysis, we find that variation in the acceleration curve is essentially three-dimensional in nature. Evidence for a small growth spurt between the ages of six and eight, reported for data collected in Switzerland, is examined, and little support is found for the existence of this phenomenon in the Fels data.     

RATES OF CONVERGENCE IN SEMI-PARAMETRIC MODELLING OF LONGITUDINAL DATA

We consider the problem of semi-parametric regression modelling when the data consist of a collection of short time series for which measurements within series are correlated. The objective is to estimate a regression function of the form E[Y(t) | x] =x'ß+μ(t), where μ(.) is an arbitrary, smooth function of time t, and x is a vector of explanatory variables which may or may not vary with t. For the non-parametric part of the estimation we use a kernel estimator with fixed bandwidth h. When h is chosen without reference to the data we give exact expressions for the bias and variance of the estimators for β and μ(t) and an asymptotic analysis of the case in which the number of series tends to infinity whilst the number of measurements per series is held fixed. We also report the results of a small-scale simulation study to indicate the extent to which the theoretical results continue to hold when h is chosen by a data-based cross-validation method.     

Hierarchical bayesian curve fitting and smoothing

Estimation of a smooth function is considered when observations on this function added with Gaussian errors are observed. The problem is formulated as a general linear model, and a hierarchical Bayesian approach is then used to study it. Credible bands are also developed for the function. Sensitivity analysis is conducted to determine the influence of the choice of priors on hyperparameters. Finally, the methodology is illustrated using real and simulated examples where it is compared with classical cubic splines. It is also shown that our approach provides a Bayesian solution to some problems in discrete time series.     

On Smoothing Sparse Multinomial Data

Asymptotic theory is developed for the problem of smoothing sparse multinomial data, with emphasis on the criterion of mean summed square error of estimators of the probability mass function. If the data are not too sparse, in a well-defined sense, then the optimal rate of convergence is that achieved by the unsmoothed cell proportions. Otherwise, this rate can be improved upon by smoothing. Explicit results, including formulae for the optimal smoothing parameter, are presented for a kernel-type estimator. Also for this case, a cross-validatory choice procedure is shown to be asymptotically optimal.     

Local Influence Analysis for Penalized Gaussian Likelihood Estimators in Partially Linear Models

Abstract. Partially linear models are extensions of linear models to include a non-parametric function of some covariate. They have been found to be useful in both cross-sectional and longitudinal studies. This paper provides a convenient means to extend Cook's local influence analysis to the penalized Gaussian likelihood estimator that uses a smoothing spline as a solution to its non-parametric component. Insight is also provided into the interplay of the influence or leverage measures between the linear and the non-parametric components in the model. The diagnostics are applied to a mouthwash data set and a longitudinal hormone study with informative results.