Asymptotic Properties (asymptotic + property)

Distribution by Scientific Domains

Selected Abstracts

Asymptotic properties of the QR factorization of banded Hessenberg,Toeplitz matrices

Xiao-Wen Chang
Abstract We consider Givens QR factorization of banded Hessenberg,Toeplitz matrices of large order and relatively small bandwidth. We investigate the asymptotic behaviour of the R factor and Givens rotation when the order of the matrix goes to infinity, and present some interesting convergence properties. These properties can lead to savings in the computation of the exact QR factorization and give insight into the approximate QR factorizations of interest in preconditioning. The properties also reveal the relation between the limit of the main diagonal elements of R and the largest absolute root of a polynomial. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Regression Calibration in Semiparametric Accelerated Failure Time Models

BIOMETRICS, Issue 2 2010
Menggang Yu
Summary In large cohort studies, it often happens that some covariates are expensive to measure and hence only measured on a validation set. On the other hand, relatively cheap but error-prone measurements of the covariates are available for all subjects. Regression calibration (RC) estimation method (Prentice, 1982,,Biometrika,69, 331,342) is a popular method for analyzing such data and has been applied to the Cox model by Wang et al. (1997,,Biometrics,53, 131,145) under normal measurement error and rare disease assumptions. In this article, we consider the RC estimation method for the semiparametric accelerated failure time model with covariates subject to measurement error. Asymptotic properties of the proposed method are investigated under a two-phase sampling scheme for validation data that are selected via stratified random sampling, resulting in neither independent nor identically distributed observations. We show that the estimates converge to some well-defined parameters. In particular, unbiased estimation is feasible under additive normal measurement error models for normal covariates and under Berkson error models. The proposed method performs well in finite-sample simulation studies. We also apply the proposed method to a depression mortality study. [source]

Marginal Hazards Regression for Retrospective Studies within Cohort with Possibly Correlated Failure Time Data

BIOMETRICS, Issue 2 2009
Sangwook Kang
Summary A retrospective dental study was conducted to evaluate the degree to which pulpal involvement affects tooth survival. Due to the clustering of teeth, the survival times within each subject could be correlated and thus the conventional method for the case,control studies cannot be directly applied. In this article, we propose a marginal model approach for this type of correlated case,control within cohort data. Weighted estimating equations are proposed for the estimation of the regression parameters. Different types of weights are also considered for improving the efficiency. Asymptotic properties of the proposed estimators are investigated and their finite sample properties are assessed via simulations studies. The proposed method is applied to the aforementioned dental study. [source]

Case,Cohort Analysis with Accelerated Failure Time Model

BIOMETRICS, Issue 1 2009
Lan Kong
Summary In a case,cohort design, covariates are assembled only for a subcohort that is randomly selected from the entire cohort and any additional cases outside the subcohort. This design is appealing for large cohort studies of rare disease, especially when the exposures of interest are expensive to ascertain for all the subjects. We propose statistical methods for analyzing the case,cohort data with a semiparametric accelerated failure time model that interprets the covariates effects as to accelerate or decelerate the time to failure. Asymptotic properties of the proposed estimators are developed. The finite sample properties of case,cohort estimator and its relative efficiency to full cohort estimator are assessed via simulation studies. A real example from a study of cardiovascular disease is provided to illustrate the estimating procedure. [source]

Sequential Estimation of Dynamic Discrete Games: A Comment

ECONOMETRICA, Issue 2 2010
Martin Pesendorfer
Recursive procedures which are based on iterating on the best response mapping have difficulties converging to all equilibria in multi-player games. We illustrate these difficulties by revisiting the asymptotic properties of the iterative nested pseudo maximum likelihood method for estimating dynamic games introduced by Aguirregabiria and Mira (2007). An example shows that the iterative method may not be consistent. [source]

Estimation and Confidence Regions for Parameter Sets in Econometric Models,

ECONOMETRICA, Issue 5 2007
Victor Chernozhukov
This paper develops a framework for performing estimation and inference in econometric models with partial identification, focusing particularly on models characterized by moment inequalities and equalities. Applications of this framework include the analysis of game-theoretic models, revealed preference restrictions, regressions with missing and corrupted data, auction models, structural quantile regressions, and asset pricing models. Specifically, we provide estimators and confidence regions for the set of minimizers ,I of an econometric criterion function Q(,). In applications, the criterion function embodies testable restrictions on economic models. A parameter value ,that describes an economic model satisfies these restrictions if Q(,) attains its minimum at this value. Interest therefore focuses on the set of minimizers, called the identified set. We use the inversion of the sample analog, Qn(,), of the population criterion, Q(,), to construct estimators and confidence regions for the identified set, and develop consistency, rates of convergence, and inference results for these estimators and regions. To derive these results, we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification. [source]

Confidence intervals for the calibration estimator with environmental applications

I. Müller
Abstract The article investigates different estimation techniques in the simple linear controlled calibration model and provides different types of confidence limits for the calibration estimator. In particular, M-estimation and bootstrapping techniques are implemented to obtain estimates of regression parameters during the training stage. Moreover, bootstrap is used to construct several types of confidence intervals that are compared to the classical approach based on the assumption of normality. For some of these intervals, the second order asymptotic properties can be established by means of Edgeworth expansions. Two data sets,one on space debris and the other on bacteriological counts in water samples,are used to illustrate the method's environmental applications. Copyright © 2002 John Wiley & Sons, Ltd. [source]

A mode II weight function for subsurface cracks in a two-dimensional half-space

ABSTRACT The general properties of a mode II Weight Function for a subsurface crack in a two-dimensional half-space are discussed. A general form for the WF is proposed, and its analytical expression is deduced from the asymptotic properties of the displacements field near the crack tips and from some reference cases obtained by finite elements models. Although the WF has general validity, the main interest is on its application to the study of rolling contact fatigue: its properties are explored for a crack depth range within which the most common failure phenomena in rolling contact are experimentally observed, and for a crack length range within the field of short cracks. The accuracy is estimated by comparison with several results obtained by FEM models, and its validity in the crack depth range explored is shown. [source]

An Anisotropic Model for Spatial Processes

Minfeng Deng
One of the key assumptions in spatial econometric modeling is that the spatial process is isotropic, which means that direction is irrelevant in the specification of the spatial structure. On the one hand, this assumption largely reduces the complexity of the spatial models and facilitates estimation and interpretation; on the other hand, it appears rather restrictive and hard to justify in many empirical applications. In this article a very general anisotropic spatial model, which allows for a high level of flexibility in the spatial structure, is proposed. This new model can be estimated using maximum likelihood and its asymptotic properties are derived at length. When the model is applied to the well-known 1970 Boston housing prices data, it significantly outperforms the isotropic spatial lag model. It also provides interesting additional insights into the price determination process in the properties market. Finally, a Monte Carlo simulation study is used to confirm the optimal properties of the model. [source]

Averaging probability judgments: Monte Carlo analyses of asymptotic diagnostic value

Timothy R. Johnson
Abstract Wallsten et al. (1997) developed a general framework for assessing the quality of aggregated probability judgments. Within this framework they presented a theorem regarding the effects of pooling multiple probability judgments regarding unique binary events. The theorem states that under reasonable conditions, and assuming conditional pairwise independence of the judgments, the average probability estimate is asymptotically perfectly diagnostic of the true event state as the number of estimates pooled goes to infinity. The purpose of the present study was to examine by simulation (1) the rate of convergence of averaged judgments to perfect diagnostic value under various conditions and (2) the robustness of the theorem to violations of its assumption that the covert probability judgments are conditionally pairwise independent. The results suggest that while the rate of convergence is sensitive to violations of the conditional pairwise independence, the asymptotic properties remain relatively robust under a large variety of conditions. The practical implications of these results are discussed. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Economic Efficiency and Frontier Techniques

Luis R. Murillo-Zamorano
Abstract. Most of the literature related to the measurement of economic efficiency has based its analysis either on parametric or on non-parametric frontier methods. The choice of estimation method has been an issue of debate, with some researchers preferring the parametric and others the non-parametric approach. The aim of this paper is to provide a critical and detailed review of both core frontier methods. In our opinion, no approach is strictly preferable to any other. Moreover, a careful consideration of their main advantages and disadvantages, of the data set utilized, and of the intrinsic characteristics of the framework under analysis will help us in the correct implementation of these techniques. Recent developments in frontier techniques and economic efficiency measurement such as Bayesian techniques, bootstrapping, duality theory and the analysis of sampling asymptotic properties are also considered in this paper. [source]

Failure time regression with continuous covariates measured with error

Halbo Zhou
We consider failure time regression analysis with an auxiliary variable in the presence of a validation sample. We extend the nonparametric inference procedure of Zhou and Pepe to handle a continuous auxiliary or proxy covariate. We estimate the induced relative risk function with a kernel smoother and allow the selection probability of the validation set to depend on the observed covariates. We present some asymptotic properties for the kernel estimator and provide some simulation results. The method proposed is illustrated with a data set from an on-going epidemiologic study. [source]

On the properties of the periodogram of a stationary long-memory process over different epochs with applications

Valdério A. Reisen
Primary 60G10; 60K35; Secondary 60G18 This article studies the asymptotic properties of the discrete Fourier transforms (DFT) and the periodogram of a stationary long-memory time series over different epochs. The main theoretical result is a novel bound for the covariance of the DFT ordinates evaluated on two distinct epochs, which depends explicitly on the Fourier frequencies and the gap between the epochs. This result is then applied to obtain the limiting distribution of some nonlinear functions of the periodogram over different epochs, under the additional assumption of gaussianity. We then apply this result to construct an estimator of the memory parameter based on the regression in a neighbourhood of the zero-frequency of the logarithm of the averaged periodogram, obtained by computing the empirical mean of the periodogram over adjacent epochs. It is shown that replacing the periodogram by its average has an effect similar to the frequency domain pooling to reduce the variance of the estimate. We also propose a simple procedure to test the stationarity of the memory coefficient. A limited Monte Carlo experiment is presented to support our findings. [source]

Semiparametric inference on a class of Wiener processes

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

Subsampling in testing autocovariance for periodically correlated time series

Ukasz Lenart
Abstract., The main purpose of this article was to describe the asymptotic properties of subsampling procedure applied to nonstationary, periodically correlated time series. We present the conditions under which the subsampling version for the estimator of Fourier coefficient of autocovariance function is consistent. Our result provides new tools in statistical inference methods for nonstationary, periodically correlated time series. For example, it enables to construct consistent subsampling test which successfully distinguishes the period of the series. [source]

Efficient use of higher-lag autocorrelations for estimating autoregressive processes

The Yule,Walker estimator is commonly used in time-series analysis, as a simple way to estimate the coefficients of an autoregressive process. Under strong assumptions on the noise process, this estimator possesses the same asymptotic properties as the Gaussian maximum likelihood estimator. However, when the noise is a weak one, other estimators based on higher-order empirical autocorrelations can provide substantial efficiency gains. This is illustrated by means of a first-order autoregressive process with a Markov-switching white noise. We show how to optimally choose a linear combination of a set of estimators based on empirical autocorrelations. The asymptotic variance of the optimal estimator is derived. Empirical experiments based on simulations show that the new estimator performs well on the illustrative model. [source]

Large Sample Properties of Parameter Estimates for Periodic ARMA Models

I. V. Basawa
This paper studies the asymptotic properties of parameter estimates for causal and invertible periodic autoregressive moving-average (PARMA) time series models. A general limit result for PARMA parameter estimates with a moving-average component is derived. The paper presents examples that explicitly identify the limiting covariance matrix for parameter estimates from a general periodic autoregression (PAR), a first-order periodic moving average (PMA(1)), and the mixed PARMA(1,1) model. Some comparisons and contrasts to univariate and vector autoregressive moving-average sequences are made. [source]

Functional Coefficient Autoregressive Models: Estimation and Tests of Hypotheses

Rong Chen
In this paper, we study nonparametric estimation and hypothesis testing procedures for the functional coefficient AR (FAR) models of the form Xt=f1(Xt,d)Xt, 1+ ... +fp(Xt,d)Xt,p+,t, first proposed by Chen and Tsay (1993). As a direct generalization of the linear AR model, the FAR model is a rich class of models that includes many useful parametric nonlinear time series models such as the threshold AR models of Tong (1983) and exponential AR models of Haggan and Ozaki (1981). We propose a local linear estimation procedure for estimating the coefficient functions and study its asymptotic properties. In addition, we propose two testing procedures. The first one tests whether all the coefficient functions are constant, i.e. whether the process is linear. The second one tests if all the coefficient functions are continuous, i.e. if any threshold type of nonlinearity presents in the process. The results of some simulation studies as well as a real example are presented. [source]

MSM Estimators of European Options on Assets with Jumps

João Amaro de Matos
This paper shows that, under some regularity conditions, the method of simulated moments estimator of European option pricing models developed by Bossaerts and Hillion (1993) can be extended to the case where the prices of the underlying asset follow Lévy processes, which allow for jumps, with no losses on their asymptotic properties, still allowing for the joint test of the model. [source]

On a class of PDEs with nonlinear distributed in space and time state-dependent delay terms

Alexander V. Rezounenko
Abstract A new class of nonlinear partial differential equations with distributed in space and time state-dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state-dependent delay and discuss advantages of this class. Local and long-time asymptotic properties, including the existence of global attractor and a principle of linearized stability, are studied. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Resonance phenomena in compound cylindrical waveguides

Günter Heinzelmann
Abstract We study the large time asymptotics of the solutions u(x,t) of the Dirichlet and the Neumann initial boundary value problem for the wave equation with time-harmonic right-hand side in domains , which are composed of a finite number of disjoint half-cylinders ,1,,,,r with cross-sections ,,1,,,,,r and a bounded part (,compound cylindrical waveguides'). We show that resonances of orders t and t1/2 may occur at a finite or countable discrete set of frequencies ,, while u(x,t) is bounded as t,, for the remaining frequencies. A resonance of order t occurs at , if and only if ,2 is an eigenvalue of the Laplacian ,, in , with regard to the given boundary condition u=0 or ,u/,n=0, respectively. A resonance of order t1/2 occurs at , if and only if (i) ,2 is an eigenvalue of at least one of the Laplacians for the cross-sections ,,1,,,,r, with regard to the respective boundary condition and (ii) the respective homogeneous boundary value problem for the reduced wave equation ,U+,2U=0 in , has non-trivial solutions with suitable asymptotic properties as | x | ,, (,standing waves'). Copyright © 2006 John Wiley & Sons, Ltd. [source]

A random evolution related to a Fisher,Wright,Moran model with mutation, recombination and drift

Adam Bobrowski
Abstract The paper deals with a model of the genetic process of recombination, one of the basis mechanisms of generating genetic variability. Mathematically, the model can be represented by the so-called random evolution of Griego and Hersch, in which a random switching process selects from among several possible modes of operation of a dynamical system. The model, introduced by Polanska and Kimmel, involves mutations in the form of a time-continuous Markov chain and genetic drift. We demonstrate asymptotic properties of the model under different demographic scenarios for the population in which the process evolves. Copyright © 2003 John Wiley & Sons, Ltd. [source]

The Milne problem for the linear Fokker,Planck operator with a force term

Brigitte Lucquin-Desreux
Abstract This paper deals with the mathematical analysis of the linear stationary Fokker,Planck equation in a half-space (also called ,Milne' problem), in presence of an external electrostatic force field. We prove existence, uniqueness and asymptotic properties of the solution. Copyright © 2003 John Wiley & Sons, Ltd. [source]

On a linear differential equation with a proportional delay

Abstract This paper deals with the delay differential equation We impose some growth conditions on c, under which we are able to give a precise description of the asymptotic properties of all solutions of this equation. Although we naturally have to distinguish the cases c eventually positive and c eventually negative, we show a certain resemblance between the asymptotic formulae corresponding to both cases. Moreover, using the transformation approach we generalize these results to the equation with a general form of a delay. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Nonparametric covariate adjustment for receiver operating characteristic curves

Fang Yao
Abstract The accuracy of a diagnostic test is typically characterized using the receiver operating characteristic (ROC) curve. Summarizing indexes such as the area under the ROC curve (AUC) are used to compare different tests as well as to measure the difference between two populations. Often additional information is available on some of the covariates which are known to influence the accuracy of such measures. The authors propose nonparametric methods for covariate adjustment of the AUC. Models with normal errors and possibly non-normal errors are discussed and analyzed separately. Nonparametric regression is used for estimating mean and variance functions in both scenarios. In the model that relaxes the assumption of normality, the authors propose a covariate-adjusted Mann,Whitney estimator for AUC estimation which effectively uses available data to construct working samples at any covariate value of interest and is computationally efficient for implementation. This provides a generalization of the Mann,Whitney approach for comparing two populations by taking covariate effects into account. The authors derive asymptotic properties for the AUC estimators in both settings, including asymptotic normality, optimal strong uniform convergence rates and mean squared error (MSE) consistency. The MSE of the AUC estimators was also assessed in smaller samples by simulation. Data from an agricultural study were used to illustrate the methods of analysis. The Canadian Journal of Statistics 38:27,46; 2010 © 2009 Statistical Society of Canada La précision d'un test diagnostique est habituellement établie en utilisant les courbes caracté-ristiques de fonctionnement du récepteur (« ROC »). Des statistiques telles que l'aire sous la courbe ROC (« AUC ») sont utilisées afin de comparer différents tests et pour mesurer la différence entre deux populations. Souvent de l'information supplémentaire est disponible sur quelques covariables dont l'influence sur de telles statistiques est connue. Les auteurs suggèrent des méthodes non paramétriques afin d'ajuster la statistique AUC pour prendre en compte les covariables. Des modèles avec des erreurs gaussiennes et même non gaussiennes sont présentés et analysés séparément. Une régression non paramétrique est utilisée afin d'estimer les fonctions moyenne et variance dans les deux scénarios. Pour le modèle sans l'hypothèse de normalité, les auteurs proposent un estimateur de Mann-Whithney tenant compte des covariables pour l'AUC qui utilise l'information disponible dans les données afin de construire des échantillons d'analyse pour n'importe quelle valeur des covariables. Cet estimateur est implanté, car il est calculable de façon efficace. Il généralise l'approche de Mann-Whitney pour comparer deux populations en considérant l'effet des covariables. Les auteurs obtiennent les propriétés asymptotiques des estimateurs AUC pour les deux scénarios incluant la normalité asymptotique, les vitesses optimales de convergence uniforme forte et la convergence en erreur quadratique moyenne (« MSE »). Le MSE de l'estimateur de l'AUC est aussi étudié pour les petits échantillons à l'aide de simulations. Des données provenant d'une étude dans le domaine agricole sont utilisées afin d'illustrer les méthodes d'analyse. La revue canadienne de statistique 38: 27,46; 2010 © 2009 Sociètè statistique du Canada [source]

Bootstrapping data with multiple levels of variation

Christopher A. Field
Abstract The authors consider general estimators for the mean and variance parameters in the random effect model and in the transformation model for data with multiple levels of variation. They show that these estimators have different distributions under the two models unless all the variables have Gaussian distributions. They investigate the asymptotic properties of bootstrap procedures designed for the two models. They also report simulation results and illustrate the bootstraps using data on red spruce trees. Rééchantillonnage de données à variation multiniveau Les auteurs s'intéressent à des estimateurs généraux des paramètres de moyenne et variance dans le modèle à effets aléatoires et le modèle de transformation pour des données à variation multiniveau. Ils montrent que la loi de ces estimateurs dépend du modèle sauf si toutes les variables sont gaussiennes. Ils explorent les propriétés asymptotiques de procédures bootstrap propres aux deux modèles. Ils présentent des résultats de simulation et illustrent l'emploi de ces bootstraps à l'aide de données sur l'épinette rouge. [source]

Doubly adaptive biased coin designs with delayed responses

Feifang Hu
Abstract In clinical studies, patients are usually accrued sequentially. Response-adaptive designs are then useful tools for assigning treatments to incoming patients as a function of the treatment responses observed thus far. In this regard, doubly adaptive biased coin designs have advantageous properties under the assumption that their responses can be obtained immediately after testing. However, it is a common occurrence that responses are observed only after a certain period of time. The authors examine the effect of delayed responses on doubly adaptive biased coin designs and derive some of their asymptotic properties. It turns out that these designs are relatively insensitive to delayed responses under widely satisfied conditions. This is illustrated with a simulation study. Plans aléatoires non équilibrés doublement adaptatifs avec délais de réponse Dans les études cliniques, les patients arrivent souvent séquentiellement. Aussi souhaite-t-on disposer de plans adaptatifs permettant d'assigner des traitements aux nouveaux patients au vu des résultats observés jusque-là. À cet égard, les plans aléatoires non équilibrés doublement adaptatifs sont avantageux dans la mesure où un test permet d'observer sans délai l'effet d'un traitement. Toutefois, il arrive aussi qu'un traitement ne fasse effet qu'après un temps. Les auteurs examinent les répercussions de tels délais sur les plans aléatoires non équilibrés doublement adaptatifs, dont ils dégagent certaines propriétés asymptotiques. Il s'avère que sous des conditions assez générales, ces plans sont plutôt insensibles aux délais. Une étude de simulation illustre le propos. [source]

Nonresponse weighting adjustment using estimated response probability

Jae Kwang Kim
Abstract To reduce nonresponse bias in sample surveys, a method of nonresponse weighting adjustment is often used which consists of multiplying the sampling weight of the respondent by the inverse of the estimated response probability. The authors examine the asymptotic properties of this estimator. They prove that it is generally more efficient than an estimator which uses the true response probability, provided that the parameters which govern this probability are estimated by maximum likelihood. The authors discuss variance estimation methods that account for the effect of using the estimated response probability; they compare their performances in a small simulation study. They also discuss extensions to the regression estimator. Correction de la non-réponse par repondération au moyen d'une estimation de la probabilité de réponse Pour réduire le biais d, à la non-réponse dans les enqu,tes, on fait souvent appel à une méthode d'ajustement dans laquelle le poids de sondage de chaque répondant est multiplié par l'inverse d'une estimation de la probabilité de réponse. Les auteurs étudient les propriétés asymptotiques de cet estimateur. Ils démontrent qu'il est généralement plus efficace que celui qui fait intervenir la probabilité de réponse théorique, pourvu que les paramètres qui régissent cette probabilité soient estimés par vraisemblance maximale. Les auteurs évoquent diverses méthodes d'estimation de la variance qui tiennent compte du fait que la probabilité de réponse est estimée; ils en comparent la performance dans le cadre d'une petite étude de simulation. Ils étendent aussi leurs résultats à l'estimateur obtenu par régression. [source]

Smoothness adaptive average derivative estimation

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


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