Spatial scan statistics are commonly used to detect clustering. We present a Bayesian spatial scan statistic for multinomial data. After validating our method with a simulation study, we use it to detect clusters of SARS-CoV-2 infection/immunity in South Carolina.

We consider identification and inference about a counterfactual outcome mean when there is unmeasured confounding using tools from proximal causal inference (Miao et al., 2018, Tchetgen Tchetgen et al., 2020). Proximal causal inference requires existence of solutions to at least one of two integral equations. We motivate the existence of solutions to the integral equations from proximal causal inference by demonstrating that, assuming the existence of a solution to one of the integral equations, n-estimability of a linear functional (such as its mean) of that solution requires the existence of a solution to the other integral equation. Solutions to the integral equations may not be unique, which complicates estimation and inference. We construct a consistent estimator for the solution set for one of the integral equations and then adapt the theory of extremum estimators to find from the estimated set a consistent estimator for a uniquely defined solution. A debiased estimator for the counterfactual mean is shown to be root-n consistent, regular, and asymptotically semiparametrically locally efficient under additional regularity conditions.

Recently, Wang and Tchetgen Tchetgen (2018) showed that the global average treatment effect is identifiable even in the presence of unmeasured confounders so long as they do not modify the instrument’s additive effect on the treatment. We use a simple and direct method to show that this no-interaction assumption allows identification of the entire outcome distribution, which leads to multiply robust estimation procedures for nonlinear functionals like the quantile and Mann–Whitney treatment effects. Similarly, we can bound these causal estimands through the outcome distribution in sensitivity analysis against confounder–instrument interaction.

Neural networks have become increasingly popular in the field of machine learning and have been successfully used in many applied fields (e.g., imaging recognition). With more and more research has been conducted on neural networks, we have a better understanding of the statistical proprieties of neural networks. While many studies focus on bounding the prediction error of neural network estimators, limited research has been done on the statistical inference of neural networks. From a statistical point of view, it is of great interest to investigate the statistical inference of neural networks as it could facilitate hypothesis testing in many fields (e.g., genetics, epidemiology, and medical science). In this paper, we propose a goodness-of-fit test statistic based on neural network sieve estimators. The test statistic follows an asymptotic distribution, which makes it easy to use in practice. We have also verified the theoretical asymptotic results via simulation studies and a real data application.

Generalized rank regression, which is a class of weighted rank regression with weights based on factor space, provides a powerful tool for conducting robust estimation. In this article, we first establish the asymptotic properties of generalized rank regression under local model misspecification. We then apply the generalized rank regression to the focus information criterion and frequentist model averaging and establish their properties.

One-parameter natural exponential family (NEF) plays fundamental roles in probability and statistics. This article contains two independent results: (a) A conjecture of Bar-Lev, Bshouty and Enis states that a polynomial with a simple root at $0$ and a complex root with positive imaginary part is the variance function of some NEF with mean domain $\left(0,\infty\right)$ if and only if the real part of the complex root is not positive. This conjecture is resolved. The positive answer to this conjecture enlarges existing family of polynomials that are able to generate NEFs, and it helps prevent practitioners from choosing incompatible functions as variance functions for statistical modeling using NEFs. (b) if a random variable $\xi$ has parametric distributions that form a infinitely divisible NEF whose induced measure is absolutely continuous with respect to its basis measure, then there exists a deterministic function $h$, called function, such that $\mathbb{E} \left(h\left(\xi\right)\right)=\mathbb{V}\left(\xi\right)$, i.e., $h\left(\xi\right)$ is an unbiased estimator of the variance of $\xi$. The reduction function has applications to estimating latent, low-dimensional structures and to dimension reduction in the first and/or second moments in high-dimensional data.

Abstract Several unbiased length-intensity estimators for stationary segment processes in the d -dimensional Euclidean space are considered. The variances of the estimators are compared. The asymptotic behaviour is investigated if the sampling window increases unboundedly in all directions. The segments are assumed to be independent, identically distributed and independent of the locations. Special attention is devoted to the particular case of stationary Poisson segment processes. For the planar case and rectangular sampling windows, Neyman–Scott segment processes are studied in more detail.

Abstract In this note we prove that the spaces ( S ) , ( S ) ∗ and G , G ∗ are invariant under a certain class of translations of the underlying Brownian motion. This problem arises naturally in dealing with anticipating stochastic differential equations, in particular when the Girsanov theorem is involved. The proofs are based on a Bayes formula for second-quantization operators that was derived by Lanconelli [Lanconelli, A., 2006a. Bayes’ formula for second quantization operators. Stoch. Dyn. 6 (2), 245–253] and on the properties of the translation operators.

In this paper the problem of testing a multivariate point hypothesis is considered. Of interest is the relationship between the p -value and the posterior probability. A Bayesian test for simple H 0 : θ = θ 0 versus bilateral H 0 : θ ≠ θ 0 , with a mixed prior distribution for the parameter θ , is developed. The methodology consists of fixing a sphere of radius δ around θ 0 and assigning a prior mass, π 0 , to H 0 by integrating the density π ( θ ) over this sphere and spreading the remainder, 1 − π 0 , over H 1 according to π ( θ ) . A theorem that shows when the frequentist and Bayesian procedures can give rise to the same decision is proved. Then, some examples are revisited.

We present non-linear wavelet methods to compute simultaneous confidence intervals for f(x) when f is a functional parameter issued from a non-parametric model. The levels of the intervals are at least γ, and we prove that they achieve the minimum diameter up to a logarithmic term. The procedure is data-driven and the adaptation is made via the Lepskii's algorithm.