Dependence Estimation for High‐frequency Sampled Multivariate CARMA Models

The paper considers high‐frequency sampled multivariate continuous‐time autoregressive moving average (MCARMA) models and derives the asymptotic behaviour of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behaviour of the cross‐covariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous‐time and in the discrete‐time model. As a special case, we consider a CARMA (one‐dimensional MCARMA) process. For a CARMA process, we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models; only the sums in the discrete‐time model are exchanged by integrals in the continuous‐time model. Finally, we present limit results for multivariate MA processes as well, which are not known in this generality in the multivariate setting yet.     

Optimal covariance estimation of discrete-time locally self-similar processes in time-scale and ambiguity domains

This paper investigates the optimal estimate of the covariance function in the sense of mean-square of errors, for the class of discrete-time locally self-similar processes. The covariance function is estimated in time-scale and ambiguity domains. Since the class of estimators is completely characterized in terms of kernels, the problem is reduced to finding the optimal kernel, which is obtained in time-scale domain. Also, the optimal kernel is computed for two classes of discrete-time locally self-similar and locally self-similar chirp processes. Furthermore, it is shown that the proposed method gives more accurate estimate than the ordinary methods for non stationary processes.     

Quotient spaces and statistical models

The purpose of this paper is to draw attention to the widespread occurrence of quotient spaces in statistical work. Quotient spaces are intrinsic to probability distributions, residuals and interaction in linear models, covariance functions and variograms of stochastic processes, etc. The theme is that explicit recognition of the quotient space can offer surprising conceptual simplification. The advantages of working directly with the quotient space are hard to describe in general. As the examples demonstrate, the answer lies partly in directness of approach.     

Marginally restricted D-optimal designs for correlated observations

Two practical degrees of complexity may arise when designing an experiment for a model of a real life case. First, some explanatory variables may not be under the control of the practitioner. Secondly, the responses may be correlated. In this paper three real life cases in this situation are considered. Different covariance structures are studied and some designs are computed adapting the theory of marginally restricted designs for correlated observations. An exchange algorithm given by Brimkulov's algorithm is also adapted to marginally restricted D–optimality and it is applied to a complex situation.     

Recent advances to model anisotropic space–time data

Building new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to present an up-to-date overview of recent spatio-temporal covariance models taking into account the problem of spatial anisotropy. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability. In particular, we focus on the problem of modelling anisotropy through isotropy within components. We present the Bernstein class, and a generalisation of Gneiting’s approach (2002a) to obtain new classes of space–time covariance functions which are spatially anisotropic. We also discuss some methods for building covariance functions that attain negative values. We finally present several differentiation and integration operators acting on particular space–time covariance classes.      

Computation aspects of the parameter estimates of linear mixed effects model in multivariate repeated measures set-up

The number of parameters mushrooms in a linear mixed effects (LME) model in the case of multivariate repeated measures data. Computation of these parameters is a real problem with the increase in the number of response variables or with the increase in the number of time points. The problem becomes more intricate and involved with the addition of additional random effects. A multivariate analysis is not possible in a small sample setting. We propose a method to estimate these many parameters in bits and pieces from baby models, by taking a subset of response variables at a time, and finally using these bits and pieces at the end to get the parameter estimates for the mother model, with all variables taken together. Applying this method one can calculate the fixed effects, the best linear unbiased predictions (BLUPs) for the random effects in the model, and also the BLUPs at each time of observation for each response variable, to monitor the effectiveness of the treatment for each subject. The proposed method is illustrated with an example of multiple response variables measured over multiple time points arising from a clinical trial in osteoporosis.     

Random sampling of long-memory stationary processes

This paper investigates the second order properties of a stationary process after random sampling. While a short memory process gives always rise to a short memory one, we prove that long-memory can disappear when the sampling law has heavy enough tails. We prove that under rather general conditions the existence of the spectral density is preserved by random sampling. We also investigate the effects of deterministic sampling on seasonal long-memory.     

Bias in Small-Sample Inference With Count-Data Models

Abstract

Both Poisson and negative binomial regression can provide quasi-likelihood estimates for coefficients in exponential-mean models that are consistent in the presence of distributional misspecification. It has generally been recommended, however, that inference be carried out using asymptotically robust estimators for the parameter covariance matrix. As with linear models, such robust inference tends to lead to over-rejection of null hypotheses in small samples. Alternative methods for estimating coefficient estimator variances are considered. No one approach seems to remove all test bias, but the results do suggest that the use of the jackknife with Poisson regression tends to be least biased for inference.     

A control chart based on sample ranges for monitoring the covariance matrix of the multivariate processes

For the univariate case, the R chart and the S ² chart are the most common charts used for monitoring the process dispersion. With the usual sample size of 4 and 5, the R chart is slightly inferior to the S ² chart in terms of efficiency in detecting process shifts. In this article, we show that for the multivariate case, the chart based on the standardized sample ranges, we call the RMAX chart, is substantially inferior in terms of efficiency in detecting shifts in the covariance matrix than the VMAX chart, which is based on the standardized sample variances. The user's familiarity with sample ranges is a point in favor of the RMAX chart. An example is presented to illustrate the application of the proposed chart.     

On some generalized farlie-gumbel-morgenstern distributions-II regression,correlation and further generalizations

Regression and correlation properties of the generalized Farlie-Gumbel-Morgenstem distributions introduced in Johnson and Kotz (1975) are studied. Further generalizations of these distributions are considered.