Generalized information matrix tests for copulas

A test when the Fisher information may be infinite, exemplified by a test for marginal independence in extreme value distributions

The authors examine the ability of the Johansen (1991) test to estimate the number of unit roots in high dimensional systems. They use data based Monte Carlo methods as a simple means of evaluating the validity of inference using asymptotic critical values. These simulations for a typical annual post-World War II dataset illustrate how the estimated number of unit roots change in a nonmonotone fashion with the dimension of the system, and with the number of lags in the VAR representation. The authors find that overparametrization in high dimensions is as bad as underparametrization. The Bayes information criteria outperforms the Akaike information criteria in their setup. Copyright 1996 by MIT Press.

Testing for monotonicity under endogeneity: An application to the reservation wage function

Statistical tests of a simple energy balance equation in a synthetic model of cotrending and cointegration

The asymptotic distributions of the recursive out-of-sample forecast accuracy test statistics depend on stochastic integrals of Brownian motion when the models under comparison are nested. This often complicates their implementation in practice because the computation of their asymptotic critical values is costly. Hansen and Timmermann (2015, Econometrica) propose a Wald approximation of the commonly used recursive F-statistic and provide a simple characterization of the exact density of its asymptotic distribution. However, this characterization holds only when the larger model has one extra predictor or the forecast errors are homoscedastic. No such closed-form characterization is readily available when the nesting involves more than one predictor and heteroskedasticity is present. We first show both the recursive F-test and its Wald approximation have poor finite-sample properties, especially when the forecast horizon is greater than one. We then propose a hybrid bootstrap method consisting of a block moving bootstrap (which is nonparametric) and a residual based bootstrap for both statistics, and establish its validity. Simulations show that our hybrid bootstrap has good finite-sample performance, even in multi-step ahead forecasts with heteroscedastic or autocorrelated errors, and more than one predictor. The bootstrap method is illustrated on forecasting core inflation and GDP growth.

In the reliability analysis of safety critical complex engineering structures, a very large number of the system parameters can be considered as random variables. The difficulty in computing the failure probability using the classical first- and second-order reliability methods (FORM and SORM) increases rapidly with the number of variables or ‘dimension’. There are mainly two reasons behind this. The first is the increase in computational time with the increase in the number of random variables. In principle, this problem can be handled with superior computational tools. The second reason, which is perhaps more fundamental, is that there are some conceptual difficulties typically associated with high dimensions. This means that even when one manages to carry out the necessary computations, the application of existing FORM and SORM may still lead to incorrect results in high dimensions. This paper is aimed at addressing this issue. Based on the asymptotic distribution of quadratic form in Gaussian random variables, two formulations for the case when the number of random variables n →∞ is provided. The first is called ‘strict asymptotic formulation’ and the second is called ‘weak asymptotic formulation’. Both approximations result in simple closed-form expressions for the probability of failure of an engineering structure. The proposed asymptotic approximations are compared with existing approximations and Monte Carlo simulations using numerical examples.

Monte Carlo experiments have shown that tests based on generalized-method-ofmoments estimators often have true levels that differ greatly from their nominal levels when asymptotic critical values are used. This paper gives conditions under which the bootstrap provides asymptotic refinements to the critical values of t tests and the test of overidentifying restrictions. Particular attention is given to the case of dependent data. It is shown that with such data, the bootstrap must sample blocks of data and that the formulae for the bootstrap versions of test statistics differ from the formulae that apply with the original data. The results of Monte Carlo experiments on the numerical performance of the bootstrap show that it usually reduces the errors in level that occur when critical values based on first-order asymptotic theory are used. The bootstrap also provides an indication of the accuracy of critical values obtained from first-order asymptotic theory.

Any consistent and asymptotically normal estimator with a variance-covariance matrix of the stabilizing transformation attaining the Cramer-Rao efficiency bound is said to be asymptotically efficient (cf. Amemiya, 1985, p. 124). It is well known that the Cramer-Rao bound is given by the inverse of the information matrix. Throughout this chapter, let J(θ0) denote the information matrix for a single observation, evaluated at the true parameter vector, defined as $$ J\left( {\theta _0 } \right) \equiv - E\left[ {\frac{{\partial ^2 \ln f\left( {Z|\theta _0 } \right)}} {{\partial \theta \partial \theta '}}} \right], $$ (5.1.1) where ∂2 lnf (z | θ )/∂θ∂θ′ is the Hessian matrix for a single observation containing the second derivatives of its loglikelihood contribution ln f(z | θ). Let S(θ) ≡ ∂ln f (z | θ)/∂θ denote the vector of first derivatives of the loglikelihood contribution of a single observation, henceforth referred to as the score. Using the information matrix equality at the individual level, (5.1.1) can be rewritten as $$ J\left( {\theta _0 } \right) = E\left[ {S(\theta _0 )S(\theta _0 )'} \right] = V\left[ {S(\theta _0 )} \right], $$ (5.1.2) which will be more convenient for the results stated in the following two sections.

In this paper, we suggest a new set of regression‐based statistics for testing the seasonal unit root null hypothesis. These tests are based on combining conventional Hylleberg et al. (1990) ‐type seasonal unit root test statistics calculated from both forward and reverse estimation of the auxiliary regression equation. We derive the asymptotic distributions of the new test statistics under the seasonal unit root null hypothesis. We provide finite sample critical values appropriate for the case of quarterly data together with asymptotic critical values, the latter appropriate for any seasonal aspect. Monte Carlo simulation of the finite‐sample size and power properties of the new tests reveals that, overall, they perform rather better than extant tests of the seasonal unit root hypothesis.

In the application of extreme value analysis it is usually assumed that the size of the samples from which the extreme values are obtained is sufficiently large for the asymptotic extreme value distribution to be used. The necessary sample size depends upon the population distribution and this is generally not known; but assuming a Weibull distribution, which is often fitted to wind speed and wave height data, it is shown that the rate of convergence is rapid and that the asymptotic distribution may be used for a sample size as small as ten. An ‘exponential approximation’ for the distribution of maxima is sometimes confused with the extreme value distribution. This approximate form is derived for the Weibull distribution and the essential difference between it and the asymptotic extreme value distribution is explained.

We give an apparently new characterization of the equality of two positive definite matrices A and B; and we use this to propose several new computationally convenient statistical tests for the equality of two unknown positive definite matrices. Our primary focus is on testing the information matrix equality (e.g., White, 1982, 1994). We characterize the asymptotic behavior of our new trace-determinant information matrix test statistics under the null and the alternative and investigate their finite-sample performance for a variety of models: linear regression, exponential duration, probit, and Tobit. The parametric bootstrap suggested by Horowitz (1994) delivers critical values that provide admirable level behavior, even in samples as small asn = 50: Our new tests often have better power than the parametric-bootstrap version of the traditional IMT; when they don’t, they nevertheless perform respectably.

Perron and Yabu (2009a) consider the problem of testing for a break occurring at an unknown date in the trend function of a univariate time series when the noise component can be either stationary or integrated. This article extends their work by proposing a sequential test that allows one to test the null hypothesis of, say, l breaks versus the alternative hypothesis of (l + 1) breaks. The test enables consistent estimation of the number of breaks. In both stationary and integrated cases, it is shown that asymptotic critical values can be obtained from the relevant quantiles of the limit distribution of the test for a single break. Monte Carlo simulations suggest that the procedure works well in finite samples.

Gaussian process (GP) regression or kriging has been extensively applied in the engineering literature for the purposes of building a cheap-to-evaluate surrogate, within the contexts of multi-fidelity modeling, model calibration, and design optimization. With the ongoing automation of manufacturing and industrial practices as a part of Industry 4.0, there has been a greater need for advancing GP regression techniques to handle challenges such as high input dimensionality, data paucity or big data problems, these consist primarily of proposing efficient design of experiments, optimal data acquisition strategies, sparsifying covariance kernels, and other mathematical tricks. In this work, our attention is focused on the challenges of efficiently training a GP model, which, to the authors opinion, has attracted very little attention and is to-date poorly addressed. The performance of widely used training approaches such as maximum likelihood estimation and Markov Chain Monte Carlo (MCMC) sampling can deteriorate significantly in high-dimensional and big data problems and can lead to cost deficient implementations of critical importance to many industrial applications. Here, we compare an Adaptive Sequential Monte Carlo (ASMC) sampling algorithm to classic MCMC sampling strategies and we demonstrate the effectiveness of our implementation on several mathematical problems and challenging industry applications of varying complexity. The computational time savings of the ASMC approach manifest in large-scale problems helping us to push the boundaries of applicability and scalability of GPs for model calibration in various domains of the industry, including but not limited to design automation, design engineering, smart manufacturing, predictive maintenance, and supply chain manufacturing.

The IOS test of Presnell and Boss (J. Am. Stat. Assoc. 2004; 99(465):216-227) is a general-purpose goodness-of-fit test based on the ratio of in-sample and out-of-sample likelihoods. For large samples, the IOS statistic can be approximated by a multiplicative contrast between two estimates of the information matrix, and in this way the IOS test is connected to White's (Econometrica 1982; 50:1-26) information matrix test, or IM test, which is based directly on the difference of two estimates of the information matrix. In this paper, we compare the performance of IOS to that of the IM test and of other goodness-of-fit tests for binomial and beta-binomial models, in both examples and simulations. Our findings suggest that IOS is strongly competitive, not only against the IM test but also against tests designed for specific binomial and beta-binomial models.

The maximum absolute internally studentized residual is a regular diagnostic measure for identification of a single outlying observation in the response variable in linear regression models. However, due to the daunting and formidable nature of the probability density function of this statistic, exact critical values are tough to compute. The Bonferroni inequality and intensive simulations are the only tools for determining its critical values as a means for detecting a single outlying observation in a linear regression model. In this paper, we present a straightforward alternative technique for obtaining asymptotic critical values of this statistic. The technique can be applied to any linear regression model and is convenient for routine use. The asymptotic distribution of this statistic is derived and used in obtaining the upper bounds for its critical values. It is shown that the proposed technique does not depend on the number of independent variables or the number of regression parameters in the model. Thus, the computational cumbersomeness and tedium imposed by the complexity associated with the distribution of this statistic and the use of the Bonferroni inequality are circumvented. The main advantages of the proposed procedure are its computational simplicity and efficiency to handle large datasets in high dimension. The asymptotic critical values of this statistic obtained by the proposed method are almost identical to those obtained by other authors, even though the techniques and principles employed in this work are entirely different from that employed by them.