Estimation of the offspring mean in a branching process with non stationary immigration

© 2016, © Taylor & Francis Group, LLC. In the paper, we consider a natural estimator of the offspring mean of a branching process with non stationary immigration based on observation of population sizes and number of immigrating individuals to each generation. We demonstrate that using a central limit theorem for multiple sums of dependent random variables it is possible to derive asymptotic distributions for the estimator without prior knowledge about the behavior (criticality) of the reproduction process. Before the three cases of criticality have been considered separately. Assuming that the immigration mean and variance vary regularly, conditions guaranteeing the strong consistency of the proposed estimator is also derived

What we look at in paintings: A comparison between experienced and inexperienced art viewers

How do people look at art? Are there any differences between how experienced and inexperienced art viewers look at a painting? We approach these questions by analyzing and modeling eye movement data from a cognitive art research experiment, where the eye movements of twenty test subjects, ten experienced and ten inexperienced art viewers, were recorded while they were looking at paintings. Eye movements consist of stops of the gaze as well as jumps between the stops. Hence, the observed gaze stop locations can be thought of as a spatial point pattern, which can be modeled by a spatio-temporal point process. We introduce some statistical tools to analyze the spatio-temporal eye movement data, and compare the eye movements of experienced and inexperienced art viewers. In addition, we develop a stochastic model, which is rather simple but fits quite well to the eye movement data, to further investigate the differences between the two groups through functional summary statistics

Asymptotic normality of quadratic forms of martingale differences

We establish the asymptotic normality of a quadratic form QnQn in martingale difference random variables ηtηt when the weight matrix A of the quadratic form has an asymptotically vanishing diagonal. Such a result has numerous potential applications in time series analysis. While for i.i.d. random variables ηtηt, asymptotic normality holds under condition ||A||sp=o(||A||)||A||sp=o(||A||), where ||A||sp||A||sp and ||A|| are the spectral and Euclidean norms of the matrix A, respectively, finding corresponding sufficient conditions in the case of martingale differences ηtηt has been an important open problem. We provide such sufficient conditions in this paper

Statistical tests for large tree-structured data

We develop a general statistical framework for the analysis and inference of large tree-structured data, with a focus on developing asymptotic goodness-of-fit tests. We first propose a consistent statistical model for binary trees, from which we develop a class of invariant tests. Using the model for binary trees, we then construct tests for general trees by using the distributional properties of the Continuum Random Tree, which arises as the invariant limit for a broad class of models for tree-structured data based on conditioned Galton–Watson processes. The test statistics for the goodness-of-fit tests are simple to compute and are asymptotically distributed as χ2 and F random variables. We illustrate our methods on an important application of detecting tumour heterogeneity in brain cancer. We use a novel approach with tree-based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients

Detecting the direction of a signal on high-dimensional spheres: Non-null and Le Cam optimality results

We consider one of the most important problems in directional statistics, namely the problem of testing the null hypothesis that the spike direction $\theta$ of a Fisher-von Mises-Langevin distribution on the $p$-dimensional unit hypersphere is equal to a given direction $\theta_0$. After a reduction through invariance arguments, we derive local asymptotic normality (LAN) results in a general high-dimensional framework where the dimension $p_n$ goes to infinity at an arbitrary rate with the sample size $n$, and where the concentration $\kappa_n$ behaves in a completely free way with $n$, which offers a spectrum of problems ranging from arbitrarily easy to arbitrarily challenging ones. We identify various asymptotic regimes, depending on the convergence/divergence properties of $(\kappa_n)$, that yield different contiguity rates and different limiting experiments. In each regime, we derive Le Cam optimal tests under specified $\kappa_n$ and we compute, from the Le Cam third lemma, asymptotic powers of the classical Watson test under contiguous alternatives. We further establish LAN results with respect to both spike direction and concentration, which allows us to discuss optimality also under unspecified $\kappa_n$. To investigate the non-null behavior of the Watson test outside the parametric framework above, we derive its local asymptotic powers through martingale CLTs in the broader, semiparametric, model of rotationally symmetric distributions. A Monte Carlo study shows that the finite-sample behaviors of the various tests remarkably agree with our asymptotic results.Comment: 47 pages, 4 figure

Generalized Whittle-MatEˊ\acute{\text{E}}rn random field as a model of correlated fluctuations

This paper considers a generalization of Gaussian random field with covariance function of Whittle-Mat$\acute{\text{e}}$rn family. Such a random field can be obtained as the solution to the fractional stochastic differential equation with two fractional orders. Asymptotic properties of the covariance functions belonging to this generalized Whittle-Mat$\acute{\text{e}}$rn family are studied, which are used to deduce the sample path properties of the random field. The Whittle-Mat$\acute{\text{e}}$rn field has been widely used in modeling geostatistical data such as sea beam data, wind speed, field temperature and soil data. In this article we show that generalized Whittle-Mat$\acute{\text{e}}$rn field provides a more flexible model for wind speed data.Comment: 22 pages, 10 figures, accepted by Journal of Physics

Determining the best population-level alcohol consumption model and its impact on estimates of alcohol-attributable harms

BACKGROUND: The goals of our study are to determine the most appropriate model for alcohol consumption as an exposure for burden of disease, to analyze the effect of the chosen alcohol consumption distribution on the estimation of the alcohol Population- Attributable Fractions (PAFs), and to characterize the chosen alcohol consumption distribution by exploring if there is a global relationship within the distribution. METHODS: To identify the best model, the Log-Normal, Gamma, and Weibull prevalence distributions were examined using data from 41 surveys from Gender, Alcohol and Culture: An International Study (GENACIS) and from the European Comparative Alcohol Study. To assess the effect of these distributions on the estimated alcohol PAFs, we calculated the alcohol PAF for diabetes, breast cancer, and pancreatitis using the three above-named distributions and using the more traditional approach based on categories. The relationship between the mean and the standard deviation from the Gamma distribution was estimated using data from 851 datasets for 66 countries from GENACIS and from the STEPwise approach to Surveillance from the World Health Organization. RESULTS: The Log-Normal distribution provided a poor fit for the survey data, with Gamma and Weibull distributions providing better fits. Additionally, our analyses showed that there were no marked differences for the alcohol PAF estimates based on the Gamma or Weibull distributions compared to PAFs based on categorical alcohol consumption estimates. The standard deviation of the alcohol distribution was highly dependent on the mean, with a unit increase in alcohol consumption associated with a unit increase in the mean of 1.258 (95% CI: 1.223 to 1.293) (R2 = 0.9207) for women and 1.171 (95% CI: 1.144 to 1.197) (R2 = 0. 9474) for men. CONCLUSIONS: Although the Gamma distribution and the Weibull distribution provided similar results, the Gamma distribution is recommended to model alcohol consumption from population surveys due to its fit, flexibility, and the ease with which it can be modified. The results showed that a large degree of variance of the standard deviation of the alcohol consumption Gamma distribution was explained by the mean alcohol consumption, allowing for alcohol consumption to be modeled through a Gamma distribution using only average consumption