The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.

In this article we study a small random perturbation of a linear recurrence equation. If all the roots of its corresponding characteristic equation have modulus strictly less than one, the random linear recurrence goes exponentially fast to its limiting distribution in the total variation distance as time increases. By assuming that all the roots of its corresponding characteristic equation have modulus strictly less than one and rather mild conditions, we prove that this convergence happens as a switch-type, i.e., there is a sharp transition in the convergence to its limiting distribution. This fact is known as a cut-off phenomenon in the context of stochastic processes.

The Central Limit Theorem (CLT) for additive functionals of Markov chains is a well-known result with a long history. In this paper, we present applications to two finite-memory versions of the Elephant Random Walk, solving a problem from Gut and Stadtmüeller (2018). We also present a derivation of the CLT for additive functionals of finite state Markov chains, which is based on positive recurrence, the CLT for IID sequences and some elementary linear algebra, and which focuses on characterization of the variance.

Motivated by models in engineering and also biology we determine in closed form the probability density function of the angle shaped by two random chords in a fixed disc. Our main result focus on the event in which the intersection locates inside the fixed disc and establishes a sine law.

For Λ-n-coalescents with mutation, we analyse the minimal observable clade size ${\mathit{O}}_{\mathit{n}}$. For a given individual $\mathit{i}\in \{1,\dots ,\mathit{n}\}$, the minimal observable clade size ${\mathit{O}}_{\mathit{n}}(\mathit{i})$ is 1 plus the number of other individuals j such that every non-singleton mutation inherited by i is also inherited by j. We provide asymptotics of ${\mathit{O}}_{\mathit{n}}$ as $\mathit{n}\to \infty$ and a recursion for all moments of ${\mathit{O}}_{\mathit{n}}$ for finite n. This statistic gives an upper bound for the minimal clade size (Advances in Applied Probability 37 (2005) 647–662) which is not observable in real data (in contrast to ${\mathit{O}}_{\mathit{n}}$). In applications to genetics, it has been shown to be useful to lower classification errors in genealogical model selection (Freund and Siri-Jégousse, 2020).

Recently, there has been a growing interest in integer-valued time series models, including integer-valued autoregressive (INAR) models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models, but only a few of them can deal with data on the full set of integers, that is, $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$. Although some attempts have been made to deal with $\mathbb{Z}$-valued time series, these models do not provide enough flexibility in modeling some specific integers (e.g., 0, $\pm 1$). A symmetric Skellam INGARCH$(1,1)$ model was proposed in the literature, but it only considered zero-mean processes, which limits its application. We first extend the symmetric Skellam INGARCH model to an asymmetric version, which can deal with non-zero-mean processes. Then we propose a modified Skellam model which adopts a careful treatment on integers 0 and $\pm 1$ to satisfy a special feature of the data. Our models are easy-to-use and flexible. The maximum likelihood method is used to estimate unknown parameters and the log-likelihood ratio test statistic is provided for testing the asymmetric model against the modified one. Simulation studies are given to evaluate performances of the parametric estimation and log-likelihood ratio test. A real data example is also presented to demonstrate good performances of newly proposed models.

In this paper, we consider a semiparametric regression model where the error follows a scale mixture of Gaussian distributions. The purpose is to estimate the target function which is assumed to belong to some class of functions using the EM algorithm and approximations via P-splines and B-splines. We illustrate the proposed methodology through several simulation studies. Other forms of function approximation are also studied, namely Fourier and wavelet expansions.

In many applications, data exhibit skewness and in this paper we present a new family of density functions modeling skewness based on a transformation, analogous to those of location and scale. Here we note that location will always refer to mode. Hence, in order to model data to include shape, we need only to find a family of densities exhibiting a variety of shapes, since we can obtain the other three properties via the transformations. The chosen class of densities with the variety of shape is, we argue, the simplest available. Illustrations including regression and time series models are given.

In this paper, we establish some results for multivariate selection scale-mixtures of normal distributions with arbitrary mixing variable. First, we discuss their stochastic representation in terms of multivariate selection normal distributions. Next, the conditional distributions as well as the first two moments of multivariate selection scale-mixtures of normal distributions are obtained when the selection set is an arbitrary rectangle in the q-dimensional Euclidean space of ${\mathbb{R}}^{\mathit{q}}$. The unified skew-scale mixture of normal (SUSMN) distributions are subsequently discussed as a special case. As a subclass of SUSMN distributions, the class of unified skew-symmetric generalized hyperbolic (SUSGH) distributions are studied in detail. Finally, we show that our results can be used to obtain moments of L-statistics and of multivariate concomitants from multivariate scale-mixtures of normal distributions.

This work deals with problem of estimating the odds using judgment post stratification (JPS) sampling design. Several estimators of the odds are described and the asymptotic normality of each of them is established. Monte Carlo simulation study is then used to compare different estimators of the odds in the JPS with the standard estimator in simple random sampling (SRS) with replacement for both perfect/imperfect ranking and for both JPS data with/without empty strata. The comparison results indicate that the estimators developed here can be highly more efficient than their SRS counterpart in some certain circumstances. Finally, a real dataset from the third National Health and Nutrition Examination Survey (NHANES III) is employed for illustration purposes.

Suppose ${\mathit{X}}_1,\dots ,{\mathit{X}}_{\mathit{n}}$ is a random sample from a bounded and decreasing density ${\mathit{f}}_0$ on $[0,\infty )$. We are interested in estimating such ${\mathit{f}}_0$, with special interest in ${\mathit{f}}_0(0)$. This problem is encountered in various statistical applications and has gained quite some attention in the statistical literature. It is well known that the maximum likelihood estimator is inconsistent at zero. This has led several authors to propose alternative estimators which are consistent. As any decreasing density can be represented as a scale mixture of uniform densities, a Bayesian estimator is obtained by endowing the mixture distribution with the Dirichlet process prior. Assuming this prior, we derive contraction rates of the posterior density at zero by carefully revising arguments presented in Salomond (Electronic Journal of Statistics 8 (2014) 1380–1404). Several choices of base measure are numerically evaluated and compared. In a simulation various frequentist methods and a Bayesian estimator are compared. Finally, the Bayesian procedure is applied to current durations data described in Slama et al. (Human Reproduction 27 (2012) 1489–1498).

A Bayesian nonparametric estimator to entropy is proposed. The derivation of the new estimator relies on using the Dirichlet process and adapting the well-known frequentist estimators of Vasicek (Journal of Royal Statistical Society B 38 (1976) 54–59) and Ebrahimi, Pflughoeft and Soofi (Statistics & Probability Letters 20 (1994) 225–234). Several theoretical properties, such as consistency, of the proposed estimator are obtained. The quality of the proposed estimator has been investigated through several examples, in which it exhibits excellent performance.