Asymptotic efficiency in an instrumental variable model

Esta dissertação se propõe ao estudo de inferência usando estimação por método generalizado dos momentos (GMM) baseado no uso de instrumentos. A motivação para o estudo está no fato de que sob identificação fraca dos parâmetros, a inferência tradicional pode levar a resultados enganosos. Dessa forma, é feita uma revisão dos mais usuais testes para superar tal problema e uma apresentação dos arcabouços propostos por Moreira (2002) e Moreira & Moreira (2013), e Kleibergen (2005). Com isso, o trabalho concilia as estatísticas utilizadas por eles para realizar inferência e reescreve o teste score proposto em Kleibergen (2005) utilizando as estatísticas de Moreira & Moreira (2013), e é obtido usando a teoria assintótica em Newey & McFadden (1984) a estatística do teste score ótimo. Além disso, mostra-se a equivalência entre a abordagem por GMM e a que usa sistema de equações e verossimilhança para abordar o problema de identificação fraca.

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n(-1/2) and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes. (C) 2011 Elsevier B.V. All rights reserved.

The local power of the gradient test

The asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n(-1/2), n being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property. The power performance of all four criteria in one-parameter exponential family is examined.

A Bootstrap Test for the Probability of Ruin in the Compound Poisson Risk Process

In this article we propose a bootstrap test for the probability of ruin in the compound Poisson risk process. We adopt the P-value approach, which leads to a more complete assessment of the underlying risk than the probability of ruin alone. We provide second-order accurate P-values for this testing problem and consider both parametric and nonparametric estimators of the individual claim amount distribution. Simulation studies show that the suggested bootstrap P-values are very accurate and outperform their analogues based on the asymptotic normal approximation.

Power Calculations for Preclinical Studies Using a K-Sample Rank Test and the Lehmann Alternative Hypothesis

Power calculations in a small sample comparative study, with a continuous outcome measure, are typically undertaken using the asymptotic distribution of the test statistic. When the sample size is small, this asymptotic result can be a poor approximation. An alternative approach, using a rank based test statistic, is an exact power calculation. When the number of groups is greater than two, the number of calculations required to perform an exact power calculation is prohibitive. To reduce the computational burden, a Monte Carlo resampling procedure is used to approximate the exact power function of a k-sample rank test statistic under the family of Lehmann alternative hypotheses. The motivating example for this approach is the design of animal studies, where the number of animals per group is typically small.

A Diagnostic Test for the Mixing Distribution in a Generalised Linear Mixed Model

We introduce a diagnostic test for the mixing distribution in a generalised linear mixed model. The test is based on the difference between the marginal maximum likelihood and conditional maximum likelihood estimates of a subset of the fixed effects in the model. We derive the asymptotic variance of this difference, and propose a test statistic that has a limiting chi-square distribution under the null hypothesis that the mixing distribution is correctly specified. For the important special case of the logistic regression model with random intercepts, we evaluate via simulation the power of the test in finite samples under several alternative distributional forms for the mixing distribution. We illustrate the method by applying it to data from a clinical trial investigating the effects of hormonal contraceptives in women.

Optimal Test for Parameter Instability When the Unstable Process and the Error Distribution are Unknown

This paper proposes asymptotically optimal tests for unstable parameter process under the feasible circumstance that the researcher has little information about the unstable parameter process and the error distribution, and suggests conditions under which the knowledge of those processes does not provide asymptotic power gains. I first derive a test under known error distribution, which is asymptotically equivalent to LR tests for correctly identified unstable parameter processes under suitable conditions. The conditions are weak enough to cover a wide range of unstable processes such as various types of structural breaks and time varying parameter processes. The test is then extended to semiparametric models in which the underlying distribution in unknown but treated as unknown infinite dimensional nuisance parameter. The semiparametric test is adaptive in the sense that its asymptotic power function is equivalent to the power envelope under known error distribution.

Rank test of location optimal for hyperbolic secant distribution

There are at least two reasons for a symmetric, unimodal, diffuse tailed hyperbolic secant distribution to be interesting in real-life applications. It displays one of the common types of non normality in natural data and is closely related to the logistic and Cauchy distributions that often arise in practice. To test the difference in location between two hyperbolic secant distributions, we develop a simple linear rank test with trigonometric scores. We investigate the small-sample and asymptotic properties of the test statistic and provide tables of the exact null distribution for small sample sizes. We compare the test to the Wilcoxon two-sample test and show that, although the asymptotic powers of the tests are comparable, the present test has certain practical advantages over the Wilcoxon test.

Asymptotic theory of time varying networks with memory and heterogeneous activation pattern

In this thesis work we develop a new generative model of social networks belonging to the family of Time Varying Networks. The importance of correctly modelling the mechanisms shaping the growth of a network and the dynamics of the edges activation and inactivation are of central importance in network science. Indeed, by means of generative models that mimic the real-world dynamics of contacts in social networks it is possible to forecast the outcome of an epidemic process, optimize the immunization campaign or optimally spread an information among individuals. This task can now be tackled taking advantage of the recent availability of large-scale, high-quality and time-resolved datasets. This wealth of digital data has allowed to deepen our understanding of the structure and properties of many real-world networks. Moreover, the empirical evidence of a temporal dimension in networks prompted the switch of paradigm from a static representation of graphs to a time varying one. In this work we exploit the Activity-Driven paradigm (a modeling tool belonging to the family of Time-Varying-Networks) to develop a general dynamical model that encodes fundamental mechanism shaping the social networks' topology and its temporal structure: social capital allocation and burstiness. The former accounts for the fact that individuals does not randomly invest their time and social interactions but they rather allocate it toward already known nodes of the network. The latter accounts for the heavy-tailed distributions of the inter-event time in social networks. We then empirically measure the properties of these two mechanisms from seven real-world datasets and develop a data-driven model, analytically solving it. We then check the results against numerical simulations and test our predictions with real-world datasets, finding a good agreement between the two. Moreover, we find and characterize a non-trivial interplay between burstiness and social capital allocation in the parameters phase space. Finally, we present a novel approach to the development of a complete generative model of Time-Varying-Networks. This model is inspired by the Kaufman's adjacent possible theory and is based on a generalized version of the Polya's urn. Remarkably, most of the complex and heterogeneous feature of real-world social networks are naturally reproduced by this dynamical model, together with many high-order topological properties (clustering coefficient, community structure etc.).

Suboptimal Multistage Nonparametric Hypotheses Test

2000 Mathematics Subject Classification: 62L10, 62L15.

Asymptotic Change-Point Analysis of the Dependencies in Time Series

In recent papers, Wied and his coauthors have introduced change-point procedures to detect and estimate structural breaks in the correlation between time series. To prove the asymptotic distribution of the test statistic and stopping time as well as the change-point estimation rate, they use an extended functional Delta method and assume nearly constant expectations and variances of the time series. In this thesis, we allow asymptotically infinitely many structural breaks in the means and variances of the time series. For this setting, we present test statistics and stopping times which are used to determine whether or not the correlation between two time series is and stays constant, respectively. Additionally, we consider estimates for change-points in the correlations. The employed nonparametric statistics depend on the means and variances. These (nuisance) parameters are replaced by estimates in the course of this thesis. We avoid assuming a fixed form of these estimates but rather we use "blackbox" estimates, i.e. we derive results under assumptions that these estimates fulfill. These results are supplement with examples. This thesis is organized in seven sections. In Section 1, we motivate the issue and present the mathematical model. In Section 2, we consider a posteriori and sequential testing procedures, and investigate convergence rates for change-point estimation, always assuming that the means and the variances of the time series are known. In the following sections, the assumptions of known means and variances are relaxed. In Section 3, we present the assumptions for the mean and variance estimates that we will use for the mean in Section 4, for the variance in Section 5, and for both parameters in Section 6. Finally, in Section 7, a simulation study illustrates the finite sample behaviors of some testing procedures and estimates.