Extremum estimators and stochastic optimization methods

Submitted in partial fulfillment for the Requirements for the Degree of PhD in Mathematics, in the Speciality of Statistics in the Faculdade de Ciências e Tecnologia

Hausman tests for inefficient estimators: application to demand for health care service (revised)

The Hausman (1978) test is based on the vector of differences of two estimators. It is usually assumed that one of the estimators is fully efficient, since this simplifies calculation of the test statistic. However, this assumption limits the applicability of the test, since widely used estimators such as the generalized method of moments (GMM) or quasi maximum likelihood (QML) are often not fully efficient. This paper shows that the test may easily be implemented, using well-known methods, when neither estimator is efficient. To illustrate, we present both simulation results as well as empirical results for utilization of health care services.

Inference with the lognormal distribution

Several estimators of the expectation, median and mode of the lognormal distribution are derived. They aim to be approximately unbiased, efficient, or have a minimax property in the class of estimators we introduce. The small-sample properties of these estimators are assessed by simulations and, when possible, analytically. Some of these estimators of the expectation are far more efficient than the maximum likelihood or the minimum-variance unbiased estimator, even for substantial samplesizes.

Robust accelerated failure time regression

Robust estimators for accelerated failure time models with asymmetric (or symmetric) error distribution and censored observations are proposed. It is assumed that the error model belongs to a log-location-scale family of distributions and that the mean response is the parameter of interest. Since scale is a main component of mean, scale is not treated as a nuisance parameter. A three steps procedure is proposed. In the first step, an initial high breakdown point S estimate is computed. In the second step, observations that are unlikely under the estimated model are rejected or down weighted. Finally, a weighted maximum likelihood estimate is computed. To define the estimates, functions of censored residuals are replaced by their estimated conditional expectation given that the response is larger than the observed censored value. The rejection rule in the second step is based on an adaptive cut-off that, asymptotically, does not reject any observation when the data are generat ed according to the model. Therefore, the final estimate attains full efficiency at the model, with respect to the maximum likelihood estimate, while maintaining the breakdown point of the initial estimator. Asymptotic results are provided. The new procedure is evaluated with the help of Monte Carlo simulations. Two examples with real data are discussed.

Biometrical analysis and selection of tetraploid progenies of Panicum maximum using mixed model methods

The objectives of this work were to estimate the genetic and phenotypic parameters and to predict the genetic and genotypic values of the selection candidates obtained from intraspecific crosses in Panicum maximum as well as the performance of the hybrid progeny of the existing and projected crosses. Seventy-nine intraspecific hybrids obtained from artificial crosses among five apomictic and three sexual autotetraploid individuals were evaluated in a clonal test with two replications and ten plants per plot. Green matter yield, total and leaf dry matter yields and leaf percentage were evaluated in five cuts per year during three years. Genetic parameters were estimated and breeding and genotypic values were predicted using the restricted maximum likelihood/best linear unbiased prediction procedure (REML/BLUP). The dominant genetic variance was estimated by adjusting the effect of full-sib families. Low magnitude individual narrow sense heritabilities (0.02-0.05), individual broad sense heritabilities (0.14-0.20) and repeatability measured on an individual basis (0.15-0.21) were obtained. Dominance effects for all evaluated characteristics indicated that breeding strategies that explore heterosis must be adopted. Less than 5% increase in the parameter repeatability was obtained for a three-year evaluation period and may be the criterion to determine the maximum number of years of evaluation to be adopted, without compromising gain per cycle of selection. The identification of hybrid candidates for future cultivars and of those that can be incorporated into the breeding program was based on the genotypic and breeding values, respectively. The prediction of the performance of the hybrid progeny, based on the breeding values of the progenitors, permitted the identification of the best crosses and indicated the best parents to use in crosses.

The Gaussian assumption in second-order estimation problems in digital communications

This paper deals with the goodness of the Gaussian assumption when designing second-order blind estimationmethods in the context of digital communications. The low- andhigh-signal-to-noise ratio (SNR) asymptotic performance of the maximum likelihood estimator—derived assuming Gaussiantransmitted symbols—is compared with the performance of the optimal second-order estimator, which exploits the actualdistribution of the discrete constellation. The asymptotic study concludes that the Gaussian assumption leads to the optimalsecond-order solution if the SNR is very low or if the symbols belong to a multilevel constellation such as quadrature-amplitudemodulation (QAM) or amplitude-phase-shift keying (APSK). On the other hand, the Gaussian assumption can yield importantlosses at high SNR if the transmitted symbols are drawn from a constant modulus constellation such as phase-shift keying (PSK)or continuous-phase modulations (CPM). These conclusions are illustrated for the problem of direction-of-arrival (DOA) estimation of multiple digitally-modulated signals.

On the inclusion of channel's time dependence in a hidden Markov model for blind channel estimation

In this paper, the theory of hidden Markov models (HMM) isapplied to the problem of blind (without training sequences) channel estimationand data detection. Within a HMM framework, the Baum–Welch(BW) identification algorithm is frequently used to find out maximum-likelihood (ML) estimates of the corresponding model. However, such a procedureassumes the model (i.e., the channel response) to be static throughoutthe observation sequence. By means of introducing a parametric model fortime-varying channel responses, a version of the algorithm, which is moreappropriate for mobile channels [time-dependent Baum-Welch (TDBW)] isderived. Aiming to compare algorithm behavior, a set of computer simulationsfor a GSM scenario is provided. Results indicate that, in comparisonto other Baum–Welch (BW) versions of the algorithm, the TDBW approachattains a remarkable enhancement in performance. For that purpose, onlya moderate increase in computational complexity is needed.

Joint array combining and MLSE for single-user receivers in multipath Gaussian multiuser channels

The well-known structure of an array combiner along with a maximum likelihood sequence estimator (MLSE) receiveris the basis for the derivation of a space-time processor presentinggood properties in terms of co-channel and intersymbol interferencerejection. The use of spatial diversity at the receiver front-endtogether with a scalar MLSE implies a joint design of the spatialcombiner and the impulse response for the sequence detector. Thisis faced using the MMSE criterion under the constraint that thedesired user signal power is not cancelled, yielding an impulse responsefor the sequence detector that is matched to the channel andcombiner response. The procedure maximizes the signal-to-noiseratio at the input of the detector and exhibits excellent performancein realistic multipath channels.

Waveform-independent frame-timing acquisition for UWB signals

In this paper, the problem of frame-level symboltiming acquisition for UWB signals is addressed. The main goalis the derivation of a frame-level timing estimator which does notrequire any prior knowledge of neither the transmitted symbolsnor the received template waveform. The independence withrespect to the received waveform is of special interest in UWBcommunication systems, where a fast and accurate estimation ofthe end-to-end channel response is a challenging and computationallydemanding task. The proposed estimator is derived under theunconditional maximum likelihood criterion, and because of thelow power of UWB signals, the low-SNR assumption is adopted. Asa result, an optimal frame-level timing estimator is derived whichoutperforms existing acquisition methods in low-SNR scenarios.

Second-order parameter estimation

This work provides a general framework for the design of second-order blind estimators without adopting anyapproximation about the observation statistics or the a prioridistribution of the parameters. The proposed solution is obtainedminimizing the estimator variance subject to some constraints onthe estimator bias. The resulting optimal estimator is found todepend on the observation fourth-order moments that can be calculatedanalytically from the known signal model. Unfortunately,in most cases, the performance of this estimator is severely limitedby the residual bias inherent to nonlinear estimation problems.To overcome this limitation, the second-order minimum varianceunbiased estimator is deduced from the general solution by assumingaccurate prior information on the vector of parameters.This small-error approximation is adopted to design iterativeestimators or trackers. It is shown that the associated varianceconstitutes the lower bound for the variance of any unbiasedestimator based on the sample covariance matrix.The paper formulation is then applied to track the angle-of-arrival(AoA) of multiple digitally-modulated sources by means ofa uniform linear array. The optimal second-order tracker is comparedwith the classical maximum likelihood (ML) blind methodsthat are shown to be quadratic in the observed data as well. Simulationshave confirmed that the discrete nature of the transmittedsymbols can be exploited to improve considerably the discriminationof near sources in medium-to-high SNR scenarios.

Inférence robuste à la présence des valeurs aberrantes dans les enquêtes

Cette thèse comporte trois articles dont un est publié et deux en préparation. Le sujet central de la thèse porte sur le traitement des valeurs aberrantes représentatives dans deux aspects importants des enquêtes que sont : l’estimation des petits domaines et l’imputation en présence de non-réponse partielle. En ce qui concerne les petits domaines, les estimateurs robustes dans le cadre des modèles au niveau des unités ont été étudiés. Sinha & Rao (2009) proposent une version robuste du meilleur prédicteur linéaire sans biais empirique pour la moyenne des petits domaines. Leur estimateur robuste est de type «plugin», et à la lumière des travaux de Chambers (1986), cet estimateur peut être biaisé dans certaines situations. Chambers et al. (2014) proposent un estimateur corrigé du biais. En outre, un estimateur de l’erreur quadratique moyenne a été associé à ces estimateurs ponctuels. Sinha & Rao (2009) proposent une procédure bootstrap paramétrique pour estimer l’erreur quadratique moyenne. Des méthodes analytiques sont proposées dans Chambers et al. (2014). Cependant, leur validité théorique n’a pas été établie et leurs performances empiriques ne sont pas pleinement satisfaisantes. Ici, nous examinons deux nouvelles approches pour obtenir une version robuste du meilleur prédicteur linéaire sans biais empirique : la première est fondée sur les travaux de Chambers (1986), et la deuxième est basée sur le concept de biais conditionnel comme mesure de l’influence d’une unité de la population. Ces deux classes d’estimateurs robustes des petits domaines incluent également un terme de correction pour le biais. Cependant, ils utilisent tous les deux l’information disponible dans tous les domaines contrairement à celui de Chambers et al. (2014) qui utilise uniquement l’information disponible dans le domaine d’intérêt. Dans certaines situations, un biais non négligeable est possible pour l’estimateur de Sinha & Rao (2009), alors que les estimateurs proposés exhibent un faible biais pour un choix approprié de la fonction d’influence et de la constante de robustesse. Les simulations Monte Carlo sont effectuées, et les comparaisons sont faites entre les estimateurs proposés et ceux de Sinha & Rao (2009) et de Chambers et al. (2014). Les résultats montrent que les estimateurs de Sinha & Rao (2009) et de Chambers et al. (2014) peuvent avoir un biais important, alors que les estimateurs proposés ont une meilleure performance en termes de biais et d’erreur quadratique moyenne. En outre, nous proposons une nouvelle procédure bootstrap pour l’estimation de l’erreur quadratique moyenne des estimateurs robustes des petits domaines. Contrairement aux procédures existantes, nous montrons formellement la validité asymptotique de la méthode bootstrap proposée. Par ailleurs, la méthode proposée est semi-paramétrique, c’est-à-dire, elle n’est pas assujettie à une hypothèse sur les distributions des erreurs ou des effets aléatoires. Ainsi, elle est particulièrement attrayante et plus largement applicable. Nous examinons les performances de notre procédure bootstrap avec les simulations Monte Carlo. Les résultats montrent que notre procédure performe bien et surtout performe mieux que tous les compétiteurs étudiés. Une application de la méthode proposée est illustrée en analysant les données réelles contenant des valeurs aberrantes de Battese, Harter & Fuller (1988). S’agissant de l’imputation en présence de non-réponse partielle, certaines formes d’imputation simple ont été étudiées. L’imputation par la régression déterministe entre les classes, qui inclut l’imputation par le ratio et l’imputation par la moyenne sont souvent utilisées dans les enquêtes. Ces méthodes d’imputation peuvent conduire à des estimateurs imputés biaisés si le modèle d’imputation ou le modèle de non-réponse n’est pas correctement spécifié. Des estimateurs doublement robustes ont été développés dans les années récentes. Ces estimateurs sont sans biais si l’un au moins des modèles d’imputation ou de non-réponse est bien spécifié. Cependant, en présence des valeurs aberrantes, les estimateurs imputés doublement robustes peuvent être très instables. En utilisant le concept de biais conditionnel, nous proposons une version robuste aux valeurs aberrantes de l’estimateur doublement robuste. Les résultats des études par simulations montrent que l’estimateur proposé performe bien pour un choix approprié de la constante de robustesse.

De lo micro a lo macro prudencial: Un análisis de la supervisión bancaria desde la econometría espacial

La crisis que se desató en el mercado hipotecario en Estados Unidos en 2008 y que logró propagarse a lo largo de todo sistema financiero, dejó en evidencia el nivel de interconexión que actualmente existe entre las entidades del sector y sus relaciones con el sector productivo, dejando en evidencia la necesidad de identificar y caracterizar el riesgo sistémico inherente al sistema, para que de esta forma las entidades reguladoras busquen una estabilidad tanto individual, como del sistema en general. El presente documento muestra, a través de un modelo que combina el poder informativo de las redes y su adecuación a un modelo espacial auto regresivo (tipo panel), la importancia de incorporar al enfoque micro-prudencial (propuesto en Basilea II), una variable que capture el efecto de estar conectado con otras entidades, realizando así un análisis macro-prudencial (propuesto en Basilea III).

A capture–recapture approach for screening using two diagnostic tests with availability of disease status for the test positives only

The article considers screening human populations with two screening tests. If any of the two tests is positive, then full evaluation of the disease status is undertaken; however, if both diagnostic tests are negative, then disease status remains unknown. This procedure leads to a data constellation in which, for each disease status, the 2 × 2 table associated with the two diagnostic tests used in screening has exactly one empty, unknown cell. To estimate the unobserved cell counts, previous approaches assume independence of the two diagnostic tests and use specific models, including the special mixture model of Walter or unconstrained capture–recapture estimates. Often, as is also demonstrated in this article by means of a simple test, the independence of the two screening tests is not supported by the data. Two new estimators are suggested that allow associations of the screening test, although the form of association must be assumed to be homogeneous over disease status. These estimators are modifications of the simple capture–recapture estimator and easy to construct. The estimators are investigated for several screening studies with fully evaluated disease status in which the superior behavior of the new estimators compared to the previous conventional ones can be shown. Finally, the performance of the new estimators is compared with maximum likelihood estimators, which are more difficult to obtain in these models. The results indicate the loss of efficiency as minor.

Asymptotic normality in mixtures of power series distributions

The problem of estimating the individual probabilities of a discrete distribution is considered. The true distribution of the independent observations is a mixture of a family of power series distributions. First, we ensure identifiability of the mixing distribution assuming mild conditions. Next, the mixing distribution is estimated by non-parametric maximum likelihood and an estimator for individual probabilities is obtained from the corresponding marginal mixture density. We establish asymptotic normality for the estimator of individual probabilities by showing that, under certain conditions, the difference between this estimator and the empirical proportions is asymptotically negligible. Our framework includes Poisson, negative binomial and logarithmic series as well as binomial mixture models. Simulations highlight the benefit in achieving normality when using the proposed marginal mixture density approach instead of the empirical one, especially for small sample sizes and/or when interest is in the tail areas. A real data example is given to illustrate the use of the methodology.

A likelihood ratio appropach to family-based association studies with covariates

We introduce a procedure for association based analysis of nuclear families that allows for dichotomous and more general measurements of phenotype and inclusion of covariate information. Standard generalized linear models are used to relate phenotype and its predictors. Our test procedure, based on the likelihood ratio, unifies the estimation of all parameters through the likelihood itself and yields maximum likelihood estimates of the genetic relative risk and interaction parameters. Our method has advantages in modelling the covariate and gene-covariate interaction terms over recently proposed conditional score tests that include covariate information via a two-stage modelling approach. We apply our method in a study of human systemic lupus erythematosus and the C-reactive protein that includes sex as a covariate.