Alberto Contreras-Cristán

A Note on Whittle's Likelihood

The approximate likelihood function introduced by Whittle has been used to estimate the spectral density and certain parameters of a variety of time series models. In this note we attempt to empirically quantify the loss of efficiency of Whittles method in nonstandard settings. A recently developed representation of some first-order non-Gaussian stationary autoregressive process allows a direct comparison of the true likelihood function with that of Whittle. The conclusion is that Whittles likelihood can produce unreliable estimates in the non-Gaussian case, even for moderate sample sizes. Moreover, for small samples, and if the autocorrelation of the process is high, Whittles approximation is not efficient even in the Gaussian case. While these facts are known to some extent, the present study sheds more light on the degree of efficiency loss incurred by using Whittles likelihood, in both Gaussian and non-Gaussian cases.

On the construction of stationary AR(1) models via random distributions

We explore a method for constructing first-order stationary autoregressive-type models with given marginal distributions. We impose the underlying dependence structure in the model using Bayesian non-parametric predictive distributions. This approach allows for nonlinear dependency and at the same time works for any choice of marginal distribution. In particular, we look at the case of discrete-valued models; that is the marginal distributions are supported on the non-negative integers.

Inference Using Latent Variables for Mixtures of Distributions for Censored Data with Partial Identification

Abstract In this article two methods are proposed to make inferences about the parameters of a finite mixture of distributions in the context of partially identifiable censored data. The first method focuses on a mixture of location and scale models and relies on an asymptotic approximation to a suitably constructed augmented likelihood; the second method provides a full Bayesian analysis of the mixture based on a Gibbs sampler. Both methods make explicit use of latent variables and provide computationally efficient procedures compared to other methods which deal directly with the likelihood of the mixture. This may be crucial if the number of components in the mixture is not small. Our proposals are illustrated on a classical example on failure times for communication devices first studied by Mendenhall and Hader (Mendenhall, W., Hader, R. J. (1958). Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45:504–520.). In addition, we study the coverage of the confidence intervals obtained from each of the methods by means of a small simulation exercise.

A Statistical Method for the Determination of the Appropriate Order in a General Class of Time Series Models

Abstract We propose a method to determine the order q of a model in a general class of time series models. For the subset of linear moving average models (MA(q)), our method is compared with that of the sample autocorrelations. Since the sample autocorrelation is meant to detect a linear structure of dependence between random variables, it turns out to be more suitable for the linear case. However, our method presents a competitive option in that case, and for nonlinear models (NLMA(q)) it is shown to work better. The main advantages of our approach are that it does not make assumptions on the existence of moments and on the distribution of the noise involved in the moving average models. We also include an example with real data corresponding to the daily returns of the exchange rate process of mexican pesos and american dollars.

A Nonparametric Test for Symmetry Based on Freeman and Halton's Ideas on Contingency Tables

In this article, we propose a nonparametric method to test for symmetry in bivariate data. By using the extension of Fishers exact treatment for 2 × 2 contingency tables proposed by Freeman and Halton (1951), we can test the hypothesis of equal distribution for two samples of integer valued variables. Then, by counting the number of observations belonging to each cell of a symmetric, appropriately built grid, we can produce the two samples of integers required to use this test for equal distribution. The resulting test for symmetry is potentially extendible to higher dimensions. A simulation study is performed to compare with some known tests (Bowker, 1948; Hollander, 1971; and its improvement given in Krampe and Kuhnt, 2007). Our proposal represents a competitive option as a test for symmetry.

Model Selection Using Conditional Densities

In this article we propose a new method to select a discrete model f(x; θ), based on the conditional density of a sample given the value of a sufficient statistic for θ. The main idea is to work with a broad family of discrete distributions, called the family of power series distribution, for which there is a common sufficient statistic for the parameter of interest. The proposed method uses the maximum conditional density in order to select the best model. We compare our proposal with the usual methodology based on Bayes factors. We provide several examples that show that our proposal works fine in most instances. Bayes factors are strongly dependent on the prior information about the parameters. Since our method does not require the specification of a prior distribution, it provides a useful alternative to Bayes factors.