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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.

This paper obtains some specific density functions for aggregated discounted claims where the claim amounts are dependent, or the inter-claim times are dependent, or the claim amounts and the claim arrival process are both dependent. The dependence is structured through mixing, and the claim arrival process studied is either an ordinary Poisson process or a mixed Poisson process. Closed form densities are obtained for gamma, generalized exponential, generalized Pareto and beta mixing, and an important use of them is illustrated.

Generalized method of moment (GMM) is an important estimation method for econometric models. However, it is highly sensitive to the outliers and influential observations. This paper studies the detection of influential observations using GMM estimation and establishes some useful diagnostic tools, such as residual and leverage measures. The case deletion technique is employed to derive diagnostic measures. Under linear moment conditions, an exact deletion formula is derived, and under nonlinear moment condition an approximate formula is suggested. The results are applied to efficient instrumental variable estimation and dynamic panel data models. In addition, generalized residuals and leverage measure for GMM estimator are defined and discussed. Two real data sets are used for illustration and a simulation study is conducted to confirm the usefulness of the proposed 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.

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.

Empirical martingale simulation (EMS) was proposed by Duan and Simonato (Duan, J.-C., J.-G. Simonato. 1998. Empirical martingale simulation for asset prices. Management Sci. 44(9) 1218–1233) as an adjustment to the standard Monte Carlo simulation to reduce simulation errors. The EMS price estimator of derivative contracts was shown to be asymptotically normally distributed in Duan et al. (Duan, J.-C., G. Gauthier, J.-G. Simonato. 2001. Asymptotic distribution of the EMS option price estimator. Management Sci. 47(8) 1122–1132) when the payoffs are piecewise linear and continuous. In this paper, we extend the asymptotic normality result to more general continuous payoffs, and for discontinuous payoffs we make a conjecture.

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.

Correlation is an important and widely used concept, and it is usually taught through word explanation or graphic description accompanied with a numeric measure if such a measure exists. In this note, we explore graphing one of the commonly used correlation measures, Kendall's @t, in such a way that we display how Kendall's @t is calculated and see how Kendall's @t is influenced by data features.

The generalized Pareto distribution (GPD) is important in the analysis of extreme values, especially in modeling exceedances over thresholds. Most of the existing methods for estimating the scale and shape parameters of the GPD suffer from theoretical and/or computational problems. A new hybrid estimation method is proposed in this article, which minimizes a goodness-of-fit measure and incorporates some useful likelihood information. Compared with the maximum likelihood method and other leading methods, our new hybrid estimation method retains high efficiency, reduces the estimation bias, and is computation friendly.

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.

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.

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.

This paper presents three versions of the Lagrange multiplier (LM) tests of transformation in nonlinear regression: (i) LM test based on expected information, (ii) LM test based on Hessian, and (iii) the LM test based on gradient. All three tests can be easily implemented through a nonlinear least squares procedure. Simulation results show that, in terms of finite sample performance, the LM test based on expected information is the best, followed by the LM test based on Hessian and then the LM test based on gradient. The LM test based on gradient can perform rather poorly. An example is given for illustration.

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..

Insight into the causes and consequences of changes in aquatic biodiversity requires an improved understanding of the nature of the relationships between species richness and ecosystem function over a much longer temporal perspective than we currently possess. We used high-resolution paleoecological records from two prairie lakes to show that diatom species richness (as fossil frustules) was negatively correlated ( r 2 5 0.09‐0.24, p , 0.001) with diatom production (as fossil pigments) during the past 2,000 yr. By comparing analyses from intervals of fresh and saline waters, we demonstrate that these significant richness‐production relationships arose during freshwater periods (r 2 5 0.13‐0.45, p , 0.001) and could be eliminated (r 2 , 0.02, p . 0.1) by abiotic disturbances such as droughts. Procrustes analyses of the concordance of species change within freshwater communities and the change in richness‐ production relationships through time revealed that shifts in diatom community composition could have a large influence in determining the negative relationship between richness and production. Finally, significant correlations (r 2 5 0.09‐0.24, p , 0.0001) between past diatom species richness and ratios of stable isotopes (primarily d 15 N) suggested that C and N biogeochemical cycles are also linked to changes in algal biodiversity. Taken together, these analyses suggest that the ongoing disruption of climate and biogeochemical systems by humans may obscure the relationship between aquatic biodiversity and ecosystem function in the future.

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.

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..

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.

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.

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.