Generalized fiducial inference on the mean of zero-inflated Poisson and Poisson hurdle models

Zero-inflated and hurdle models are widely applied to count data possessing excess zeros, where they can simultaneously model the process from how the zeros were generated and potentially help mitigate the effects of overdispersion relative to the assumed count distribution. Which model to use depends on how the zeros are generated: zero-inflated models add an additional probability mass on zero, while hurdle models are two-part models comprised of a degenerate distribution for the zeros and a zero-truncated distribution. Developing confidence intervals for such models is challenging since no closed-form function is available to calculate the mean. In this study, generalized fiducial inference is used to construct confidence intervals for the means of zero-inflated Poisson and Poisson hurdle models. The proposed methods are assessed by an intensive simulation study. An illustrative example demonstrates the inference methods.

A NOTE ON THE ASYMPTOTIC BEHAVIOR OF THE HEIGHT FOR A BIRTH-AND-DEATH PROCESS

Recently, the asymptotic mean value of the height for a birth-and-death process is given in Videla [Videla, L.A. (2020)]. We consider the asymptotic variance of the height in the case when the number of states tends to infinity. Further, we prove that the heights exhibit a cutoff phenomenon and that the normalized height converges to a degenerate distribution.

Regulating gene expression to achieve temporal precision

Cellular response to an environmental change is often triggered by accumulation of an appropriate gene product up to a critical threshold. How do cells regulate gene expression to achieve precision in timing of such responses is a question of interest. Earlier work has shown that for a stable gene product, a constant rate of accumulation provides minimum noise in the time of response, provided that initial gene product distribution is degenerate. Here, we show that this strategy is no longer optimal if the initial gene product level is drawn from a non-degenerate distribution. Finally, we discuss biological relevance of these findings.

Note on History of Age Replacement Policies

This paper tries to trace our research history briefly from Barlow and Proschan to attain general replacement models. We begin with a random age replacement policy that is planned at a random time Y and call it as random replacement. When the distribution of Y becomes a degenerate distribution placing unit mass at T, age replacement is formulated. We obtain the general formulas for optimum replacement times. We next suppose the unit works for a job with random works, and replacement policies with N cycles are discussed. As follows, we combine age and random replacement models and discuss replacement first, replacement last, replacement overtime, replacement overtime first and replacement overtime last. By formulating the distributions of replacement times with n variables, general replacement models with n replacement times are obtained.

Einstein-Hilbert type action on spacetimes

The mixed gravitational field equations have been recently introduced for codimension one foliated manifolds, e.g. stably causal and globally hyperbolic spacetimes. These Euler-Lagrange equations for the total mixed scalar curvature (as analog of Einstein-Hilbert action) involve a new kind of Ricci curvature (called the mixed Ricci curvature). In the work, we derive Euler-Lagrange equations of the action for any spacetime, in fact, for a pseudo-Riemannian manifold endowed with a non-degenerate distribution. The obtained equations are presented in the classical form of Einstein field equation with the new Ricci type curvature instead of Ricci curvature

From the divergence between two measures to the shortest path between two observables

We consider two independent and stationary measures over $\unicode[STIX]{x1D712}^{\mathbb{N}}$, where $\unicode[STIX]{x1D712}$ is a finite or countable alphabet. For each pair of $n$-strings in the product space we define $T_{n}^{(2)}$ as the length of the shortest path connecting one of them to the other. Here the paths are generated by the underlying dynamic of the measures. If they are ergodic and have positive entropy we prove that, for almost every pair of realizations $(\mathbf{x},\mathbf{y})$, $T_{n}^{(2)}/n$ is concentrated in one, as $n$ diverges. Under mild extra conditions we prove a large-deviation principle. We also show that the fluctuations of $T_{n}^{(2)}$ converge (only) in distribution to a non-degenerate distribution. These results are all linked to a quantity that computes the similarity between those two measures. This is the so-called divergence between two measures, which is also introduced. Several examples are provided.

On the range of the transient frog model on ℤ

We observe the frog model, an infinite system of interacting random walks, on ℤ with an asymmetric underlying random walk. For certain initial frog distributions we construct an explicit formula for the moments of the leftmost visited site, as well as their asymptotic scaling limits as the drift of the underlying random walk vanishes. We also provide conditions in which the lower bound can be scaled to converge in probability to the degenerate distribution at 1 as the drift vanishes.

The degree profile and weight in Apollonian networks and k-trees

We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi–Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k-trees, with minor modification in the proof methods, done mutatis mutandis.