Abstract
Often, real-life problems require modelling several response variables together. This work analyses a multivariate linear regression model when the data are censored. Censoring distorts the correlation structure of the underlying variables and increases the bias of the usual estimators. Thus, we propose three methods to deal with multivariate data under left censoring, namely Expectation Maximization (EM), DataAugmentation (DA) and Gibbs Sampler with Data Augmentation (GDA). Results from a simulation study showthat both DA and GDA estimates are consistent for low and moderate correlation. Under high correlation scenarios, EM estimates present a lower bias.

Abstract

Abstract
Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The aim of this work is to contribute to the modelling of time series of counts under censoring using convolution closed infinitely divisible (CCID) models. The emphasis is on estimation and inference problems, using Bayesian approaches with Approximate Bayesian Computation (ABC) and Gibbs sampler with Data Augmentation (GDA) algorithms.

Abstract
Greenland ice core records display abrupt transitions, designated as Dansgaard-Oeschger (DO) events, characterised by episodes of rapid warming (typically decades) followed by a slower cooling. The identification of abrupt transitions is hindered by the typical low resolution and small size of paleoclimate records, and their significant temporal variability. Furthermore, the amplitude and duration of the DO events varies substantially along the last glacial period, which further hinders the objective identification of abrupt transitions from ice core records Automatic, purely data-driven methods, have the potential to foster the identification of abrupt transitions in palaeoclimate time series in an objective way, complementing the traditional identification of transitions by visual inspection of the time series.In this study we apply an algorithmic time series method, the Matrix Profile approach, to the analysis of the NGRIP Greenland ice core record, focusing on:- the ability of the method to retrieve in an automatic way abrupt transitions, by comparing the anomalies identified by the matrix profile method with the expert-based identification of DO events;- the characterisation of DO events, by classifying DO events in terms of shape and identifying events with similar warming/cooling temporal patternThe results for the NGRIP time series show that the matrix profile approach struggles to retrieve all the abrupt transitions that are identified by experts as DO events, the main limitation arising from the diversity in length of DO events and the method’s dependence on fixed-size sub-sequences within the time series. However, the matrix profile method is able to characterise the similarity of shape patterns between DO events in an objective and consistent way.

Abstract
The INteger-valued AutoRegressive (INAR) processes were introduced in the literature by Al-Osh and Alzaid [1987. First-order integer-valued autoregressive (INAR(l)) process. J. Time Ser. Anal. 8, 261-275] and McKenzie [1988. Some ARMA models for dependent sequences of Poisson counts. Adv. Appl. Probab. 20, 822-835] for modelling correlated series of counts. These processes have been considered as the discrete counter part of AR processes, but their highly nonlinear characteristics lead to some statistically challenging problems, namely in parameter estimation. Several estimation procedures have been proposed in the literature, mainly for processes of first order. For some of these estimators the asymptotic properties as well as finite sample properties have been obtained and studied. This paper considers Yule-Walker parameter estimation for the pth-order integer-valued autoregressive, INAR(p), process. In particular, the asymptotic distribution of the Yule-Walker estimator is obtained and it is shown that this estimator is asymptotically normally distributed, unbiased and consistent.

Abstract
We consider integer-valued autoregressive models of order one contaminated with innovational outliers. Assuming that the time points of the outliers are known but their sizes are unknown, we prove that Conditional Least Squares (CLS) estimators of the offspring and innovation means are strongly consistent. In contrast, CLS estimators of the outliers' sizes are not strongly consistent. We also prove that the joint CLS estimator of the offspring and innovation means is asymptotically normal. Conditionally on the values of the process at time points preceding the outliers' occurrences, the joint CLS estimator of the sizes of the outliers is asymptotically normal.