Monte Carlo Tests with Nuisance Parameters: A General Approach to Finite-Sample Inference and Nonstandard Asymptotics

The technique of Monte Carlo (MC) tests [Dwass (1957), Barnard (1963)] provides an attractive method of building exact tests from statistics whose finite sample distribution is intractable but can be simulated (provided it does not involve nuisance parameters). We extend this method in two ways: first, by allowing for MC tests based on exchangeable possibly discrete test statistics; second, by generalizing the method to statistics whose null distributions involve nuisance parameters (maximized MC tests, MMC). Simplified asymptotically justified versions of the MMC method are also proposed and it is shown that they provide a simple way of improving standard asymptotics and dealing with nonstandard asymptotics (e.g., unit root asymptotics). Parametric bootstrap tests may be interpreted as a simplified version of the MMC method (without the general validity properties of the latter).

Asymptotics of number fields and the Cohen-Lenstra heuristics

We study the asymptotics conjecture of Malle for dihedral groups Dl of order 2l, where l is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen-Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.

Asymptotics

Exercises and solutions for a third year asymptotics course.

The failure of meta-analytic asymptotics for the seemingly efficient estimator of the common risk difference

We consider the case of a multicenter trial in which the center specific sample sizes are potentially small. Under homogeneity, the conventional procedure is to pool information using a weighted estimator where the weights used are inverse estimated center-specific variances. Whereas this procedure is efficient for conventional asymptotics (e. g. center-specific sample sizes become large, number of center fixed), it is commonly believed that the efficiency of this estimator holds true also for meta-analytic asymptotics (e.g. center-specific sample size bounded, potentially small, and number of centers large). In this contribution we demonstrate that this estimator fails to be efficient. In fact, it shows a persistent bias with increasing number of centers showing that it isnot meta-consistent. In addition, we show that the Cochran and Mantel-Haenszel weighted estimators are meta-consistent and, in more generality, provide conditions on the weights such that the associated weighted estimator is meta-consistent.

(Non)differentiability and asymptotics for potential densities of subordinators

For subordinators with positive drift we extend recent results on the structure of the potential measures and the renewal densities. Applying Fourier analysis a new representation of the potential densities is derived from which we deduce asymptotic results and show how the atoms of the Lévy measure translate into points of (non)differentiability of the potential densities.

A stochastic discount factor approach to asset pricing using panel data asymptotics

Using the Pricing Equation in a panel-data framework, we construct a novel consistent estimator of the stochastic discount factor (SDF) which relies on the fact that its logarithm is the "common feature" in every asset return of the economy. Our estimator is a simple function of asset returns and does not depend on any parametric function representing preferences. The techniques discussed in this paper were applied to two relevant issues in macroeconomics and finance: the first asks what type of parametric preference-representation could be validated by asset-return data, and the second asks whether or not our SDF estimator can price returns in an out-of-sample forecasting exercise. In formal testing, we cannot reject standard preference specifications used in the macro/finance literature. Estimates of the relative risk-aversion coefficient are between 1 and 2, and statistically equal to unity. We also show that our SDF proxy can price reasonably well the returns of stocks with a higher capitalization level, whereas it shows some difficulty in pricing stocks with a lower level of capitalization.

Monotonicity and asymptotics of zeros of Sobolev type orthogonal polynomials: A general case

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials P.. (a) of degree m, defined by Boros and Moll via[GRAPHICS]approach the lemmiscate {zeta epsilon C: \zeta(2) - 1\ = Hzeta < 0}, as m --> infinity. (C) 2004 Elsevier B.V. All rights reserved.