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We propose a new sparse principal component analysis (SPCA) method in which the solutions are obtained by projecting the full cardinality principal components onto subsets of variables. The resulting components are guaranteed to explain a given proportion of variance. The computation of these solutions is very efficient. The proposed method compares well with the optimal least squares sparse components. We show that other SPCA methods fail to identify the best sparse approximations of the principal components and explain less variance than our solutions. We illustrate and compare our method with others with extensive simulations and with the analysis of the computational results for nine datasets of increasing dimensions up to 16,000 variables.

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.

This paper studies case deletion diagnostics for multilevel models. Using subset deletion, diagnostic measures for identifying influential units at any level are developed for both fixed and random parameters. Two approximate update formulae are derived. The first formula uses one-step approximation, while the second formula also includes the impact of estimating the random parameter. Two examples are used to illustrate the methodology developed.

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.

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 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).

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 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.

The authors extend the block external bootstrap to partially linear regression models with strongly mixing, nonstationary error terms. In addition to providing an approximate distribution for the semiparametric least square estimator of the parametric component, they propose a consistent estimator of the co-variance matrix of this estimator. Les auteurs etendent le bootstrap externe par bloc aux modeles partiellement lineaires dont les termes d'erreur sont fortement melangeants et non stationnaires. En plus de donner une approximation de la loi de l'estimateur des moindres carres semiparametrique de la composante parametrique, ils proposent un estimateur convergent de la matrice de covariance de cet estimateur.

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.

This paper studies general sufficient conditions for the geometric ergodicity and the existence of moments for a class of nonlinear autoregressive models with nonlinear ARCH errors. Applications of these conditions to various well-known nonlinear time series models yield specific sufficient conditions, many of which are new or generalizations of existing conditions.

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.

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 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.

A Box-Cox transformed linear model usually has the form y(λ) = μ + β1x1 +… + βpxp + oe, where y(λ) is the power transform of y. Although widely used in practice, the Fisher information matrix for the unknown parameters and, in particular, its inverse have not been studied seriously in the literature. We obtain those two important matrices to put the Box-Cox transformed linear model on a firmer ground. The question of how to make inference on β = (β1,…,βp)T when λ; is estimated from the data is then discussed for large but finite sample size by studying some parameter-based asymptotics. Both unconditional and conditional inference are studied from the frequentist point of view. Un modele lineaire transforme suivant Box-Cox a habituellement la forme y(λ) = μ+β1x1 +…+ βpxp + oϵ, ou y(λ) est est la transformation puissance de y. Malgre le fait qu'elle est communement utilisee dans la pratique, la matrice d'information de Fisher pour les parametres inconnus et, en particulier, son inverse n'ont pas ete etudiees serieusement dans la litterature. Dans cet article, nous obtenons ces deux importantes matrices ce qui permet de placer le modele lineaire transforme suivant Box-Cox sur des bases plus solides. Nous etudions I'inference au sujet de β = (β1,….βp)T lorsque λ est estime a partir des donnees, cela pour un echantillon fini de grande taille et en considerant certaines asymptotiques fondees sur des parametres. Nous etudions, d'un point de vue frequentiel, les inferences inconditionnelle et conditionnelle.

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.

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.

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.