Thomas E. Wehrly

Kernel Regression Estimation Using Repeated Measurements Data

Abstract The estimation of growth curves has been studied extensively in parametric situations. Here we consider the nonparametric estimation of an average growth curve. Suppose that there are observations from several experimental units, each following the regression model y(xi)=f(xj)+e(j=1,…,n), where e1, …, e n are correlated zero mean errors and 0≤x1<…<xn≤1 are fixed constants. We study some of the properties of a kernel estimator of f(x). Asymptotic and finite-sample results concerning the mean squared error of the estimator are obtained. In particular, the influence of correlation on the bandwidth minimizing mean squared error is discussed. A data-based method for selecting the bandwidth is illustrated in a data analysis. Most previous research on kernel regression estimators has involved uncorrelated errors. We investigate how dependence of the errors changes the behavior of a kernel estimator. Our theorems concerning the asymptotic mean squared error show that the estimator cannot be consistent un...

Some Angular-Linear Distributions and Related Regression Models

Abstract Parametric models are proposed for the joint distribution of bivariate random variables when one variable is directional and one is scalar. These distributions are developed on the basis of the maximum entropy principle and by the specification of the marginal distributions. The properties of these distributions and the statistical analysis of regression models based on these distributions are explored. One model is extended to several variables in a form that justifies the use of least squares for estimation of parameters, conditional on the observed angles.

Testing the equality of two regression curves using linear smoothers

Suppose that data (y, z) are observed from two regression models, y = f(x) + [var epsilon] and z = g(x) + [eta]. Of interest is testing the hypothesis H: f [triple bond; length as m-dash] g without assuming that f or g is in a parametric family. A test based on the difference between linear, but nonparametric, estimates of f and g is proposed. The exact distribution of the test statistic is obtained on the assumption that the errors in the two regression models are normally distributed. Asymptotic distribution theory is outlined under more general conditions on the errors. It is shown by simulation that the test based on the assumption of normal errors is reasonably robust to departures from normality. A data analysis illustrates that, in addition to being attractive descriptive devices, nonparametric smoothers can be valuable inference tools.

A geometrical method for removing edge effects from kernel-type nonparametric regression estimators

Abstract We introduce a simple geometric method for removing edge effects from kernel-type nonparametric regression estimators. It involves reflecting the data set in two estimated points, thereby generating a new data set with three times the range of the original data. The usual kernel-type estimator may be applied to the new, enlarged data set, without any danger of edge effects. This technique is applicable generally to both regularly spaced and randomly spaced designs and admits a natural analog of leave-one-out cross-validation. The new cross-validation algorithm may be extended to the very ends of the design interval, unlike its more conventional counterpart, which must be downweighted at the ends of the interval to avoid edge effects.

Bias robust estimation in finite populations using nonparametric calibration

A Standard problem in sample survey inference is that of predicting the finite population total H of a function h(y) of a random variable Y. The model-based approach to this problem first defines a working model ξ for Y and then predicts H by estimating its expectation under ξ, conditional on the sample values of Y. This approach leads to biased predictions if ξ is incorrect. We explore an automatic solution to this misspecification bias that uses nonparametric regression to define a robust (but inefficient) predictor of H, and then calibrates this predictor for its bias under ξ. An application to prediction of the finite population distribution function of a population of Australian beef farms is presented.

On some stochastic formulations and related statistical moments of pharmacokinetic models

This paper presents the deterministic and stochastic model for a linear compartment system with constant coefficients, and it develops expressions for the mean residence times (MRT) and the variances of the residence times (VRT) for the stochastic model. The expressions are relatively simple computationally, involving primarily matrix inversion, and they are elegant mathematically, in avoiding eigenvalue analysis and the complex domain. The MRT and VRT provide a set of new meaningful response measures for pharmacokinetic analysis and they give added insight into the system kinetics. The new analysis is illustrated with an example involving the cholesterol turnover in rats.

Kernel Regression When the Boundary Region is Large, with an Application to Testing the Adequacy of Polynomial Models

Abstract It is well known that kernel regression estimators are subject to so-called boundary or edge effects, a phenomenon in which the bias of an estimator increases near the endpoints of the estimation interval. When the regression curve is linear or nearly linear, the requisite amount of smoothing is so great that the boundary region is effectively the entire estimation interval. Special boundary kernels are proposed here to deal with such cases. It is shown that the proposed kernel estimator has a property also enjoyed by cubic smoothing splines; namely, as the estimators smoothing parameter becomes large, the estimator tends to a straight line. The limiting straight line is essentially the least squares line when the design points are equally spaced. A simple generalization of ideas in the linear case leads to kernel estimates that are polynomials of any given degree for large bandwidths. Such estimates are an important component of a proposed test for the adequacy of a polynomial model. The test s...

Consistency of cross-validation when the data are curves

Suppose one observes a random sample of n continuous time Gaussian processes on the interval [0, 1]; in other words, each observation is a curve. Of interest is estimating the common mean function of the processes by a kernel smoother. The bandwidth of the kernel estimator is chosen by a version of cross-validation in which deleting an observation means deleting one of the n curves. It is shown that using this form of cross-validation leads to an asymptotically optimal choice of bandwidth. This result is contrasted with the inconsistency of cross-validation in a seemingly more tractable problem.

Generalized stochastic compartmental models with Erlang transit times.

This paper considers the use of semi-Markov process models with Erlang transit times for the description of compartmental systems. The semi-Markov models seem particularly useful for systems with nonhomogeneous “poorly-stirred” compartments. The paper reviews the Markov process models with exponential transit times, and illustrates the application of such models, describing the clearance of calcium in man. The semi-Markov model with Erlang transit times is then developed, and the solutions for its concentration-time curves and residence time moments are given. The use of semi-Markov models is illustrated with the same calcium data, and the results from the two models are compared. The example demonstrates that these semi-Markov models are physiologically more realistic than standard models and may befitted to pharmacokinetic data using readily available software.

Cluster analysis of MCMI scores of suicidal psychiatric patients: Four personality profiles.

Millon Clinical Multiaxial Inventory (MCMI) scores from 299 suicidal psychiatric outpatients were cluster analyzed in hopes of identifying clinical subgroups and deriving specific treatment indications. The analysis revealed four personality profiles: Negativistic/Avoidant/Schizoid, Avoidant/Dependent/ Negativistic, Antisocial, and Histrionic/Narcissistic. This grouping was validated by examining demographics, diagnoses, and scores on several psychometric scales. Results showed few differences in demographics, diagnosis, or intelligence, but significant differences in depression severity, hopelessness, anxiety, problem-solving self-appraisal, and alcohol abuse. Implications for tailoring interventions for subtypes of suicidal patients are discussed.