Optimal block designs comparing treatments with a control when the errors are correlated

The efficient design of experiments for comparing a control with v new treatments when the data are dependent is investigated. We concentrate on generalized least-squares estimation for a known covariance structure. We consider block sizes k equal to 3 or 4 and approximate designs. This method may lead to exact optimal designs for some v, b, k, but usually will only indicate the structure of an efficient design for any particular v, b, k, and yield an efficiency bound, usually unattainable. The bound and the structure can then be used to investigate efficient finite designs.     

Combining estimating functions for volatility

Accurate estimates of volatility are needed in risk management. Generalized autoregressive conditional heteroscedastic (GARCH) models and random coefficient autoregressive (RCA) models have been used for volatility modelling. Following Heyde [1997. Quasi-likelihood and its Applications. Springer, New York], volatility estimates are obtained by combining two different estimating functions. It turns out that the combined estimating function for the parameter in autoregressive processes with GARCH errors and RCA models contains maximum information. The combination of the least squares (LS) estimating function and the least absolute deviation (LAD) estimating function with application to GARCH model error identification is discussed as an application.     

Estimating nonlinear additive models with nonstationarities and correlated errors

In this paper, we study a nonparametric additive regression model suitable for a wide range of time series applications. Our model includes a periodic component, a deterministic time trend, various component functions of stochastic explanatory variables, and an AR(p) error process that accounts for serial correlation in the regression error. We propose an estimation procedure for the nonparametric component functions and the parameters of the error process based on smooth backfitting and quasimaximum likelihood methods. Our theory establishes convergence rates and the asymptotic normality of our estimators. Moreover, we are able to derive an oracle‐type result for the estimators of the AR parameters: Under fairly mild conditions, the limiting distribution of our parameter estimators is the same as when the nonparametric component functions are known. Finally, we illustrate our estimation procedure by applying it to a sample of climate and ozone data collected on the Antarctic Peninsula.     

Quickly variable selection for varying coefficient models with missing response at random

This paper focuses on the variable selections for a varying coefficient models with missing response at random. A procedure is presented by basis function approximations with smooth-threshold estimating equations. Furthermore, the proposed method selects significant variables and estimates coefficients simultaneously avoiding the problem of solving a convex optimization, which reduced the burden of computation. Compared to existing equation based approaches, our procedure is more efficient and quick. With proper choices the regularization parameter, the resulting estimates perform an oracle property. A cross-validation for tuning parameter selection is also proposed, a numerical study confirms the performance of the proposed method.     

Optimum percentile estimating equations for nonlinear random coefficient models

In nonlinear random coefficients models, the means or variances of response variables may not exist. In such cases, commonly used estimation procedures, e.g., (extended) least-squares (LS) and quasi-likelihood methods, are not applicable. This article solves this problem by proposing an estimate based on percentile estimating equations (PEE). This method does not require full distribution assumptions and leads to efficient estimates within the class of unbiased estimating equations. By minimizing the asymptotic variance of the PEE estimates, the optimum percentile estimating equations (OPEE) are derived. Several examples including Weibull regression show the flexibility of the PEE estimates. Under certain regularity conditions, the PEE estimates are shown to be strongly consistent and asymptotic normal, and the OPEE estimates have the minimal asymptotic variance. Compared with the parametric maximum likelihood estimates (MLE), the asymptotic efficiency of the OPEE estimates is more than 98%, while the LS-type of procedures can have infinite variances. When the observations have outliers or do not follow the distributions considered in model assumptions, the article shows that OPEE is more robust than the MLE, and the asymptotic efficiency in the model misspecification cases can be above 150%.     

An exact upper limit for the variance bias in a carry-over model with correlated errors

The analysis of crossover designs assuming i.i.d. errors leads to biased variance estimates whenever the true covariance structure is not spherical. As a result, the OLS F-test for the equality of the direct effects of the treatments is not valid. Bellavance et al. [1996. Biometrics 52, 607–612] use simulations to show that a modified F-test based on an estimate of the within subjects covariance matrix allows for nearly unbiased tests. Kunert and Utzig [1993. JRSS B 55, 919–927] propose an alternative test that does not need an estimate of the covariance matrix. Instead, they correct the F-statistic by multiplying by a constant based on the worst-case scenario. However, for designs with more than three observations per subject, Kunert and Utzig (1993) only give a rough upper bound for the worst-case variance bias. This may lead to overly conservative tests. In this paper we derive an exact upper limit for the variance bias due to carry-over for an arbitrary number of observations per subject. The result holds for a certain class of highly efficient balanced crossover designs.     

On weak consistency in linear models with equi-correlated random errors

A new estimator in linear models with equi-correlated random errors is postulated. Consistency properties of the proposed estimator and the ordinary least squares estimator are studied. It is shown that the new estimator has smaller variance than the usual least squares estimator under some mild conditions. In addition, it is observed that the new estimator tends to be weakly consistent in many cases where the usual least squares estimator is not.     

Ratio and product methods of estimation of population mean in the presence of correlated measurement errors

The properties of the estimators of population mean arising from the ratio and product methods of estimation in the context of sample surveys have been analyzed in this paper when the observations on both the study and auxiliary variables are contaminated with measurement errors. The measurement errors in both the variables are also correlated. The properties of the ratio and product estimators along with the sample mean under the influence of measurement errors are derived and studied. The properties of the estimators in finite samples are studied through Monte-Carlo simulation and its findings are reported.     

The consistency of model selection for dynamic Semi-varying coefficient models with autocorrelated errors

The consistency of model selection criterion BIC has been well and widely studied for many nonlinear regression models. However, few of them had considered models with lag variables as regressors and auto-correlated errors in time series settings, which is common in both linear and nonlinear time series modeling. This paper studies a dynamic semi-varying coefficient model with ARMA errors, using an approach based on spectrum analysis of time series. The consistency property of the proposed model selection criteria is established and an implementation procedure of model selection is proposed for practitioners. Simulation studies have also been conducted to numerically show the consistency property.     

Wavelet estimation in time-varying coefficient time series models with measurement errors

The article studies a time-varying coefficient time series model in which some of the covariates are measured with additive errors. In order to overcome the bias of estimator of the coefficient functions when measurement errors are ignored, we propose a modified least squares estimator based on wavelet procedures. The advantage of the wavelet method is to avoid the restrictive smoothness requirement for varying-coefficient functions of the traditional smoothing approaches, such as kernel and local polynomial methods. The asymptotic properties of the proposed wavelet estimators are established under the α-mixing conditions and without specifying the error distribution. These results can be used to make asymptotically valid statistical inference.     

Inference for linear and nonlinear stable error processes via estimating functions

This paper describes an estimating function approach for parameter estimation in linear and nonlinear times series models with infinite variance stable errors. Joint estimates of location and scale parameters are derived for classes of autoregressive (AR) models and random coefficient autoregressive (RCA) models with stable errors, as well as for AR models with stable autoregressive conditionally heteroscedastic (ARCH) errors. Fast, on-line, recursive parametric estimation for the location parameter based on estimating functions is discussed using simulation studies. A real financial time series is also discussed in some detail.     

Adaptive estimating function inference for nonstationary determinantal point processes

Estimating function inference is indispensable for many common point process models where the joint intensities are tractable while the likelihood function is not. In this article, we establish asymptotic normality of estimating function estimators in a very general setting of nonstationary point processes. We then adapt this result to the case of nonstationary determinantal point processes, which are an important class of models for repulsive point patterns. In practice, often first‐ and second‐order estimating functions are used. For the latter, it is a common practice to omit contributions for pairs of points separated by a distance larger than some truncation distance, which is usually specified in an ad hoc manner. We suggest instead a data‐driven approach where the truncation distance is adapted automatically to the point process being fitted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies for non‐stationary determinantal point processes and by an application to a real dataset.     

Bandwidth selection for kernel regression with correlated errors

In this paper, we propose bandwidth selectors for nonparametric regression with dependent errors. The methods are based on criteria that approximate the average squared error. We show that these approximations are uniform over the bandwidth sequence. The criteria involve some constants that depend on the unknown error correlations. We propose a novel way of estimating these constants. Our numerical study shows that the method is quite efficient in a variety of error models.     

Robust modified profile estimating function with application to the generalized estimating equation

We consider methods for reducing the effect of fitting nuisance parameters on a general estimating function, when the estimating function depends on not only a vector of parameters of interest, θ$\theta$, but also on a vector of nuisance parameters, λ$\lambda$. We propose a class of modified profile estimating functions with plug-in bias reduced by two orders. A robust version of the adjustment term does not require any information about the probability mechanism beyond that required by the original estimating function. An important application of this method is bias correction for the generalized estimating equation in analyzing stratified longitudinal data, where the stratum-specific intercepts are considered as fixed nuisance parameters, the dependence of the expected outcome on the covariates is of interest, and the intracluster correlation structure is unknown. Furthermore, when the quasi-scores for θ$\theta$ and λ$\lambda$ are available, we propose an additional multiplicative adjustment term such that the modified profile estimating function is approximately information unbiased. This multiplicative adjustment term can serve as an optimal weight in the analysis of stratified studies. A brief simulation study shows that the proposed method considerably reduces the impact of the nuisance parameters.     

Estimation for two-way analysis of variance with correlated errors

We consider some estimation and distribution problems encountered in a two way analysis of variance model with only one observation per cell, errors correlated in one level, and the variances are not necessarily equal. The independence criteria for the row and interaction mean sum of squares and distribution of the maximum likelihood estimator of the correlation coefficient are given.     

Missing data in linear models with correlated errors

One common method for analyzing data in experimental designs when observations are missing was devised by Yates (1933), who developed his procedure based upon a suggestion by R. A. Fisher. Considering a linear model with independent, equi-variate errors, Yates substituted algebraic values for the missing data and then minimized the error sum of squares with respect to both the unknown parameters and the algebraic values. Yates showed that this procedure yielded the correct error sum of squares and a positively biased hypothesis sum of squares.

Others have elaborated on this technique. Chakrabarti (1962) gave a formal proof of Fisher's rule that produced a way to simplify the calculations of the auxiliary values to be used in place of the missing observations. Kshirsagar (1971) proved that the hypothesis sum of squares based on these values was biased, and developed an easy way to compute that bias. Sclove     

Statistical inferences for single-index models with measurement errors

An important factor in house prices is its location. However, measurement errors arise frequently in the process of observing variables such as the latitude and longitude of the house. The single-index models with measurement errors are used to study the relationship between house location and house price. We obtain the estimators by a SIMEX method based on the local linear method and the estimating equation. To test the significance of the index coefficient and the linearity of the link function, we establish the generalized likelihood ratio (GLR) tests for the models. We demonstrate that the asymptotic null distributions of the established GLR tests follow χ2-distributions which are independent of nuisance parameters or functions. Finally, two simulated examples and a real estate valuation data set are given to illustrate the effect of GLR tests.     

Consistent model selection and data-driven smooth tests for longitudinal data in the estimating equations approach

Summary.  Model selection for marginal regression analysis of longitudinal data is challenging owing to the presence of correlation and the difficulty of specifying the full likelihood, particularly for correlated categorical data. The paper introduces a novel Bayesian information criterion type model selection procedure based on the quadratic inference function, which does not require the full likelihood or quasi-likelihood. With probability approaching 1, the criterion selects the most parsimonious correct model. Although a working correlation matrix is assumed, there is no need to estimate the nuisance parameters in the working correlation matrix; moreover, the model selection procedure is robust against the misspecification of the working correlation matrix. The criterion proposed can also be used to construct a data-driven Neyman smooth test for checking the goodness of fit of a postulated model. This test is especially useful and often yields much higher power in situations where the classical directional test behaves poorly. The finite sample performance of the model selection and model checking procedures is demonstrated through Monte Carlo studies and analysis of a clinical trial data set.     

First-order marginalised transition random effects models with probit link function

Marginalised models, also known as marginally specified models, have recently become a popular tool for analysis of discrete longitudinal data. Despite being a novel statistical methodology, these models introduce complex constraint equations and model fitting algorithms. On the other hand, there is a lack of publicly available software to fit these models. In this paper, we propose a three-level marginalised model for analysis of multivariate longitudinal binary outcome. The implicit function theorem is introduced to approximately solve the marginal constraint equations explicitly. probit link enables direct solutions to the convolution equations. Parameters are estimated by maximum likelihood via a Fisher–Scoring algorithm. A simulation study is conducted to examine the finite-sample properties of the estimator. We illustrate the model with an application to the data set from the Iowa Youth and Families Project. The R package pnmtrem is prepared to fit the model.     

On multivariate nonlinear regression models with stationary correlated errors

In this paper we consider the statistical analysis of multivariate multiple nonlinear regression models with correlated errors, using Finite Fourier Transforms. Consistency and asymptotic normality of the weighted least squares estimates are established under various conditions on the regressor variables. These conditions involve different types of scalings, and the scaling factors are obtained explicitly for various types of nonlinear regression models including an interesting model which requires the estimation of unknown frequencies. The estimation of frequencies is a classical problem occurring in many areas like signal processing, environmental time series, astronomy and other areas of physical sciences. We illustrate our methodology using two real data sets taken from geophysics and environmental sciences. The data we consider from geophysics are polar motion (which is now widely known as “Chandlers Wobble”), where one has to estimate the drift parameters, the offset parameters and the two periodicities associated with elliptical motion. The data were first analyzed by Arato, Kolmogorov and Sinai who treat it as a bivariate time series satisfying a finite order time series model. They estimate the periodicities using the coefficients of the fitted models. Our analysis shows that the two dominant frequencies are 12 h and 410 days. The second example, we consider is the minimum/maximum monthly temperatures observed at the Antarctic Peninsula (Faraday/Vernadsky station). It is now widely believed that over the past 50 years there is a steady warming in this region, and if this is true, the warming has serious consequences on ecology, marine life, etc. as it can result in melting of ice shelves and glaciers. Our objective here is to estimate any existing temperature trend in the data, and we use the nonlinear regression methodology developed here to achieve that goal.