On the identification of ARMA echelon-form models

An identification procedure for multivariate autoregressive moving average (ARMA) echelon-form models is proposed. It is based on the study of the linear dependence between rows of the Hankel matrix of serial correlations. To that end, we define a statistical test for checking the linear dependence between vectors of serial correlations. It is shown that the test statistic t?_n considered is distributed asymptotically as a finite linear combination of independent chi-square random variables with one degree of freedom under the null hypothesis, whereas under the alternative hypothesis, t?_N/N converges in probability to a positive constant. These results allow us, in particular, to compute the asymptotic probability of making a specification error with the proposed procedure. Links to other methods based on the application of canonical analysis are discussed. A simulation experiment was done in order to study the performance of the procedure. It is seen that the graphical representation of t?_N, as a function of N, can be very useful in identifying the dynamic structure of ARMA models. Furthermore, for the model considered, the proposed identification procedure performs very well for series of 100 observations or more and reasonably well with short series of 50 observations.     

On Markov-switching periodic ARMA models

In this work, we propose a generalization of the classical Markov-switching ARMA models to the periodic time-varying case. Specifically, we propose a Markov-switching periodic ARMA (MS-PARMA) model. In addition of capturing regime switching often encountered during the study of many economic time series, this new model also captures the periodicity feature in the autocorrelation structure. We first provide some probabilistic properties of this class of models, namely the strict periodic stationarity and the existence of higher-order moments. We thus propose a procedure for computing the autocovariance function where we show that the autocovariances of the MS-PARMA model satisfy a system of equations similar to the PARMA Yule–Walker equations. We propose also an easily implemented algorithm which can be used to obtain parameter estimates for the MS-PARMA model. Finally, a simulation study of the performance of the proposed estimation method is provided.     

Sufficient reduction methods for multivariate time-dependent health surveillance data

ABSTRACT

Early detection with a low false alarm rate (FAR) is the main aim of outbreak detection as used in public health surveillance or in regard to bioterrorism. Multivariate surveillance is preferable to univariate surveillance since correlation between series (CBS) is recognized and incorporated. Sufficient reduction has proved a promising method for handling CBS, but has not previously been used when correlation within series (CWS) is present. Here we develop sufficient reduction methods for reducing a p-dimensional multivariate series to a univariate series of statistics shown to be sufficient to monitor a sudden, but persistent, shift in the multivariate series mean. Correlation both within and between series is taken into account, as public health data typically exhibit both forms of association. Simultaneous and lagged changes and different shift sizes are investigated. A one-sided exponentially weighted moving average chart is used as a tool for detection of a change. The performance of the proposed method is compared with existing sufficient reduction methods, the parallel univariate method and both VarR and Z charts. A simulation study using bivariate normal autoregressive data shows that the new method gives shorter delays and a lower FAR than other methods, which have high FARs when CWS is clearly present.     

Testing a linear ARMA model against threshold-ARMA models: A Bayesian approach

We introduce a Bayesian approach to test linear autoregressive moving-average (ARMA) models against threshold autoregressive moving-average (TARMA) models. First, the marginal posterior densities of all parameters, including the threshold and delay, of a TARMA model are obtained by using Gibbs sampler with Metropolis–Hastings algorithm. Second, reversible-jump Markov chain Monte Carlo (RJMCMC) method is adopted to calculate the posterior probabilities for ARMA and TARMA models: Posterior evidence in favor of TARMA models indicates threshold nonlinearity. Finally, based on RJMCMC scheme and Akaike information criterion (AIC) or Bayesian information criterion (BIC), the procedure for modeling TARMA models is exploited. Simulation experiments and a real data example show that our method works well for distinguishing an ARMA from a TARMA model and for building TARMA models.     

Goodness-of-link tests for multivariate regression models

ABSTRACT

This note presents an approximation to multivariate regression models which is obtained from a first-order series expansion of the multivariate link function. The proposed approach yields a variable-addition approximation of regression models that enables a multivariate generalization of the well-known goodness-of-link specification test, available for univariate generalized linear models. Application of this general methodology is illustrated with models of multinomial discrete choice and multivariate fractional data, in which context it is shown to lead to well-established approximation and testing procedures.     

Maximum likelihood estimation for arma models in the presence of ARMA errors

An ARMA(p, q) process observed with an ARMA(c, d) error has an ARMA (p + c, k) representation with k = max(c + q, p + d) whose parameters satisfy some nonlinear constraints. Identification of the model is discussed. We develop Newton-Raphson estimators for the ARMA(p + c, k) process which take advantage of the information contained in the nonlinear restrictions. Explicit expressions for the derivatives of the restrictions are derived.     

The asymptotic covariance matrix of the QMLE in ARMA models

A compact analytical representation of the asymptotic covariance matrix, in terms of model parameters directly, of the quasi maximum likelihood estimator (QMLE) is derived in autoregressive moving average (ARMA) models with possible nonzero means and non-Gaussian error terms. For model parameters excluding the error variance, it is found that the Huber (1967 Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1, pp. 221–233. [Google Scholar]) sandwich form for the asymptotic covariance matrix degenerates into the inverse of the associated information matrix. In comparison to the existing result that involves the second moments of some auxiliary variables for the case of zero-mean ARMA models, the analytical asymptotic covariance in this article has an advantage in that it can be conveniently estimated by plugging in the estimated model parameters directly.     

Sufficiency of a sufficient statistic for equivariant estimation

Suppose an estimation problem is invariant under a group of transformations and one is interested in finding an optimal equivariant estimator. The usual proactice is to confine attention to non-randomized equivariant estimators based on a minimal sufficient statistic. A justification of this restriction to a smaller clas of estimators is given in this paper under certain conditions.     

Calculating the autocovariances and the likelihood for periodic V ARMA models

This paper concerns the autocovariance calculation and likelihood evaluation for periodic vector ARMA models (PV ARMA). Based on a state space representation of PV ARMA models, we derive an algorithm for computing the PV ARMA autocovariances. The proposed method computes the autocovariances for distinct seasons separately, thereby facilitating efficient calculation for models with a large period. As a result, the obtained autocovariance calculation procedure is exploited in a periodic Chandrasekhar-type filter to evaluate the exact likelihood for Gaussian PV ARMA series. Empirical evidence shows the superiority of the periodic Chandrasekhar algorithm for likelihood evaluation over the Kalman-based one.     

On the multivariate predictive distribution of multi-dimensional effective dose: a Bayesian approach

We propose a Bayesian procedure to sample from the distribution of the multi-dimensional effective dose. This effective dose is the set of dose levels of multiple predictive factors that produce a binary response with a fixed probability.We apply our algorithms to parametric and semiparametric logistics regression models, respectively. The graphical display of random samples obtained through Markov chain Monte Carlo can provide some insight into the predictive distribution.     

Estimation of nonlinear errors-in-variables models: an approximate solution

We propose an easy to derive and simple to compute approximate least squares or maximum likelihood estimator for nonlinear errors-in-variables models that does not require the knowledge of the conditional density of the latent variables given the observables. Specific examples and Monte Carlo studies demonstrate that the bias of this approximate estimator is small even when the magnitude of the variance of measurement errors to the variance of measured covariates is large. Cheng Hsiao and Qing Wang's work was supported in part by National Science Foundation grant SeS91-22481 and SBR94-09540. Liqun Wang gratefully acknowledges the financial support from Swiss National Science Foundation. We wish to thank Professor H. Schneeweiss and a referee for helpful comments and suggestions.     

Minimum-volume confidence sets for normal linear regression models

In this article, we establish the minimum-volume confidence sets for normal linear regression models, extending the results in Zhang [Minimum volume confidence sets for parameters of normal distributions. Adv Stat Anal. 2017;101:309–320] on building the minimum-volume confidence sets for parameters of normal distributions. Compared with classical confidence sets, the proposed optimal confidence set is proved to have the smallest volume, for whatever confidence level, sample size and sample data.     

Forecasting ARMA models: a comparative study of information criteria focusing on MDIC

This paper deals with the implementation of model selection criteria to data generated by ARMA processes. The recently introduced modified divergence information criterion is used and compared with traditional selection criteria like the Akaike information criterion (AIC) and the Schwarz information criterion (SIC). The appropriateness of the selected model is tested for one- and five-step ahead predictions with the use of the normalized mean squared forecast errors (NMSFE).     

Union-Intersection testing for outliers in multivariate normal data

The union-intersection approach to multivariate test construction is used to develop an alternative to Wilks' likelihood ratio test statistic for testing for two or more outliers in multivariate normal data. It is shown that critical values of both statistics are poorly approximated by Bonferroni bounds. Simulated critical values are presented for both statistics for significance levels 1% and 5%, for sample sizes 10(5)30, 40, 50, 75 and 100 for 2, 3, 4 and 5 dimensions. A power comparison of the two tests in the slippage of the mean model for generating outliers indicates that the union-intersection test is the more powerful when the slippages are close to collinear. Although Wilks' test remains the preference for general use, the union-intersection test could be valuable when such special structure in the data is suspected.     

Pairwise likelihood estimation for multivariate mixed Poisson models generated by Gamma intensities

Estimating the parameters of multivariate mixed Poisson models is an important problem in image processing applications, especially for active imaging or astronomy. The classical maximum likelihood approach cannot be used for these models since the corresponding masses cannot be expressed in a simple closed form. This paper studies a maximum pairwise likelihood approach to estimate the parameters of multivariate mixed Poisson models when the mixing distribution is a multivariate Gamma distribution. The consistency and asymptotic normality of this estimator are derived. Simulations conducted on synthetic data illustrate these results and show that the proposed estimator outperforms classical estimators based on the method of moments. An application to change detection in low-flux images is also investigated.     

On influence diagnostics in elliptical multivariate regression models with equicorrelated random errors

In this paper we discuss estimation and diagnostic procedures in elliptical multivariate regression models with equicorrelated random errors. Two procedures are proposed for the parameter estimation and the local influence curvatures are derived under some usual perturbation schemes to assess the sensitivity of the maximum likelihood estimates (MLEs). Two motivating examples preliminarily analyzed under normal errors are reanalyzed considering appropriate elliptical distributions. The local influence approach is used to compare the sensitivity of the model estimates.     

Comparison of performance measures for multivariate discrete models

For multivariate probit models, Spiess and Tutz suggest three alternative performance measures, which are all based on the decomposition of the variation. The multivariate probit model can be seen as a special case of the discrete copula model. This paper proposes some new measures based on the value of the likelihood function and the prediction-realization table. In addition, it generalizes the measures from Spiess and Tutz for the discrete copula model. Results of a simulation study designed to compare the different measures in various situations are presented.     

On nonparametric density estimation for multivariate linear long-memory processes

We consider nonparametric estimation of the density function and its derivatives for multivariate linear processes with long-range dependence. In a first step, the asymptotic distribution of the multivariate empirical process is derived. In a second step, the asymptotic distribution of kernel density estimators and their derivatives is obtained.     

Inference for multivariate normal hierarchical models

This paper provides a new method and algorithm for making inferences about the parameters of a two-level multivariate normal hierarchical model. One has observed J p -dimensional vector outcomes, distributed at level 1 as multivariate normal with unknown mean vectors and with known covariance matrices. At level 2, the unknown mean vectors also have normal distributions, with common unknown covariance matrix A and with means depending on known covariates and on unknown regression coefficients. The algorithm samples independently from the marginal posterior distribution of A by using rejection procedures. Functions such as posterior means and covariances of the level 1 mean vectors and of the level 2 regression coefficient are estimated by averaging over posterior values calculated conditionally on each value of A drawn. This estimation accounts for the uncertainty in A , unlike standard restricted maximum likelihood empirical Bayes procedures. It is based on independent draws from the exact posterior distributions, unlike Gibbs sampling. The procedure is demonstrated for profiling hospitals based on patients' responses concerning p =2 types of problems (non-surgical and surgical). The frequency operating characteristics of the rule corresponding to a particular vague multivariate prior distribution are shown via simulation to achieve their nominal values in that setting.     

Optimal designs for multivariate logistic mixed models with longitudinal data

This paper considers the optimal design problem for multivariate mixed-effects logistic models with longitudinal data. A decomposition method of the binary outcome and the penalized quasi-likelihood are used to obtain the information matrix. The D-optimality criterion based on the approximate information matrix is minimized under different cost constraints. The results show that the autocorrelation coefficient plays a significant role in the design. To overcome the dependence of the D-optimal designs on the unknown fixed-effects parameters, the Bayesian D-optimality criterion is proposed. The relative efficiencies of designs reveal that both the cost ratio and autocorrelation coefficient play an important role in the optimal designs.