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Confidence Regions (confidence + regions)
Selected AbstractsEstimation and Confidence Regions for Parameter Sets in Econometric Models,ECONOMETRICA, Issue 5 2007Victor 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] Sample Splitting and Threshold EstimationECONOMETRICA, Issue 3 2000Bruce E. Hansen Threshold models have a wide variety of applications in economics. Direct applications include models of separating and multiple equilibria. Other applications include empirical sample splitting when the sample split is based on a continuously-distributed variable such as firm size. In addition, threshold models may be used as a parsimonious strategy for nonparametric function estimation. For example, the threshold autoregressive model (TAR) is popular in the nonlinear time series literature. Threshold models also emerge as special cases of more complex statistical frameworks, such as mixture models, switching models, Markov switching models, and smooth transition threshold models. It may be important to understand the statistical properties of threshold models as a preliminary step in the development of statistical tools to handle these more complicated structures. Despite the large number of potential applications, the statistical theory of threshold estimation is undeveloped. It is known that threshold estimates are super-consistent, but a distribution theory useful for testing and inference has yet to be provided. This paper develops a statistical theory for threshold estimation in the regression context. We allow for either cross-section or time series observations. Least squares estimation of the regression parameters is considered. An asymptotic distribution theory for the regression estimates (the threshold and the regression slopes) is developed. It is found that the distribution of the threshold estimate is nonstandard. A method to construct asymptotic confidence intervals is developed by inverting the likelihood ratio statistic. It is shown that this yields asymptotically conservative confidence regions. Monte Carlo simulations are presented to assess the accuracy of the asymptotic approximations. The empirical relevance of the theory is illustrated through an application to the multiple equilibria growth model of Durlauf and Johnson (1995). [source] A Three-step Method for Choosing the Number of Bootstrap RepetitionsECONOMETRICA, Issue 1 2000Donald W. K. Andrews This paper considers the problem of choosing the number of bootstrap repetitions B for bootstrap standard errors, confidence intervals, confidence regions, hypothesis tests, p -values, and bias correction. For each of these problems, the paper provides a three-step method for choosing B to achieve a desired level of accuracy. Accuracy is measured by the percentage deviation of the bootstrap standard error estimate, confidence interval length, test's critical value, test's p -value, or bias-corrected estimate based on B bootstrap simulations from the corresponding ideal bootstrap quantities for which B=,. The results apply quite generally to parametric, semiparametric, and nonparametric models with independent and dependent data. The results apply to the standard nonparametric iid bootstrap, moving block bootstraps for time series data, parametric and semiparametric bootstraps, and bootstraps for regression models based on bootstrapping residuals. Monte Carlo simulations show that the proposed methods work very well. [source] Non-parametric tests and confidence regions for intrinsic diversity profiles of ecological populationsENVIRONMETRICS, Issue 8 2003Tonio Di Battista Abstract Evaluation of diversity profiles is useful for ecologists to quantify the diversity of biological communities. Measures of diversity profile can be expressed as a function of the unknown abundance vector. Thus, the estimators and related confidence regions and tests of hypotheses involve aspects of multivariate analysis. In this setting, using a suitable sampling design, inference is developed assuming an asymptotic specific distribution of the profile estimator. However, in a biological framework, ecologists work with small sample sizes, and the use of any probability distribution is hazardous. Assuming that a sample belongs to the family of replicated sampling design, we show that the diversity profile estimator can be expressed as a linear combination of the ranked abundance vector estimators. Hence we are able to develop a non-parametric approach based on a bootstrap in order to build balanced simultaneous confidence sets and tests of hypotheses for diversity profiles. Finally, the proposed procedure is applied on the avian populations of four parks in Milan, Italy. Copyright © 2003 John Wiley & Sons, Ltd. [source] EXPERIMENTAL EVIDENCE FOR MULTIVARIATE STABILIZING SEXUAL SELECTIONEVOLUTION, Issue 4 2005Robert Brooks Abstract Stabilizing selection is a fundamental concept in evolutionary biology. In the presence of a single intermediate optimum phenotype (fitness peak) on the fitness surface, stabilizing selection should cause the population to evolve toward such a peak. This prediction has seldom been tested, particularly for suites of correlated traits. The lack of tests for an evolutionary match between population means and adaptive peaks may be due, at least in part, to problems associated with empirically detecting multivariate stabilizing selection and with testing whether population means are at the peak of multivariate fitness surfaces. Here we show how canonical analysis of the fitness surface, combined with the estimation of confidence regions for stationary points on quadratic response surfaces, may be used to define multivariate stabilizing selection on a suite of traits and to establish whether natural populations reside on the multivariate peak. We manufactured artificial advertisement calls of the male cricket Teleogryllus commodus and played them back to females in laboratory phonotaxis trials to estimate the linear and nonlinear sexual selection that female phonotactic choice imposes on male call structure. Significant nonlinear selection on the major axes of the fitness surface was convex in nature and displayed an intermediate optimum, indicating multivariate stabilizing selection. The mean phenotypes of four independent samples of males, from the same population as the females used in phonotaxis trials, were within the 95% confidence region for the fitness peak. These experiments indicate that stabilizing sexual selection may play an important role in the evolution of male call properties in natural populations of T. commodus. [source] Quantifying the Effects of Mask Metadata Disclosure and Multiple Releases on the Confidentiality of Geographically Masked Health DataGEOGRAPHICAL ANALYSIS, Issue 1 2008Dale L. Zimmerman The availability of individual-level health data presents opportunities for monitoring the distribution and spread of emergent, acute, and chronic conditions, as well as challenges with respect to maintaining the anonymity of persons with health conditions. Particularly when such data are mapped as point locations, concerns arise regarding the ease with which individual identities may be determined by linking geographic coordinates to digital street networks, then determining residential addresses and, finally, names of occupants at specific addresses. The utility of such data sets must therefore be balanced against the requirements of protecting the confidentiality of individuals whose identities might be revealed through the availability of precise and accurate locational data. Recent literature has pointed toward geographic masking as a means for striking an appropriate balance between data utility and confidentiality. However, questions remain as to whether certain characteristics of the mask (mask metadata) should be disclosed to data users and whether two or more distinct masked versions of the data can be released without breaching confidentiality. In this article, we address these questions by quantifying the extent to which the disclosure of mask metadata and the release of multiple masked versions may affect confidentiality, with a view toward providing guidance to custodians of health data sets. The masks considered include perturbation, areal aggregation, and their combination. Confidentiality is measured by the areas of confidence regions for individuals' locations, which are derived under the probability models governing the masks, conditioned on the disclosed mask metadata. [source] Design for model parameter uncertainty using nonlinear confidence regionsAICHE JOURNAL, Issue 8 2001William C. Rooney An accurate method presented accounts for uncertain model parameters in nonlinear process optimization problems. The model representation is considered in terms of algebraic equations. Uncertain quantity parameters are often discretized into a number of finite values that are then used in multiperiod optimization problems. These discrete values usually range between some lower and upper bound that can be derived from individual confidence intervals. Frequently, more than one uncertain parameter is estimated at a time from a single set of experiments. Thus, using simple lower and upper bounds to describe these parameters may not be accurate, since it assumes the parameters are uncorrelated. In 1999 Rooney and Biegler showed the importance of including parameter correlation in design problems by using elliptical joint confidence regions to describe the correlation among the uncertain model parameters. In chemical engineering systems, however, the parameter estimation problem is often highly nonlinear, and the elliptical confidence regions derived from these problems may not be accurate enough to capture the actual model parameter uncertainty. In this work, the description of model parameter uncertainty is improved by using confidence regions derived from the likelihood ratio test. It captures the nonlinearities efficiently and accurately in the parameter estimation problem. Several examples solved show the importance of accurately capturing the actual model parameter uncertainty at the design stage. [source] Sensitivity analysis for incomplete contingency tables: the Slovenian plebiscite caseJOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES C (APPLIED STATISTICS), Issue 1 2001Geert Molenberghs Classical inferential procedures induce conclusions from a set of data to a population of interest, accounting for the imprecision resulting from the stochastic component of the model. Less attention is devoted to the uncertainty arising from (unplanned) incompleteness in the data. Through the choice of an identifiable model for non-ignorable non-response, one narrows the possible data-generating mechanisms to the point where inference only suffers from imprecision. Some proposals have been made for assessing the sensitivity to these modelling assumptions; many are based on fitting several plausible but competing models. For example, we could assume that the missing data are missing at random in one model, and then fit an additional model where non-random missingness is assumed. On the basis of data from a Slovenian plebiscite, conducted in 1991, to prepare for independence, it is shown that such an ad hoc procedure may be misleading. We propose an approach which identifies and incorporates both sources of uncertainty in inference: imprecision due to finite sampling and ignorance due to incompleteness. A simple sensitivity analysis considers a finite set of plausible models. We take this idea one step further by considering more degrees of freedom than the data support. This produces sets of estimates (regions of ignorance) and sets of confidence regions (combined into regions of uncertainty). [source] Estimation of the expected ROCOF of a repairable system with bootstrap confidence regionQUALITY AND RELIABILITY ENGINEERING INTERNATIONAL, Issue 3 2001M. J. Phillips Abstract Bootstrap methods are presented for constructing confidence regions for the expected ROCOF of a repairable system. This is based on the work of Cowling et al. (Journal of the American Statistical Association 1996; 91: 1516,1524) for the intensity function of a NHPP. The method is applied to the failure times of a photocopier given by Baker (Technometrics 1996; 38: 256,265). Copyright © 2001 John Wiley & Sons, Ltd. [source] Estimation methods for time-dependent AUC models with survival dataTHE CANADIAN JOURNAL OF STATISTICS, Issue 1 2010Hung Hung Abstract The performance of clinical tests for disease screening is often evaluated using the area under the receiver-operating characteristic (ROC) curve (AUC). Recent developments have extended the traditional setting to the AUC with binary time-varying failure status. Without considering covariates, our first theme is to propose a simple and easily computed nonparametric estimator for the time-dependent AUC. Moreover, we use generalized linear models with time-varying coefficients to characterize the time-dependent AUC as a function of covariate values. The corresponding estimation procedures are proposed to estimate the parameter functions of interest. The derived limiting Gaussian processes and the estimated asymptotic variances enable us to construct the approximated confidence regions for the AUCs. The finite sample properties of our proposed estimators and inference procedures are examined through extensive simulations. An analysis of the AIDS Clinical Trials Group (ACTG) 175 data is further presented to show the applicability of the proposed methods. The Canadian Journal of Statistics 38:8,26; 2010 © 2009 Statistical Society of Canada La performance des tests cliniques pour le dépistage de maladie est souvent évaluée en utilisant l'aire sous la courbe caractéristique de fonctionnements du récepteur (, ROC , ), notée , AUC , . Des développements récents ont généralisé le cadre traditionnel à l'AUC avec un statut de panne binaire variant dans le temps. Sans considérer les covariables, nous commençons par proposer un estimateur non paramétrique pour l'AUC simple et facile à calculer. De plus, nous utilisons des modèles linéaires généralisés avec des coefficients dépendant du temps pour caractériser les AUC, dépendant du temps, comme fonction des covariables. Les procédures d'estimation asociées correspondantes sont proposées afin d'estimer les fonctions paramètres d'intérêt. Les processus gaussiens limites sont obtenus ainsi que les variances asymptotiques estimées afin de construire des régions de confiance approximatives pour les AUC. À l'aide de nombreuses simulations, les propriétés pour de petits échantillons des estimateurs proposés et des procédures d'inférence sont étudiées. Une analyse du groupe d'essais cliniques sur le sida 175 (ACTG 175) est aussi présentée afin de montrer l'applicabilité des méthodes proposées. La revue canadienne de statistique 38: 8,26; 2010 © 2009 Société statistique du Canada [source] Empirical likelihood confidence regions for comparison distributions and roc curvesTHE CANADIAN JOURNAL OF STATISTICS, Issue 2 2003Gerda Claeskens Abstract Abstract: The authors derive empirical likelihood confidence regions for the comparison distribution of two populations whose distributions are to be tested for equality using random samples. Another application they consider is to ROC curves, which are used to compare measurements of a diagnostic test from two populations. The authors investigate the smoothed empirical likelihood method for estimation in this context, and empirical likelihood based confidence intervals are obtained by means of the Wilks theorem. A bootstrap approach allows for the construction of confidence bands. The method is illustrated with data analysis and a simulation study. Résumé: Les auteurs déduisent de la vraisemblance empirique des régions de confiance pour la distribution comparée de deux populations dont on veut tester l'égalité en loi au moyen d'échantillons aléatoires. Une autre application qu'ils considèrent concerne les courbes ROC, qui permettent de comparer les résultats d'un test diagnostique effectué auprès de deux populations. L'estimation proposée par les auteurs dans ce contexte s'appuie sur une méthode de lissage de la vraisemblance empirique qui conduit, gr,ce au théorème de Wilks, aux intervalles de confiance recherchés. Une approche bootstrap permet en outre de construire des bandes de confiance. La méthode est illustrée au moyen de simulations et d'un jeu de données. [source] Empirical likelihood for linear regression models under imputation for missing responsesTHE CANADIAN JOURNAL OF STATISTICS, Issue 4 2001Qihua Wang Abstract The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log-likelihood ratio is asymptotically a weighted sum of chi-square variables with unknown weights. They obtain an adjusted empirical log-likelihood ratio which is asymptotically standard chi-square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log-likelihood ratio and use its distribution to approximate that of the empirical log-likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method. [source] NON-SYMMETRICAL CORRESPONDENCE ANALYSIS WITH CONCATENATION AND LINEAR CONSTRAINTSAUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 1 2010Eric J. Beh Summary Correspondence analysis is a popular statistical technique used to identify graphically the presence, and structure, of association between two or more cross-classified categorical variables. Such a procedure is very useful when it is known that there is a symmetric (two-way) relationship between the variables. When such a relationship is known not to exist, non-symmetrical correspondence analysis is more appropriate as a method of establishing the source of association. This paper highlights some tools that can be used to explore the behaviour of asymmetric categorical variables. These tools consist of confidence regions, the link between non-symmetrical correspondence analysis and the analysis of variance of categorical variables, and the effect of imposing linear constraints. We also explore the application of non-symmetrical correspondence analysis to three-way contingency tables. [source] | ||||||||||