INFERENCES FOR THE CHANGE-POINT OF THE EXPONENTIATED WEIBULL HAZARD FUNCTION

In many applications of lifetime data analysis, it is important to perform inferences about the change-point of the hazard function. The change-point could be a maximum for unimodal hazard functions or a minimum for bathtub forms of hazard functions and is usually of great interest in medical or industrial applications. For lifetime distributions where this change-point of the hazard function can be analytically calculated, its maximum likelihood estimator is easily obtained from the invariance properties of the maximum likelihood estimators. From the asymptotical normality of the maximum likelihood estimators, confidence intervals can also be obtained. Considering the exponentiated Weibull distribution for the lifetime data, we have different forms for the hazard function: constant, increasing, unimodal, decreasing or bathtub forms. This model gives great flexibility of fit, but we do not have analytic expressions for the change-point of the hazard function. In this way, we consider the use of Markov Chain Monte Carlo methods to get posterior summaries for the change-point of the hazard function considering the exponentiated Weibull distribution.

The Truncated Inflated Beta Distribution

The study of proportions is a common topic in many fields of study. The standard beta distribution or the inflated beta distribution may be a reasonable choice to fit a proportion in most situations. However, they do not fit well variables that do not assume values in the open interval (0, c), 0 < c < 1. For these variables, the authors introduce the truncated inflated beta distribution (TBEINF). This proposed distribution is a mixture of the beta distribution bounded in the open interval (c, 1) and the trinomial distribution. The authors present the moments of the distribution, its scoring vector, and Fisher information matrix, and discuss estimation of its parameters. The properties of the suggested estimators are studied using Monte Carlo simulation. In addition, the authors present an application of the TBEINF distribution for unemployment insurance data.

Statistical problems related to excitation threshold and reset value of membrane potentials

Die vorliegende Arbeit ist motiviert durch biologische Fragestellungen bezüglich des Verhaltens von Membranpotentialen in Neuronen. Ein vielfach betrachtetes Modell für spikende Neuronen ist das Folgende. Zwischen den Spikes verhält sich das Membranpotential wie ein Diffusionsprozess X der durch die SDGL dX_t= beta(X_t) dt+ sigma(X_t) dB_t gegeben ist, wobei (B_t) eine Standard-Brown'sche Bewegung bezeichnet. Spikes erklärt man wie folgt. Sobald das Potential X eine gewisse Exzitationsschwelle S überschreitet entsteht ein Spike. Danach wird das Potential wieder auf einen bestimmten Wert x_0 zurückgesetzt. In Anwendungen ist es manchmal möglich, einen Diffusionsprozess X zwischen den Spikes zu beobachten und die Koeffizienten der SDGL beta() und sigma() zu schätzen. Dennoch ist es nötig, die Schwellen x_0 und S zu bestimmen um das Modell festzulegen. Eine Möglichkeit, dieses Problem anzugehen, ist x_0 und S als Parameter eines statistischen Modells aufzufassen und diese zu schätzen. In der vorliegenden Arbeit werden vier verschiedene Fälle diskutiert, in denen wir jeweils annehmen, dass das Membranpotential X zwischen den Spikes eine Brown'sche Bewegung mit Drift, eine geometrische Brown'sche Bewegung, ein Ornstein-Uhlenbeck Prozess oder ein Cox-Ingersoll-Ross Prozess ist. Darüber hinaus beobachten wir die Zeiten zwischen aufeinander folgenden Spikes, die wir als iid Treffzeiten der Schwelle S von X gestartet in x_0 auffassen. Die ersten beiden Fälle ähneln sich sehr und man kann jeweils den Maximum-Likelihood-Schätzer explizit angeben. Darüber hinaus wird, unter Verwendung der LAN-Theorie, die Optimalität dieser Schätzer gezeigt. In den Fällen OU- und CIR-Prozess wählen wir eine Minimum-Distanz-Methode, die auf dem Vergleich von empirischer und wahrer Laplace-Transformation bezüglich einer Hilbertraumnorm beruht. Wir werden beweisen, dass alle Schätzer stark konsistent und asymptotisch normalverteilt sind. Im letzten Kapitel werden wir die Effizienz der Minimum-Distanz-Schätzer anhand simulierter Daten überprüfen. Ferner, werden Anwendungen auf reale Datensätze und deren Resultate ausführlich diskutiert.

Smooth Estimation of a Monotone Density

We investigate the interplay of smoothness and monotonicity assumptions when estimating a density from a sample of observations. The nonparametric maximum likelihood estimator of a decreasing density on the positive half line attains a rate of convergence at a fixed point if the density has a negative derivative. The same rate is obtained by a kernel estimator, but the limit distributions are different. If the density is both differentiable and known to be monotone, then a third estimator is obtained by isotonization of a kernel estimator. We show that this again attains the rate of convergence and compare the limit distributors of the three types of estimators. It is shown that both isotonization and smoothing lead to a more concentrated limit distribution and we study the dependence on the proportionality constant in the bandwidth. We also show that isotonization does not change the limit behavior of a kernel estimator with a larger bandwidth, in the case that the density is known to have more than one derivative.

On the Accelerated Failure Time Model for Current Status and Interval Censored Data

This paper introduces a novel approach to making inference about the regression parameters in the accelerated failure time (AFT) model for current status and interval censored data. The estimator is constructed by inverting a Wald type test for testing a null proportional hazards model. A numerically efficient Markov chain Monte Carlo (MCMC) based resampling method is proposed to simultaneously obtain the point estimator and a consistent estimator of its variance-covariance matrix. We illustrate our approach with interval censored data sets from two clinical studies. Extensive numerical studies are conducted to evaluate the finite sample performance of the new estimators.

Crossing borders: Ethnic minorities matter? A meta analytic examionation of the alliance-outcome relationship

Objectives: The purpose of this meta analysis was to examine the moderating impact of substance use disorder as inclusion/exclusion criterion as well as the percentage of racial/ethnic minorities on the strength of the alliance-outcome relationship in psychotherapy. It was hypothesized that the presence of a dsm axis i substance use disorders as a criterion and the presence of racial/ethnic minority as a psychosocial indicator are confounded client factors reducing the relationship between alliance and outcome. Methods: A random effects restricted maximum-likelihood estimator was used for omnibus and moderator models (k = 94). results: the presence of (a) substance use disorder and, (b) racial/ethnic minorities (overall and specific to african americans) partially moderated the alliance-outcome correlation. The percentage of substance use disorders and racial/ethnic minority status was highly correlated. Conclusions: Socio-cultural contextual variables should be considered along with dsm axis i diagnosis of substance use disorders in analyzing and interpreting mechanisms of change.

Estimadores bayesianos de la fiabilidad con muestreo censurado

El estudio de la fiabilidad de componentes y sistemas tiene gran importancia en diversos campos de la ingenieria, y muy concretamente en el de la informatica. Al analizar la duracion de los elementos de la muestra hay que tener en cuenta los elementos que no fallan en el tiempo que dure el experimento, o bien los que fallen por causas distintas a la que es objeto de estudio. Por ello surgen nuevos tipos de muestreo que contemplan estos casos. El mas general de ellos, el muestreo censurado, es el que consideramos en nuestro trabajo. En este muestreo tanto el tiempo hasta que falla el componente como el tiempo de censura son variables aleatorias. Con la hipotesis de que ambos tiempos se distribuyen exponencialmente, el profesor Hurt estudio el comportamiento asintotico del estimador de maxima verosimilitud de la funcion de fiabilidad. En principio parece interesante utilizar metodos Bayesianos en el estudio de la fiabilidad porque incorporan al analisis la informacion a priori de la que se dispone normalmente en problemas reales. Por ello hemos considerado dos estimadores Bayesianos de la fiabilidad de una distribucion exponencial que son la media y la moda de la distribucion a posteriori. Hemos calculado la expansion asint6tica de la media, varianza y error cuadratico medio de ambos estimadores cuando la distribuci6n de censura es exponencial. Hemos obtenido tambien la distribucion asintotica de los estimadores para el caso m3s general de que la distribucion de censura sea de Weibull. Dos tipos de intervalos de confianza para muestras grandes se han propuesto para cada estimador. Los resultados se han comparado con los del estimador de maxima verosimilitud, y con los de dos estimadores no parametricos: limite producto y Bayesiano, resultando un comportamiento superior por parte de uno de nuestros estimadores. Finalmente nemos comprobado mediante simulacion que nuestros estimadores son robustos frente a la supuesta distribuci6n de censura, y que uno de los intervalos de confianza propuestos es valido con muestras pequenas. Este estudio ha servido tambien para confirmar el mejor comportamiento de uno de nuestros estimadores. SETTING OUT AND SUMMARY OF THE THESIS When we study the lifetime of components it's necessary to take into account the elements that don't fail during the experiment, or those that fail by reasons which are desirable to exclude from consideration. The model of random censorship is very usefull for analysing these data. In this model the time to failure and the time censor are random variables. We obtain two Bayes estimators of the reliability function of an exponential distribution based on randomly censored data. We have calculated the asymptotic expansion of the mean, variance and mean square error of both estimators, when the censor's distribution is exponential. We have obtained also the asymptotic distribution of the estimators for the more general case of censor's Weibull distribution. Two large-sample confidence bands have been proposed for each estimator. The results have been compared with those of the maximum likelihood estimator, and with those of two non parametric estimators: Product-limit and Bayesian. One of our estimators has the best behaviour. Finally we have shown by simulation, that our estimators are robust against the assumed censor's distribution, and that one of our intervals does well in small sample situation.

Regularization for sparsity in statistical analysis and machine learning

Pragmatism is the leading motivation of regularization. We can understand regularization as a modification of the maximum-likelihood estimator so that a reasonable answer could be given in an unstable or ill-posed situation. To mention some typical examples, this happens when fitting parametric or non-parametric models with more parameters than data or when estimating large covariance matrices. Regularization is usually used, in addition, to improve the bias-variance tradeoff of an estimation. Then, the definition of regularization is quite general, and, although the introduction of a penalty is probably the most popular type, it is just one out of multiple forms of regularization. In this dissertation, we focus on the applications of regularization for obtaining sparse or parsimonious representations, where only a subset of the inputs is used. A particular form of regularization, L1-regularization, plays a key role for reaching sparsity. Most of the contributions presented here revolve around L1-regularization, although other forms of regularization are explored (also pursuing sparsity in some sense). In addition to present a compact review of L1-regularization and its applications in statistical and machine learning, we devise methodology for regression, supervised classification and structure induction of graphical models. Within the regression paradigm, we focus on kernel smoothing learning, proposing techniques for kernel design that are suitable for high dimensional settings and sparse regression functions. We also present an application of regularized regression techniques for modeling the response of biological neurons. Supervised classification advances deal, on the one hand, with the application of regularization for obtaining a na¨ıve Bayes classifier and, on the other hand, with a novel algorithm for brain-computer interface design that uses group regularization in an efficient manner. Finally, we present a heuristic for inducing structures of Gaussian Bayesian networks using L1-regularization as a filter. El pragmatismo es la principal motivación de la regularización. Podemos entender la regularización como una modificación del estimador de máxima verosimilitud, de tal manera que se pueda dar una respuesta cuando la configuración del problema es inestable. A modo de ejemplo, podemos mencionar el ajuste de modelos paramétricos o no paramétricos cuando hay más parámetros que casos en el conjunto de datos, o la estimación de grandes matrices de covarianzas. Se suele recurrir a la regularización, además, para mejorar el compromiso sesgo-varianza en una estimación. Por tanto, la definición de regularización es muy general y, aunque la introducción de una función de penalización es probablemente el método más popular, éste es sólo uno de entre varias posibilidades. En esta tesis se ha trabajado en aplicaciones de regularización para obtener representaciones dispersas, donde sólo se usa un subconjunto de las entradas. En particular, la regularización L1 juega un papel clave en la búsqueda de dicha dispersión. La mayor parte de las contribuciones presentadas en la tesis giran alrededor de la regularización L1, aunque también se exploran otras formas de regularización (que igualmente persiguen un modelo disperso). Además de presentar una revisión de la regularización L1 y sus aplicaciones en estadística y aprendizaje de máquina, se ha desarrollado metodología para regresión, clasificación supervisada y aprendizaje de estructura en modelos gráficos. Dentro de la regresión, se ha trabajado principalmente en métodos de regresión local, proponiendo técnicas de diseño del kernel que sean adecuadas a configuraciones de alta dimensionalidad y funciones de regresión dispersas. También se presenta una aplicación de las técnicas de regresión regularizada para modelar la respuesta de neuronas reales. Los avances en clasificación supervisada tratan, por una parte, con el uso de regularización para obtener un clasificador naive Bayes y, por otra parte, con el desarrollo de un algoritmo que usa regularización por grupos de una manera eficiente y que se ha aplicado al diseño de interfaces cerebromáquina. Finalmente, se presenta una heurística para inducir la estructura de redes Bayesianas Gaussianas usando regularización L1 a modo de filtro.

Interval and Point Estimators for the Location Parameter of the Three-Parameter Lognormal Distribution

The three-parameter lognormal distribution is the extension of the two-parameter lognormal distribution to meet the need of the biological, sociological, and other fields. Numerous research papers have been published for the parameter estimation problems for the lognormal distributions. The inclusion of the location parameter brings in some technical difficulties for the parameter estimation problems, especially for the interval estimation. This paper proposes a method for constructing exact confidence intervals and exact upper confidence limits for the location parameter of the three-parameter lognormal distribution. The point estimation problem is discussed as well. The performance of the point estimator is compared with the maximum likelihood estimator, which is widely used in practice. Simulation result shows that the proposed method is less biased in estimating the location parameter. The large sample size case is discussed in the paper.

Asymptotic Theory for Extended Asymmetric Multivariate GARCH Processes

The paper considers various extended asymmetric multivariate conditional volatility models, and derives appropriate regularity conditions and associated asymptotic theory. This enables checking of internal consistency and allows valid statistical inferences to be drawn based on empirical estimation. For this purpose, we use an underlying vector random coefficient autoregressive process, for which we show the equivalent representation for the asymmetric multivariate conditional volatility model, to derive asymptotic theory for the quasi-maximum likelihood estimator. As an extension, we develop a new multivariate asymmetric long memory volatility model, and discuss the associated asymptotic properties.

Realized Matrix-Exponential Stochastic Volatility with Asymmetry, Long Memory and Spillovers

The paper develops a novel realized matrix-exponential stochastic volatility model of multivariate returns and realized covariances that incorporates asymmetry and long memory (hereafter the RMESV-ALM model). The matrix exponential transformation guarantees the positivedefiniteness of the dynamic covariance matrix. The contribution of the paper ties in with Robert Basmann’s seminal work in terms of the estimation of highly non-linear model specifications (“Causality tests and observationally equivalent representations of econometric models”, Journal of Econometrics, 1988, 39(1-2), 69–104), especially for developing tests for leverage and spillover effects in the covariance dynamics. Efficient importance sampling is used to maximize the likelihood function of RMESV-ALM, and the finite sample properties of the quasi-maximum likelihood estimator of the parameters are analysed. Using high frequency data for three US financial assets, the new model is estimated and evaluated. The forecasting performance of the new model is compared with a novel dynamic realized matrix-exponential conditional covariance model. The volatility and co-volatility spillovers are examined via the news impact curves and the impulse response functions from returns to volatility and co-volatility.

Statistical modeling of protein lysate array data

The protein lysate array is an emerging technology for quantifying the protein concentration ratios in multiple biological samples. It is gaining popularity, and has the potential to answer questions about post-translational modifications and protein pathway relationships. Statistical inference for a parametric quantification procedure has been inadequately addressed in the literature, mainly due to two challenges: the increasing dimension of the parameter space and the need to account for dependence in the data. Each chapter of this thesis addresses one of these issues. In Chapter 1, an introduction to the protein lysate array quantification is presented, followed by the motivations and goals for this thesis work. In Chapter 2, we develop a multi-step procedure for the Sigmoidal models, ensuring consistent estimation of the concentration level with full asymptotic efficiency. The results obtained in this chapter justify inferential procedures based on large-sample approximations. Simulation studies and real data analysis are used to illustrate the performance of the proposed method in finite-samples. The multi-step procedure is simpler in both theory and computation than the single-step least squares method that has been used in current practice. In Chapter 3, we introduce a new model to account for the dependence structure of the errors by a nonlinear mixed effects model. We consider a method to approximate the maximum likelihood estimator of all the parameters. Using the simulation studies on various error structures, we show that for data with non-i.i.d. errors the proposed method leads to more accurate estimates and better confidence intervals than the existing single-step least squares method.

Bayesian Estimation of Small Proportions Using Binomial Group Test

Group testing has long been considered as a safe and sensible relative to one-at-a-time testing in applications where the prevalence rate p is small. In this thesis, we applied Bayes approach to estimate p using Beta-type prior distribution. First, we showed two Bayes estimators of p from prior on p derived from two different loss functions. Second, we presented two more Bayes estimators of p from prior on π according to two loss functions. We also displayed credible and HPD interval for p. In addition, we did intensive numerical studies. All results showed that the Bayes estimator was preferred over the usual maximum likelihood estimator (MLE) for small p. We also presented the optimal β for different p, m, and k.