The life distribution of a device subject to shocks governed by a homogeneous Poisson process is shown to have a bathtub failure rate average (BFRA) when the probabilities $\bar{P}_k$ of surviving k shocks possess the corresponding discrete property. We prove closure under the formation of weak limits for BFRA distributions and explore related moment convergence issues within the BFRA family. Similar results for increasing and decreasing failure rate average distributions are obtained either independently or as consequences of our results. We also establish some results outlining the positions of various non-monotonic ageing classes such as bathtub failure rate, increasing initially then decreasing mean residual life, new worse then better than used in expectation, and increasing initially then decreasing mean time to failure in the hierarchy. Finally, an open problem is posed and a partial solution provided.

We analysed the coronavirus disease 2019 epidemic curve from March to the end of April 2020 in Germany. We use statistical models to estimate the number of cases with disease onset on a given day and use back-projection techniques to obtain the number of new infections per day. The respective time series are analysed by a trend regression model with change points. The change points are estimated directly from the data. We carry out the analysis for the whole of Germany and the federal state of Bavaria, where we have more detailed data. Both analyses show a major change between 9 and 13 March for the time series of infections: from a strong increase to a decrease. Another change was found between 25 March and 29 March, where the decline intensified. Furthermore, we perform an analysis stratified by age. A main result is a delayed course of the pandemic for the age group 80 + resulting in a turning point at the end of March. Our results differ from those by other authors as we take into account the reporting delay, which turned out to be time dependent and therefore changes the structure of the epidemic curve compared to the curve of newly reported cases.

In this paper, a unique dataset of improvised explosive device attacks during “The Troubles” in Northern Ireland (NI) is analysed via a Hawkes process model. It is found that this past dependent model is a good fit to improvised explosive device attacks yielding key insights about the nature of terrorism in NI. We also present a novel approach to quantitatively investigate some of the sociological theory surrounding the Provisional Irish Republican Army which challenges previously held assumptions concerning changes seen in the organisation. Finally, we extend our use of the Hawkes process model by considering a multidimensional version which permits both self and mutual-excitations. This allows us to test how the Provisional Irish Republican Army responded to past improvised explosive device attacks on different geographical scales from which we find evidence for the autonomy of the organisation over the six counties of NI and Belfast. By incorporating a second dataset concerning British Security Force (BSF) interventions, the multidimensional model allows us to test counter-terrorism (CT) operations in NI where we find subsequent increases in violence.

Mean residual life and failure rate functions are ubiquitously employed in reliabilityanalysis. The term of useful period of lifetime distributions of bathtub-shaped failurerate functions is referred to the flat rigion of this function and has attracted authorsand researchers in reliability, actuary, and survival analysis. In recent years,considering the change points of mean residual life and failure rate functions has beenextensively utelized in determining the optimum burn-in time. In this paper we investigatethe difference between the change points of failure rate and mean residual life functionsof some generalized gamma type distributions due to the capability of these distributionsin modeling various bathtub-shaped failure rate functions.

We investigated differences between Alzheimer's disease (AD) and vascular dementia (VaD) from the appearance of the first cognitive symptoms, focusing on both time of onset and rate of accelerated decline for different cognitive functions before dementia diagnosis. Data from a longitudinal population-based study were used, including 914 participants (mean age = 82.0 years, SD = 5.0) tested with a cognitive battery (word recall and recognition, Block Design, category fluency, clock reading) on up to four occasions spanning 10 years. We fit a series of linear mixed effects models with a change point to the cognitive data, contrasting each dementia group to a control group. Significant age-related decline was observed for all five cognitive tasks. Relative to time of diagnosis, the preclinical AD persons deviated from the normal aging curve earlier (up to 9 years) compared to the preclinical VaD persons (up to 6 years). However, once the preclinical VaD persons started to decline, they deteriorated at a faster rate than the preclinical AD persons. The results have important implications for identifying the two dementia disorders at an early stage and for selecting cognitive tasks to evaluate treatment effects for persons at risk of developing AD and VaD. (JINS, 2012, 18, 191–199)

We used data from two population-based longitudinal studies to estimate time of onset and rate of accelerated decline across cognitive domains before dementia diagnosis. The H70 includes an age-homogeneous sample (127 cases and 255 non-cases) initially assessed at age 70 with 12 follow-ups over 30 years. The Kungsholmen Project (KP) includes an age-heterogeneous sample (279 cases and 562 non-cases), with an average age of 82 years at initial assessment, and 4 follow-ups spanning 13 years. We fit mixed linear models to the data and determined placement of change points by a profile likelihood method. Results demonstrated onset of accelerated decline for fluid (speed, memory) versus crystallized (verbal, clock reading) abilities occurring approximately 10 and 5 years before diagnosis, respectively. Although decline before change points was greater for fluid abilities, acceleration was more pronounced for crystallized abilities after the change points. This suggests that onset and rate of acceleration vary systematically along the fluid-crystallized ability continuum. There is early onset in fluid abilities, but these changes are difficult to detect due to substantial age-related decline. Onset occurred later and acceleration was greater in crystallized abilities, suggesting that those markers may provide more valid identification of cases in later stages of the prodromal phase. (JINS, 2011, 17, 000–000)

We analyse several aspects of a class of simple counting processes that can emerge in some fields of applications where a change point occurs. In particular, under simple conditions we prove a significant inequality for the stochastic intensity.

Jump points in curves arise when the conditions under which data are generated change suddenly, for example because of an unplanned change in a treatment. This paper suggests bootstrap methods for quantifying the error in estimates of jump points, and for constructing confidence intervals for jump points and confidence bands for the curve. These problems have the unusual feature that the sampling error of the jump-point estimator often has a highly non-normal distribution, which depends intimately on the distribution of regression errors. The methods are illustrated by a simulation study as well as by an application to data on the annual flow volume of the Nile river.

The paper studies the impact of a broadly understood trend, which includes a change point in mean and monotonic trends studied by Bhattacharya et al. (1983), on the asymptotic behaviour of a class of tests designed to detect long memory in a stationary sequence. Our results pertain to a family of tests which are similar to Lo's (1991) modified R/S test. We show that both long memory and nonstationarity (presence of trend or change points) can lead to rejection of the null hypothesis of short memory, so that further testing is needed to discriminate between long memory and some forms of nonstationarity. We provide quantitative description of trends which do or do not fool the R/S-type long memory tests. We show, in particular, that a shift in mean of a magnitude larger than N-½, where N is the sample size, affects the asymptotic size of the tests, whereas smaller shifts do not do so.

Let P(c) = P(X1 ≦ c1, · ··, Xp ≦ cp) for a random vector (X1, · ··, Xp). Bounds are considered of the form where T is a spanning tree corresponding to the bivariate probability structure and di is the degree of the vertex i in T. An optimized version of this inequality is obtained. The main result is that alwayṡ dominates certain second-order Bonferroni bounds. Conditions on the covariance matrix of a N(0,Σ) distribution are given so that this bound applies, and various applications are given.