In recent years, the demand for small area statistics has greatly increased worldwide. Small area models are formulated with random area-specific effects assumed to account for the between-area variation that is not explained by auxiliary variables. The unit level models relate the unit values of a study variable to unit-specific covariates. The main aim of this paper is to consider small area estimation under unit level models based on count data. In particular, instead of modelling the variables assuming the Poisson distribution, which is a usual choice, we consider the square root transformation of the original data. One practical advantage is that the proposed transformation achieves approximate homoscedasticity of the error variances, reducing one layer of estimation problem. Inference for the model is carried out under the hierarchical Bayes approach. The square root transformation is evaluated under a simulation study and two design-based studies with real datasets.

We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.

We present large deviations estimates in the supremum norm for a system of independent random walks superposed with a birth-and-death dynamics evolving on the discrete torus with N sites. The scaling limit considered is the so-called high density limit (see the survey in (In From Particle Systems to Partial Differential Equations (2014) 179–189 Springer) on the subject), where space, time and initial quantity of particles are rescaled. The associated rate functional here obtained is a semi-linearized version of the rate function of (Probability Theory and Related Fields 97 (1992) 339–361), which dealt with large deviations of exclusion processes superposed with birth-and-death dynamics. An important ingredient in the proof of large deviations consists in providing a limit of a suitable class of perturbations of the original process, which is precisely one of the main contributions of this work: a strategy to extend the original high density approach (as in Advances in Applied Probability 12 (1980) 367–379; Annals of Probability 19 (1991) 1440–1462; The Annals of Applied Probability 2 (1992) 131–141; Journal of Statistical Physics 149 (2012) 629–642; Annals of Probability 14 (1986) 173–193; Probability Theory and Related Fields 78 (1988) 11–37) to weakly asymmetric systems. Two cases are considered with respect to the initial quantity of particles, the power law and the (at least) exponential growth. In the first case, we present the lower bound only on a certain set of smooth profiles, while in the second case under some extra technical assumptions we provide a full large deviations principle.

Variable selection has become an indispensable part of statistical analysis for high-dimensional datasets. However, classical variable selection algorithms, such as regularization methods, are computationally high demanding when sample size and dimension of dataset are both large. Lin, Foster and Ungar (Journal of the American Statistical Association 106 (2011) 232–247) proposed a variable selection algorithm called VIF regression for massive datasets which is more computationally efficient and able to control the marginal false discovery rate. Building on the idea of VIF regression, we propose a new variable selection algorithm, Double-Gates Streamwise regression (DGS), which quickly tests whether predictors significantly reduce the prediction error in one-pass search. DGS regression has two main appealing features. First, DGS regression is computationally efficient and low demanding in the usage of memory. Second, DGS regression can control the false discovery rate, and hence improve the predictive and explanatory abilities. Its advantages relative to VIF regression and some other popular variable selection algorithms are demonstrated in extensive numerical simulated experiments and a real dataset analysis.

We consider the discrete renewal equation. The inhomogeneous term behaves like a nondecreasing submultiplicative sequence. Asymptotic properties of the solution are established depending on the asymptotics of the submultiplicative sequence.

Multivariate two-sample testing problems often arise from the statistical analysis for scientific data, especially for bioinformatics data. To detect components with different values between two mean vectors, well-known procedures are to apply Sum-of-Squares type tests, such as Hotelling’s ${\mathit{T}}^2$-test. However, such a test is not suitable to high dimensional settings because of singular covariance matrix and accumulated errors. Nowadays, a lot of test methods for high dimensional data are developed, mainly including two types, Sum-of-Squares type and Max type. The Sum-of-Squares type test statistics have poor performance against sparse alternatives. And the Max type test statistic is not powerful enough to deal with non-sparse datasets. In this paper, we propose a Max-Partial-Sum type statistic named Neyman’s Truncation test, which is conducted by maximum partial sums of marginal test statistics. Besides non-sparse datasets, Neyman’s Truncation test also has great power against dense and sparse alternatives. The asymptotic distribution of the test statistic under null hypothesis is obtained and the power of the test is analyzed. To avoid the slow convergence rate of the asymptotic distribution, we realize our method by Bootstrap procedures. Simulation studies and the analysis of leukemia dataset are carried out to verify the numerical performance.

Recently a functional limit theorem for sums of moving averages with random coefficients and i.i.d. heavy tailed innovations has been obtained under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series. The convergence takes place in the space $\mathit{D}[0,1]$ of càdlàg functions with the Skorohod ${\mathit{M}}_2$ topology. In this article, we extend this result to the case when the innovations are weakly dependent in the sense of strong mixing and local dependence condition ${\mathit{D}}^{\prime }$.

Kim (J. Korean Stat. Soc. 37 (2008) 81–87) introduced an incorrect stochastic representation (SR) for the truncated Student-t (Tt) random variable. By pointing out that the gamma mixture based on a truncated normal distribution actually cannot result in a true Tt distribution, in this paper, we first propose three correct SRs and then recalculate the corresponding moments of the Tt distribution. Different from those derived by following the invalid SR of Kim (J. Korean Stat. Soc. 37 (2008) 81–87), the correct moments of the Tt distribution play a crucial role in parameter estimations. Based on the third SR proposed and the correct expressions of truncated moments, expectation–maximization (EM) algorithms are developed for calculating the maximum likelihood estimates of parameters in the Tt distribution. Extensions to a Tt regression model and a t interval-censored regression model are provided as well. Simulated experiments are conducted to evaluate the performance of the proposed methods. Finally, two real data analyses corroborate the theoretical results.

We study the probability that one beta-distributed random variable exceeds the maximum of two others, allowing all three to have general parameters. This amounts to studying Euler transforms of products of two incomplete beta functions. We provide a closed form for the general problem in terms of Kampé de Fériet functions and a variety of simpler closed forms in special cases. The results are applied to derive the moments of the maximum of two independent beta-distributed random variables and to find inner products of incomplete beta functions. Restricted to positive integer parameters, our results are applied to determine an expected exit time for a conditioned random walk and also to a combinatorial problem of enumerating strings comprised of three different letters, subject to constraints.