We combine the selection of a statistical model with the robust parameter estimation and diagnostic properties of the Forward Search. As a result we obtain procedures that select the best model in the presence of outliers. We derive distributional properties of our method and illustrate it on data on ozone concentration. The effect of outliers on the choice of a model is revealed. Although our example is for regression, the connection with AIC is stressed.
This paper studies the nonparametric regression estimation and the prediction problem for continuous-time observations. The almost sure convergence of a kernel regression estimator and the associated predictor are given with rates depending on the regularity of the underlying sample paths.
A general method for constructing pseudo-Gaussian tests—reducing to traditional Gaussian tests under Gaussian densities but remaining valid under non-Gaussian ones—is proposed. This method provides a solution to several open problems in classical multivariate analysis. One of them is the test of the homogeneity of covariance matrices, an assumption that plays a crucial role in multivariate analysis of variance, under elliptical, and possibly heterokurtic densities with finite fourth-order moments.
The article analyses the relationship between unobserved component trend-cycle models and the Hodrick-Prescott filter. Consideration is given to the consequences of using an inappropriate smoothing constant and the effect of changing the observation interval.
The asymptotic properties of the maximum likelihood and bayesian estimators of finite dimensional parameters of any statistical model depend strongly on the regularity conditions. It is well-known that if these conditions are fulfilled then the estimators are consistent, asymptotically normal and asymptotically efficient. These regularity conditions are of the following type: the model is sufficiently smooth w.r.t. the unknown parameter, the Fisher information is a positive continuous function, the model is correct and identifiable and the unknown parameter is an interior point of the parameter set. In this work we present a review of the properties of these estimators in the situations when these regularity conditions are not fulfilled. The presented results allow us to better understand the role of regularity conditions. As the model of observations we consider the one-dimensional ergodic diffusion process.
Some limit properties for information based model selection criteria are given in the context of unit root evaluation and various assumptions about initial conditions. Allowing for a nonparametric short memory component, standard information criteria are shown to be weakly consistent for a unit root provided the penalty coefficient Cn→∞ and Cn/n→0 as n→∞. Strong consistency holds when Cn/(log logn)3→∞ under conventional assumptions on initial conditions and under a slightly stronger condition when initial conditions are infinitely distant in the unit root model. The limit distribution of the AIC criterion is obtained.
In this paper we will investigate the consequences of applying model selection methods under regularity conditions that are sufficiently general to encompass (i) stochastic models involving non-stationary processes and (ii) situations where the true structure of the process falls outside the class of models under consideration. The properties of selection criteria that use very general measures of model complexity are considered and the results are used to draw attention to the fallacy of traditional beliefs concerning commonly employed model selection criteria.
Disregarding spatial dependence can invalidate methods for analyzing cross-sectional and panel data. We discuss ongoing work on developing methods that allow for, test for, or estimate, spatial dependence. Much of the stress is on nonparametric and semiparametric methods.
Exact joint distributions of waiting times for two patterns in a sequence of l-th order time-homogeneous Markov dependent trials are studied, where the patterns are not necessarily assumed to be distinct from each other. We prove that exact joint probability generating functions, which are regarded as expectations of the corresponding random variables, are derived through calculating the conditional expectation based on conditioning by the sooner waiting time and the pattern which comes sooner. We also give illustrative numerical examples in order to demonstrate the performance of our results.
The paper outlines some aspects related to statistical model selection, focusing in particular on inference conducted in the presence of a finite set of parametric models. The point the paper emphasizes is that the basic approaches such as testing, point estimation and confidence region estimation based on a single model are extensible under pertinent modification to inference on a set of models. They are, however, replaced by plural-model testing, `point' model estimation and confidence-set construction of models.
Prof. Akaike made significant contributions in various fields of statistical science, in particular, in time series analysis in frequency domain and time domain, information criterion and Bayes modeling. In this article, his research contributions are described in order of launching period, frequency time domain analysis, time domain time series modeling, AIC and statistical modeling, and Bayes modeling.
This paper proposes efficient estimation methods of unknown parameters when frequencies as well as local moments are available in grouped data. Assuming the original data is an i.i.d. sample from a parametric density with unknown parameters, we obtain the joint density of frequencies and local moments, and propose a maximum likelihood (ML) estimator. We further compare it with the generalized method of moments (GMM) estimator and prove these two estimators are asymptotically equivalent in the first order. Based on the ML method, we propose to use the Akaike information criterion (AIC) for model selection. Monte Carlo experiments show that the estimators perform remarkably well, and AIC selects the right model with high frequency.
This paper deals with nonstationary autoregressive (AR) models with complex roots on the unit circle. We examine the asymptotic properties of the least squares estimators (LSEs) in the model. We also extend the model to the case where the error term follows a stationary linear process. We show that the limiting distribution of the LSE of the unit root parameter has a property comparable to that of the LSE in the standard nonstationary seasonal model with period two. Percent points and moments of the limiting distribution are computed by numerical integration.
The problem of fitting a parametric model of time series with time varying parameters attracts our attention. We evaluate a goodness of time varying spectral models from an information theoretic point of view. We propose model selection criteria for locally stationary processes based on nonlinear functionals of a time varying spectral density without assuming that the true time varying spectral density belongs to the model. Also, we obtain a sufficient condition such that our information criteria coincide with Akaike's information criterion.
The asymptotic expansion method for ε-Markov processes with a mixing property is briefly reviewed. It is illustrated by a point process marked by a diffusion process. As a typical application, the expansion formula for the M-estimator based on ε-Markov data is exhibited.