Search

This paper studies the estimation and inference for a class of varying-coefficient regression models with error-prone covariates. The authors focus on the situation where the covariates are unobserved, there are no repeated measurements, and the covariance matrix of the measurement errors is unknown, but some auxiliary information is available. The authors propose an instrumental variable type local polynomial estimator for the unknown varying-coefficient functions, and show that the estimator achieves the optimal nonparametric convergence rate, is asymptotically normal, and avoids using undersmoothing to allow the bandwidths to be selected using data-driven methods. A simulation is carried out to study the finite sample performance of the proposed estimator, and a real date set is analyzed to illustrate the usefulness of the developed methodology.

In this paper we study a class of multivariate partially linear regression models. Various estimators for the parametric component and the nonparametric component are constructed and their asymptotic normality established. In particular, we propose an estimator of the contemporaneous correlation among the multiple responses and develop a test for detecting the existence of such contemporaneous correlation without using any nonparametric estimation. The performance of the proposed estimators and test is evaluated through some simulation studies and an analysis of a real data set is used to illustrate the developed methodology. The Canadian Journal of Statistics 41: 1–22; 2013 © 2013 Statistical Society of Canada Dans cet article, nous etudions une classe de modeles de regression multivariee partiellement lineaires. Nous developpons plusieurs estimateurs des composantes parametrique et non parametrique et etablissons leur normalite asymptotique. Nous proposons notamment un estimateur de la correlation contemporaine parmi les reponses multiples et nous construisons un test pour detecter l'existence d'une telle correlation sans utiliser aucune estimation non parametrique. La performance des estimateurs et du test proposes est evaluee par le biais d'etudes de simulation et la methodologie elaboree est illustree par l'analyse d'un ensemble de donnees reelles. La revue canadienne de statistique 41: 1–22; 2013 © 2013 Societe statistique du Canada

Testing the equality of two independent normal populations is a perfect case of the two-sample problem, yet it is not treated in the main text of any textbook or handbook. In this article, we derive the exact distribution of the likelihood ratio test and implement this test with an R function. This article has supplementary materials online.

The authors derive the analytic expressions for the mean and variance of the log-likelihood ratio for testing equality of k (k ≥ 2) normal populations, and suggest a chi-square approximation and a gamma approximation to the exact null distribution. Numerical comparisons show that the two approximations and the original beta approximation of Neyman and Pearson (1931) are all accurate, and the gamma approximation is the most accurate.

Deletion, replacement and mean-shift model are three approaches frequently used to detect influential observations and outliers. For general linear model with known covariance matrix, it is known that these three approaches lead to the same update formulae for the estimates of the regression coefficients. However if the covariance matrix is indexed by some unknown parameters which also need to be estimated, the situation is unclear. In this paper, we show under a common subclass of linear mixed models that the three approaches are no longer equivalent. For maximum likelihood estimation, replacement is equivalent to mean-shift model but both are not equivalent to case deletion. For restricted maximum likelihood estimation, mean-shift model is equivalent to case deletion but both are not equivalent to replacement. We also demonstrate with real data that misuse of replacement and mean-shift model in place of case deletion can lead to incorrect results.

After pointing out a drawback in Bartlett's chi-square approximation, we suggest a simple modification and a Gamma approximation to improve Bartlett's M test for homogeneity of variances.

This paper is concerned with inference of panel data varying-coefficient partially linear models with a one-way error structure. The model is a natural extension of the well-known panel data linear model (due to Baltagi 1995) to the setting of semiparametric regressions. The authors propose a weighted profile least squares estimator (WPLSE) and a weighted local polynomial estimator (WLPE) for the parametric and nonparametric components, respectively. It is shown that the WPLSE is asymptotically more efficient than the usual profile least squares estimator (PLSE), and that the WLPE is also asymptotically more efficient than the usual local polynomial estimator (LPE). The latter is an interesting result. According to Ruckstuhl, Welsh and Carroll (2000) and Lin and Carroll (2000), ignoring the correlation structure entirely and “pretending” that the data are really independent will result in more efficient estimators when estimating nonparametric regression with longitudinal or panel data. The result in this paper shows that this is not true when the design points of the nonparametric component have a closeness property within groups. The asymptotic properties of the proposed weighted estimators are derived. In addition, a block bootstrap test is proposed for the goodness of fit of models, which can accommodate the correlations within groups. Some simulation studies are conducted to illustrate the finite sample performances of the proposed procedures.

The authors study the local influence of observations in multilevel regression models. To this end, they perturb simultaneously the variances, responses and design matrix. To measure the local change caused by these perturbations, they use generalized Cook statistics for the fixed and random parameter estimates. Closed form local influence measures also allow them to assess the joint influence of various observations. They suggest a simple computation method and illustrate their results using two examples. Etude de I'influence locale dans les modeles multi-niveaux Les auteurs etudient l'influence locale des observations dans les modeles de regression multi-niveaux. A cette fin, ils perturbent a la fois les variances, les variables endogenes et la matrice d'incidence. Pour mesurer le changement local cause par ces perturbations, ils se servent des statistiques de Cook generalisees associees aux estimations des effets fixes et aleatoires. Des formules explicites leur permettent aussi de mesurer l'influence locale conjuguee de diverses observations. Ils suggerent une methode de calcul simple et illustrent leurs resultats a l'aide de deux exemples.

In this article, we propose an empirical likelihood-based test to check the existence of serial correlation in partial linear errors-in-variables models. A nonparametric version of Wilk' theorem is derived, which says that our proposed test has an asymptotic chi-square distribution. Simulation results reveal that the finite sample performance of our proposed test is satisfactory in both size and power.

The authors consider a semiparametric partially linear regression model with serially correlated errors. They propose a new way of estimating the error structure which has the advantage that it does not involve any nonparametric estimation. This allows them to develop an inference procedure consisting of a bandwidth selection method, an efficient semiparametric generalized least squares estimator of the parametric component, a goodness-of-fit test based on the bootstrap, and a technique for selecting significant covariates in the parametric component. They assess their approach through simulation studies and illustrate it with a concrete application. L'inference dans le cadre d'un modele de regression semiparametrique partiellement lineaire a termes d'erreur correles en serie Les auteurs s'interessent a un modele de regression semiparametrique partiellement lineaire a termes d'erreur correles en serie. Ils proposent une facon originale d'estimer la structure d'erreur qui a l'avantage de ne faire intervenir aucune estimation non parametrique. Ceci leur permet de developper une procedure d'inference comportant un choix de fenětre, l'emploi de la methode des moindres carres generalises pour l'estimation semiparametrique efficace de la composante parametrique, un test d'adequation fonde sur le reechantillonnage et une technique de selection des covariables significatives de la composante parametrique. Ils evaluent leur approche par voie de simulation et en donnent une illustration concrete.

The authors study a heteroscedastic partially linear regression model and develop an inferential procedure for it. This includes a test of heteroscedasticity, a two-step estimator of the heteroscedastic variance function, semiparametric generalized least-squares estimators of the parametric and nonparametric components of the model, and a bootstrap goodness of fit test to see whether the nonparametric component can be parametrized.

This paper is concerned with the estimation problem in partially linear regression models with serially correlated errors. The authors propose a semiparametric generalized least squares estimator (SGLSE) for the parametric component and show that it is asymptotically more efficient than the semiparametric ordinary least squares estimator (SOLSE) in terms of asymptotic covariance matrix. Other properties of this SGLSE including the asymptotic normality and the law of the iterated logarithm are established as well. A simulation study is conducted to examine the finite-sample properties of the proposed estimator and an empirical example is discussed.

The quantile regression has several useful features and therefore is gradually developing into a comprehensive approach to the statistical analysis of linear and nonlinear response models, but it cannot deal effectively with the data with a hierarchical structure. In practice, the existence of such data hierarchies is neither accidental nor ignorable, it is a common phenomenon. To ignore this hierarchical data structure risks overlooking the importance of group effects, and may also render many of the traditional statistical analysis techniques used for studying data relationships invalid. On the other hand, the hierarchical models take a hierarchical data structure into account and have also many applications in statistics, ranging from overdispersion to constructing min-max estimators. However, the hierarchical models are virtually the mean regression, therefore, they cannot be used to characterize the entire conditional distribution of a dependent variable given high-dimensional covariates. Furthermore, the estimated coefficient vector (marginal effects) is sensitive to an outlier observation on the dependent variable. In this article, a new approach, which is based on the Gauss-Seidel iteration and taking a full advantage of the quantile regression and hierarchical models, is developed. On the theoretical front, we also consider the asymptotic properties of the new method, obtaining the simple conditions for an n1/2-convergence and an asymptotic normality. We also illustrate the use of the technique with the real educational data which is hierarchical and how the results can be explained.

The authors study a varying-coefficient regression model in which some of the covariates are measured with additive errors. They find that the usual local linear estimator (LLE) of the coefficient functions is biased and that the usual correction for attenuation fails to work. They propose a corrected LLE and show that it is consistent and asymptotically normal, and they also construct a consistent estimator for the model error variance. They then extend the generalized likelihood technique to develop a goodness of fit test for the model. They evaluate these various procedures through simulation studies and use them to analyze data from the Framingham Heart Study. Estimation polynomiale locale corrigee dans les modeles a coefficients variables comportant des erreurs de mesure Les auteurs s'interessent a un modele de regression a coefficients variables dont certaines cova-riables sont entachees d'erreurs additives. Ils montrent que l'estimateur localement lineaire (ELL) usuel des coefficients fonctionnels est biaise et que le facteur de correction habituel du phenomene d'attenuation est inefficace. Ils proposent une version corrigee de l'ELL qui s'avere convergente et asymptotiquement normale; ils suggerent aussi une estimation convergente de la variance du terme d'erreur du modele. Une adaptation de la technique de vraisemblance generalisee leur permet en outre d'elaborer un test d'adequation du modele. Ils evaluent ces diverses procedures par voie de simulation et s'en servent pour analyser des donnees issues de l'etude Framingham sur les risques cardiometaboliques.

The authors define a class of "partially linear single-index" survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single-index form. It is then pos sible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single-index component. The par tially linear hazards model and the single-index hazards model are special cases of the proposed model. The authors develop a likelihood-based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite-sample behaviour through simulation, and use a set of real data to illustrate their approach. Une classe de modeles de survie partiellement lineaires a indice simple Rgsum=: Les auteurs d6finissent une classe de modeles de survie dits "partiellement lineaires a indice simple" qui s'averent plus flexibles que les modeles de regression a risques proportionnels classiques dans le traitement des covariables. Ces dernieres entrent dans le modele propose soit sous une forme lin6aire param6trique, soit sous une forme a indice simple non parametrique. I1 est alors possible de modeliser a la fois des effets lineaires et fonctionnels de covariables sur le logarithme du risque, et de reduire au besoin la dimension de covariables multiples par l'intermediaire de la composante a indice simple. Les modeles a risques partiellement lineaires et a indice simple sont des cas speciaux du modele propose. Les auteurs developpent une methode d'inf6rence vraisemblantiste pour l'estimation des composantes du modele au moyen d'un algorithme it6ratif. Ils d6terminent la loi asymptotique des estimateurs proposes, en etudient le comportement a taille finie par voie de simulation et illustrent leur approche a l'aide de donnees reelles.

The authors propose a block empirical likelihood procedure to accommodate the within-group correlation in longitudinal partially linear regression models. This leads them to prove a nonparametric version of the Wilks theorem. In comparison with normal approximations, their method does not require a consistent estimator for the asymptotic covariance matrix, which makes it easier to conduct inference on the parametric component of the model. An application to a longitudinal study on fluctuations of progesterone level in a menstrual cycle is used to illustrate the procedure developed here. Une vraisemblance empirique par bloc pour les modeles de regression partiellernent lineaires longitudinaux Les auteurs proposent une procedure a base de vraisemblance empirique par bloc pour tenir compte de la correlation intra-groupe dans des modeles de regression partiellement lineaires longitudinaux. Ceci les ameneo a demontrer une version non parametrique du theoreme de Wilks. A la difference des approximations normales, leur methode ne fait pas appel a un estimateur convergent de la matrice des cova-riances asymptotiques, ce qui facilite l'inference concernant la composante parametrique du modele. Une etude longitudinale sur la fluctuation du niveau de progesterone pendant le cycle menstruel sert a illustrer le propos.

This paper uses the wild bootstrap technique in the estimation of a heteroscedastic partially linear regression model. We show that this approach provides reliable approximation to the asymptotic distribution of the semiparametric least-square estimators of the linear regression coefficients and consistent estimators of the asymptotic covariance matrices even when the error variances are unequal. In comparison, this robustness property is not shared by the bootstrap estimation proposed in Liang et al. (2000. Bootstrap approximation in a partially linear regression model. J. Statist. Plann. Inference, 91, 413–426).

We study the estimation problems for a partly linear regression model with a nonlinear time series error structure. The model consists of a parametric linear component for the regression coefficients and a nonparametric nonlinear component. The random errors are unobservable and modeled by a first-order Markov bilinear process. Based on a B-spline series approximation of the nonlinear component function, we propose a semiparametric ordinary least squares estimator and a semiparametric generalized least squares estimator of the regression coefficients, a least squares estimator of the autoregression parameter for the errors, and a B-spline series estimator of the nonparametric component function. The asymptotic properties of these estimators are investigated and their asymptotic distributions are derived. We also provide a consistent estimator for the asymptotic covariance matrix of the semiparametric generalized least squares estimator of the regression coefficients. Our results can be used to make asymptotically efficient statistical inferences. In addition, a small simulation is conducted to evaluate the performance of the proposed estimators, which shows that the semiparametric generalized least squares estimator of the regression coefficients is more efficient than the semiparametric ordinary least squares estimator.

Efficient inference for regression models requires that heteroscedasticity be taken into account if it exists. For partially linear regression models, however, the problem of detecting heteroscedasticity has received very little attention. The aim of this paper is to propose a test of heteroscedasticity for partially linear regression models.

Consider a repeated measurement partially linear regression model with an unknown vector parameter β 1, an unknown function g(·), and unknown heteroscedastic error variances. In order to improve the semiparametric generalized least squares estimator (SGLSE) of β, we propose an iterative weighted semiparametric least squares estimator (IWSLSE) and show that it improves upon the SGLSE in terms of asymptotic covariance matrix. An adaptive procedure is given to determine the number of iterations. We also show that when the number of replicates is less than or equal to two, the IWSLSE can not improve upon the SGLSE. These results are generalizations of those in [2] to the case of semiparametric regressions.

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.

A multivariate exponentially weighted moving average single control chart is developed in this article. This chart is capable of monitoring simultaneously the process mean vector and the process covariance matrix. Our average run length comparison shows that this new chart performs better than the combination of the χ2 chart and the |S| chart when small changes in the process parameters are of interest.

In this paper, we extend the exponentially weighted moving average (EWMA) technique to double exponentially weighted moving average (DEWMA) technique. We show that DEWMA mean charts perform better than EWMA mean charts in detecting small mean shifts ranging from 0.1 to 0.5 of the process standard deviation, and that the two types of charts perform similarly when mean shifts are larger than 0.5 standard deviation. The design of DEWMA mean charts is also discussed.

Lower-sided and upper-sided exponentially weighted moving average (EWMA) charts are developed for detecting mean changes when censoring occurs at a fixed level in processes characterized by Weibull distributions. The lower-sided EWMA chart performs bett..

The MaxEWMA chart has recently been introduced as an alternative to control charting procedures that are designed to jointly detect changes in the mean and standard deviation of a normally distributed process. Here, a method for computing both in-control and out-of-control average run lengths for purposes of effectively designing this chart is offered. Design strategies are considered, numerical results to aid the design effort are given, and examples are discussed.

A new control chart, which employs the exponentially weighted moving average (EWMA) technique, is proposed. The statistic for the chart defines the area below a straight line as the control region, which makes the charting procedure easier than the usual approach. This chart can effectively monitor the process mean and the increased process variability simultaneously, and can detect the source and the direction of a change easily.

Consider a partially linear regression model with an unknown vector parameter β, an unknown function g(·), and unknown heteroscedastic error variances. Chen, You[23] proposed a semiparametric generalized least squares estimator (SGLSE) for β, which takes the heteroscedasticity into account to increase efficiency. For inference based on this SGLSE, it is necessary to construct a consistent estimator for its asymptotic covariance matrix. However, when there exists within-group correlation, the traditional delta method and the delete-1 jackknife estimation fail to offer such a consistent estimator. In this paper, by deleting grouped partial residuals a delete-group jackknife method is examined. It is shown that the delete-group jackknife method indeed can provide a consistent estimator for the asymptotic covariance matrix in the presence of within-group correlations. This result is an extension of that in [21].

Consider the partly linear regression model $$ y_{i} = {x}'_{i} \beta + g{\left( {t_{i} } \right)} + \varepsilon _{i} ,\;\;{\kern 1pt} 1 \leqslant i \leqslant n $$ , where y i ’s are responses, $$ x_{i} = {\left( {x_{{i1}} ,x_{{i2}} , \cdots ,x_{{ip}} } \right)}^{\prime } \;\;\;{\text{and}}\;\;\;t_{i} \in {\cal T} $$ are known and nonrandom design points, $$ {\cal T} $$ is a compact set in the real line $$ {\cal R} $$ , β = (β 1, ··· , β p )' is an unknown parameter vector, g(·) is an unknown function and {e i } is a linear process, i.e., $$ \varepsilon _{i} {\kern 1pt} = {\kern 1pt} {\sum\limits_{j = 0}^\infty {\psi _{j} e_{{i - j}} ,{\kern 1pt} \;\psi _{0} {\kern 1pt} = {\kern 1pt} 1,\;{\kern 1pt} {\sum\limits_{j = 0}^\infty {{\left| {\psi _{j} } \right|} < \infty } }} } $$ , where e j are i.i.d. random variables with zero mean and variance $$ \sigma ^{2}_{e} $$ . Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {e i }. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {e i } are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coeffcients of the process. Moreover, our result can be used to construct the asymptotically effcient estimators for parameters in the ARMA error process.

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.

The authors provide a rigorous large sample theory for linear models whose response variable has been subjected to the Box-Cox transformation. They provide a continuous asymptotic approximation to the distribution of estimators of natural parameters of the model. They show, in particular, that the maximum likelihood estimator of the ratio of slope to residual standard deviation is consistent and relatively stable. The authors further show the importance for inference of normality of the errors and give tests for normality based on the estimated residuals. For non-normal errors, they give adjustments to the log-likelihood and to asymptotic standard errors.

We prove that the standard EWMA mean chart with asymptotic control limits and the EWMA mean chart with time-varying control limits for monitoring mean changes in a normal process with known mean and known variance are ARL-unbiased. Using the results derived we discuss the effects of estimation of the process mean on ARL.

Threshold autoregressive models are widely used in time-series applications. When building or using such a model, it is important to know whether conditional heteroscedasticity exists. The authors propose a nonparametric test of this hypothesis. They develop the large-sample theory of a test of nonlinear conditional heteroscedasticity adapted to nonlinear autoregressive models and study its finite-sample properties through simulations. They also provide percentage points for carrying out this test, which is found to have very good power overall.

Two EWMA charts are usually required to monitor both process mean and variability. In this paper, we propose a new EWMA chart which effectively combines the usual two EWMA charts into one chart. In...

The authors give easy-to-check sufficient conditions for the geometric ergodicity and the finiteness of the moments of a random process xt = ϕ(xt-1,…, xt-p) + ϵtσ(xt-1,…, xt-q) in which ϕ: Rp R, σ Rq R and (ϵt) is a sequence of independent and identically distributed random variables. They deduce strong mixing properties for this class of nonlinear autoregressive models with changing conditional variances which includes, among others, the ARCH(p), the AR(p)-ARCH(p), and the double-threshold autoregressive models. Les auteurs enoncent des conditions faciles a verifier qui garantissent l'ergodicite geometrique et la finitude des moments d'un processus aleatoire xt = ϵ(xt-1,…, xt-p) + ϵtσ(xt-1,…, xt-q) defini a partir de fonctions ϕ: Rp R, σ: Rq R et d'une suite (ϵt) d'aleas independants de měme loi. Ils en deduisent des proprietes de melangeance forte pour cette classe de modeles autoregressifs non lineaires a variances conditionnelles variables qui englobe, entre autres, les modeles ARCH(p), les modeles AR(p)-ARCH(p), ainsi que les modeles autoregressifs a double seuil.

We give simple and concise expressions of influence measures for general linear models with correlated errors. We show that the commonly used influence measures can be expressed as functions of the generalized leverages and residuals and have the same forms as those in standard linear regression models.

The p chart for monitoring the fraction, p, of nonconforming products requires very large sample sizes to achieve any power in detecting decreases in p, and the false alarm for probabilities differ considerably from the nomial normal values. Two alterna..

The u chart for monitoring λ, the average number of defects per inspection unit, requires λ to be large to have positive lower control limits, and the false alarm probabilities differ considerably from the nominal normal values. Based on nonlinear transformations, some alternative charts, such as the chart, can achieve large improvement over the u chart. We show in this paper that with very simple adjustments to the control limits of the u chart, one can achieve equal or even better improvement while still working on the original scale. In particular, using the very simple adjustments is equivalent to using the exact Poisson distribution to design a chart. Some guidelines for choosing an appropriate chart are given.

Very often in regression analysis, a particular functional form connecting known covariates and unknown parameters is either suggested by previous work or demanded by theoretical considerations so that the deterministic part of the responses has a known form. However, the underlying error structure is often less well understood. In this case, the transform-both-sides (TBS) models are appropriate. In this paper we generalize the usual TBS models and develop tests to assess goodness of fit when fitting TBS or GTBS models. Parameter estimation is discussed, and tests based on the Cramer-von Mises statistic and the Anderson-Darling statistic are presented with a table suitable for finite-sample applications. Tres souvent, lors de l'analyse de la regression, une forme fonctionnelle particuliere reliant des “covariates” connus et des parametres inconnus est soit suggeree par des travaux anterieurs, soit exigee par des considerations theoriques afin que la partie deterministe des reponses ait une forme connue. Toutefois, la structure d'erreur sous-jacente est souvent moins bien comprise. Dans ce cas, les modeles TBS (“transform-both-sides”) sont appropries. Cet article generalise les modeles TBS habituels et developpe des tests pour evaluer la validite de l'ajustement lorsqu'on ajuste des modeles TBS ou GTBS. Nous discutons l'estimation des parametres et presentons des tests fondes sur la statistique Cramer-von Mises et la statistique Anderson-Darling avec un tableaux convenant a des applications a echantillon fini.

Skewed distributions play an important role in the analysis of data from quality and reliability experiments. Very often unknown parameters must be estimated from the sample data in order to test whether the data has come from a certain family of distri..

A specific scale must be used to collect data in an experiment, but this scale may or may not be the appropriate scale to analyze the data thus collected. In this paper, we study a family of three-parameter distributions which includes the log-normal distribution as a special case, and one of the parameters plays the role of choosing a suitable scale for subsequent analysis. Some basic properties and maximum likelihood estimation of the unknown parameters are discussed, and an easy-to-use goodness-of-fit test is developed. A real data set, to which the usual Weibull, log-normal and gamma distributions cannot be fitted, is used to demonstrate the potential applicability of this family of distributions.

In this paper we consider the estimation of the degree of differencing d in the fractionally integrated autoregressive moving-average time series model ARFIMA (p, d, q). Using lag window spectral density estimators we develop a regression type estimator of d which is easy to calculate and does not require prior knowledge of p and q. Some large sample properties of the estimator are studied and the performance of the estimator for small samples is investigated using the simulation method for a range of commonly used lag windows. Some practical recommendations on the choice of lag windows and the choice of the window parameters are provided.

Weak convergence results are obtained in this paper for the residual empirical processes in linear models, when the normal error assumption is replaced by a number of other error assumptions such as the Gamma family and the inverse Gaussian family. Application to asymptotic power calculations for a class of tests of fit is discussed.

The shape of the commonly used distributions in extreme value analysis is determined by either a specific functional form or a shape parameter. In this paper, we develop a family of distributions which uses two parameters to determine the shape: a balance parameter to control symmetry and a tail order parameter to control the thickness of the tail and modes. Each member in this family has closed forms for its density, cumulative distribution, and quantile function. We discuss the maximum likelihood and minimum distance estimation of the unknown parameters and illustrate the usefulness of the new distribution in extreme value analysis with a real data set.

The performance of an X-bar chart is usually studied under the assumption that the process standard deviation is well estimated and does not change. This is, of course, not always the case in practice. We find that X-bar charts are not robust against errors in estimating the process standard deviation or changing standard deviation. In this paper we discuss the use of a t chart and an exponentially weighted moving average (EWMA) t chart to monitor the process mean. We determine the optimal control limits for the EWMA t chart and show that this chart has the desired robustness property. Copyright © 2009 John Wiley & Sons, Ltd.

When monitoring the process variability, it is a common practice that a Phase I data set is used to estimate the unknown in-control process standard deviation σ0 or variance [Formula: see text] to set up the control limits, then monitoring proceeds. Once the process is considered to be in-control, the estimated control limits are assumed as fixed. This practice ignores the effect of estimating the unknown in-control process variance [Formula: see text]. In this paper, we derive the exact run length distribution of the S2 control chart when the in-control process variance [Formula: see text] is estimated and find that m = 200 or more Phase I samples are needed to neglect the effect of using estimated control limits. New control limits when m is small are also derived.

This paper studies the estimation of a varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model [Fan and Huang, Manuscript, University of North Carolina, Chapel Hill, USA, 2002]. We focus on the case where some covariates are measured with additive errors. The usual profile least squares and local polynomial estimations lead to biased estimators of the parametric and nonparametric components, respectively, when measurement errors are ignored. By correcting the attenuation we propose a modified profile least squares estimator for the parametric component and a local polynomial estimator for the nonparametric component. We show that the former is consistent, asymptotically normal and achieves the rate in the law of the iterated logarithm, and the latter achieves the optimal strong convergence rate of the usual nonparametric regression. In addition, a consistent estimator is also developed for the error variance. These results can be used to make asymptotically valid statistical inferences. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.

When fitting, by least squares, a linear model (with an intercept term) with $p$ parameters to $n$ data points, the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided $p^3 \log^2 (p) /n \to 0$ for any error density having finite variance and a bounded first derivative. No further conditions are imposed on the sequence of design matrices. The result is extended to more general estimates with the property that the average error and average squared error in the fitted values are on the same order as for least squares.

The importance of the independence between a standard normal and a chi-square random variable in constructing a t random variable is explored. When independence is absent it is demonstrated that the usual t approximation can lead to misleading results.

In this paper jackknifing technique is examined for functions of the parametric component in a partially linear regression model with serially correlated errors. By deleting partial residuals a jackknife-type estimator is proposed. It is shown that the jackknife-type estimator and the usual semiparametric least-squares estimator (SLSE) are asymptotically equivalent. However, simulation shows that the former has smaller biases than the latter when the sample size is small or moderate. Moreover, since the errors are correlated, both the Tukey type and the delta type jackknife asymptotic variance estimators are not consistent. By introducing cross-product terms, a consistent estimator of the jackknife asymptotic variance is constructed and shown to be robust against heterogeneity of the error variances. In addition, simulation results show that confidence interval estimation based on the proposed jackknife estimator has better coverage probability than that based on the SLSE, even though the latter uses the information of the error structure, while the former does not.

This paper is concerned with a partial linear regression model with serially correlated random errors which are unobservable and modeled by a moving-average process of infinite order. We study a class of estimators for the linear regression coefficients as well as the function characterizing the non-linear part of the model, constructed based on general kernel smoothing and least squares methods. The law of iterated logarithm and strong convergence rates of these estimator are derived by truncating the moving-average error process, a procedure widely applied in the analysis of time series. Our results can be used to establish uniform strong convergence rate of the estimators of autocovariance and autocorrelation functions of the error process.