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Mean Function (mean + function)

Distribution by Scientific Domains
Mathematics and Statistics75%
Business, Economics, Finance and Accounting16%

Selected Abstracts

Estimation and Inference for a Spline-Enhanced Population Pharmacokinetic Model

BIOMETRICS, Issue 3 2002
Lang Li
Summary. This article is motivated by an application where subjects were dosed three times with the same drug and the drug concentration profiles appeared to be the lowest after the third dose. One possible explanation is that the pharmacokinetic (PK) parameters vary over time. Therefore, we consider population PK models with time-varying PK parameters. These time-varying PK parameters are modeled by natural cubic spline functions in the ordinary differential equations. Mean parameters, variance components, and smoothing parameters are jointly estimated by maximizing the double penalized log likelihood. Mean functions and their derivatives are obtained by the numerical solution of ordinary differential equations. The interpretation of PK parameters in the model and its flexibility are discussed. The proposed methods are illustrated by application to the data that motivated this article. The model's performance is evaluated through simulation. [source]

An Adaptive, Rate-Optimal Test of a Parametric Mean-Regression Model Against a Nonparametric Alternative

ECONOMETRICA, Issue 3 2001
Joel L. Horowitz
We develop a new test of a parametric model of a conditional mean function against a nonparametric alternative. The test adapts to the unknown smoothness of the alternative model and is uniformly consistent against alternatives whose distance from the parametric model converges to zero at the fastest possible rate. This rate is slower than n,1/2. Some existing tests have nontrivial power against restricted classes of alternatives whose distance from the parametric model decreases at the rate n,1/2. There are, however, sequences of alternatives against which these tests are inconsistent and ours is consistent. As a consequence, there are alternative models for which the finite-sample power of our test greatly exceeds that of existing tests. This conclusion is illustrated by the results of some Monte Carlo experiments. [source]

Space,time modeling of 20 years of daily air temperature in the Chicago metropolitan region

ENVIRONMETRICS, Issue 5 2009
Hae-Kyung Im
Abstract We analyze 20 years of daily minimum and maximum air temperature data in the Chicago metropolitan region and propose a parsimonious model that describes their mean function and the space,time covariance structure. The mean function contains a long-term trend, annual and semiannual harmonics, and physical covariates such as latitude, distance to the Lake Michigan, and winds, each interacted with the harmonic terms, thus allowing the effects of physical covariates to vary smoothly over time. The temporal correlation at a given location is described using an ARMA(1,2) model. The residuals (innovations) from this models are treated as independent replications of a spatial process with covariance structure in the Matérn class. The space,time covariance structure parameters are allowed to vary seasonally. Using the estimated covariance structure, we interpolate the temperature to a fine grid in the Chicago metropolitan region. This procedure borrows information from temporally and spatially adjacent data. The methods presented in this paper should be useful to approach other environmental problems where the data are discrete and regular in time but irregular in space. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Identification of continuous-time nonlinear systems by using a gaussian process model

IEEJ TRANSACTIONS ON ELECTRICAL AND ELECTRONIC ENGINEERING, Issue 6 2008
Tomohiro Hachino Member
Abstract This paper deals with a nonparametric identification of continuous-time nonlinear systems by using a Gaussian process model. Genetic algorithm is applied to train the Gaussian process prior model by minimizing the negative log marginal likelihood of the identification data. The nonlinear term of the objective system is estimated as the predictive mean function of the Gaussian process, and the confidence measure of the estimated nonlinear function is given by the predictive covariance function of the Gaussian process. Copyright © 2008 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [source]

Detecting changes in the mean of functional observations

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 5 2009
István Berkes
Summary., Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as Xi(t)=,(t)+,1,l<,,i, l+ vl(t), where , is the common mean, vl are the eigenfunctions of the covariance operator and the ,i, l are the scores. Inferential procedures assume that the mean function ,(t) is the same for all values of i. If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of principal component analysis are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from England. [source]

Penalized spline models for functional principal component analysis

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 1 2006
Fang Yao
Summary., We propose an iterative estimation procedure for performing functional principal component analysis. The procedure aims at functional or longitudinal data where the repeated measurements from the same subject are correlated. An increasingly popular smoothing approach, penalized spline regression, is used to represent the mean function. This allows straightforward incorporation of covariates and simple implementation of approximate inference procedures for coefficients. For the handling of the within-subject correlation, we develop an iterative procedure which reduces the dependence between the repeated measurements that are made for the same subject. The resulting data after iteration are theoretically shown to be asymptotically equivalent (in probability) to a set of independent data. This suggests that the general theory of penalized spline regression that has been developed for independent data can also be applied to functional data. The effectiveness of the proposed procedure is demonstrated via a simulation study and an application to yeast cell cycle gene expression data. [source]

Dynamic models for spatiotemporal data

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 4 2001
Jonathan R. Stroud
We propose a model for non-stationary spatiotemporal data. To account for spatial variability, we model the mean function at each time period as a locally weighted mixture of linear regressions. To incorporate temporal variation, we allow the regression coefficients to change through time. The model is cast in a Gaussian state space framework, which allows us to include temporal components such as trends, seasonal effects and autoregressions, and permits a fast implementation and full probabilistic inference for the parameters, interpolations and forecasts. To illustrate the model, we apply it to two large environmental data sets: tropical rainfall levels and Atlantic Ocean temperatures. [source]

Semiparametric inference on a class of Wiener processes

JOURNAL OF TIME SERIES ANALYSIS, Issue 2 2009
Xiao Wang
Abstract., This article studies the estimation of a nonhomogeneous Wiener process model for degradation data. A pseudo-likelihood method is proposed to estimate the unknown parameters. An attractive algorithm is established to compute the estimator under this pseudo-likelihood formulation. We establish the asymptotic properties of the estimator, including consistency, convergence rate and asymptotic distribution. Random effects can be incorporated into the model to represent the heterogeneity of degradation paths by letting the mean function be random. The Wiener process model is extended naturally to a normal inverse Gaussian process model and similar pseudo-likelihood inference is developed. A score test is used to test the presence of the random effects. Simulation studies are conducted to validate the method and we apply our method to a real data set in the area of health structure monitoring. [source]

Smoothness adaptive average derivative estimation

THE ECONOMETRICS JOURNAL, Issue 1 2010
Marcia M. A. Schafgans
Summary, Many important models utilize estimation of average derivatives of the conditional mean function. Asymptotic results in the literature on density weighted average derivative estimators (ADE) focus on convergence at parametric rates; this requires making stringent assumptions on smoothness of the underlying density; here we derive asymptotic properties under relaxed smoothness assumptions. We adapt to the unknown smoothness in the model by consistently estimating the optimal bandwidth rate and using linear combinations of ADE estimators for different kernels and bandwidths. Linear combinations of estimators (i) can have smaller asymptotic mean squared error (AMSE) than an estimator with an optimal bandwidth and (ii) when based on estimated optimal rate bandwidth can adapt to unknown smoothness and achieve rate optimality. Our combined estimator minimizes the trace of estimated MSE of linear combinations. Monte Carlo results for ADE confirm good performance of the combined estimator. [source]

ERROR VARIANCE ESTIMATION FOR THE SINGLE-INDEX MODEL

AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 2 2010
K. B. Kulasekera
Summary Single-index models provide one way of reducing the dimension in regression analysis. The statistical literature has focused mainly on estimating the index coefficients, the mean function, and their asymptotic properties. For accurate statistical inference it is equally important to estimate the error variance of these models. We examine two estimators of the error variance in a single-index model and compare them with a few competing estimators with respect to their corresponding asymptotic properties. Using a simulation study, we evaluate the finite-sample performance of our estimators against their competitors. [source]

A GRAPHICAL DIAGNOSTIC FOR VARIANCE FUNCTIONS

AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 3 2007
Iain Pardoe
Summary This paper proposes diagnostic plots for regression variance functions. It shows how to extend graphical methodology that uses Bayesian sampling for checking the regression mean function to also check the variance function. Plots can be constructed quickly and easily for any model of interest. These plots help to identify model weaknesses and can suggest ways to make improvements. The proposed methodology is illustrated with two examples: a simple linear regression model to fix ideas, and a more complex study involving count data to demonstrate the potential for wide application. [source]

Assessment of Agreement under Nonstandard Conditions Using Regression Models for Mean and Variance

BIOMETRICS, Issue 1 2006
Pankaj K. Choudhary
Summary The total deviation index of Lin (2000, Statistics in Medicine19, 255,270) and Lin et al. (2002, Journal of the American Statistical Association97, 257,270) is an intuitive approach for the assessment of agreement between two methods of measurement. It assumes that the differences of the paired measurements are a random sample from a normal distribution and works essentially by constructing a probability content tolerance interval for this distribution. We generalize this approach to the case when differences may not have identical distributions,a common scenario in applications. In particular, we use the regression approach to model the mean and the variance of differences as functions of observed values of the average of the paired measurements, and describe two methods based on asymptotic theory of maximum likelihood estimators for constructing a simultaneous probability content tolerance band. The first method uses bootstrap to approximate the critical point and the second method is an analytical approximation. Simulation shows that the first method works well for sample sizes as small as 30 and the second method is preferable for large sample sizes. We also extend the methodology for the case when the mean function is modeled using penalized splines via a mixed model representation. Two real data applications are presented. [source]