Statistics & Probability Letters最新文献

In the paper, upper bounds for the convergence rate in the limit theorems for random sums of $m$-orthogonal random variables are estimated using the $K$-functional method. Our results are extensions of some known results related to random sums.

Estimating hidden Markov models (HMMs) with unknown number of states is a challenging task. In this paper, we propose a new penalized composite likelihood approach for simultaneously estimating both the number of states and the parameters in an overfitted HMM. We prove the order selection consistency and asymptotic normality of the resultant estimator. Simulation studies and an application demonstrate the finite sample performance of the proposed method.

We prove a Berry–Esseen bound in de Jong’s classical CLT for normalized, completely degenerate $U$-statistics, which says that the convergence of the fourth moment sequence to three and a Lindeberg–Feller type negligibility condition are sufficient for asymptotic normality. Our bound is of the same optimal order as the bound on the Wasserstein distance to normality that has recently been proved by Döbler and Peccati (2017).

The Random Utility Model, central in stochastic choice theory, is equivalent to assume that a probability vector belongs to a convex cone. We investigate its underlying geometry, introduce two new testing procedures, and compare them by simulation.

In this paper, we study the coarse Ricci curvature of inhomogeneous random graph on vertex set $\left[n\right]≔\{1,2,\dots ,n\}$. In this graph, each pair of vertices $(i,j)$ forms an edge independently with probability $\frac{\lambda }{n^{\alpha }}g\left(\frac{i}{n},\frac{j}{n}\right)$ for some function $g(x,y)\in (0,1]$, constants $\lambda >0$ and $\alpha \in [0,1]$. We derive the asymptotic coarse Ricci curvature of this random graph. Phase transition phenomenon exists as $\alpha$ varies from zero to one.

For the discrete Weibull probability distribution we prove that it is only infinitely divisible if the shape parameter lies in the range $0<\beta \le 1$ . The proof is based on some results of Steutel and van Harn (2004). For this case we construct the corresponding compound Poisson distribution and thus the related Lévy process.

Let $X=(X_1,\dots ,X_n)$ and $Y=(Y_1,\dots ,Y_n)$ be two random vectors with common Archimedean copula with generator function $\varphi$, where, for $i=1,\dots ,n$, $X_i$ is an exponential random variable with hazard rate ${\lambda }_i$ and $Y_i$ is an exponential random variable with hazard rate $\lambda$. In this paper we prove that under some sufficient conditions on the function $\varphi$, the largest order statistic corresponding to $X$ is larger than that of $Y$ according to the dispersive ordering and hazard rate ordering. The new results generalized the results in Dykstra et al. (1997) and Khaledi and Kochar (2000). We show that the new results can be applied to some well known Archimedean copulas.

The need to accurately quantify dependence between random variables is a growing concern across various academic disciplines. Current correlation coefficients are typically intended for one of two purposes: testing independence or measuring relationship strength. Despite some attempts to address both aspects, the performance of these measures is still easily affected by oscillation and local noise. To address these limitations, we propose a new coefficient of correlation called the Adapted Chatterjee Correlation Coefficient $\left(A{C}^3\right)$. $A{C}^3$ is designed to accurately identify both independence and functional dependence between variables, even in the presence of noise. We establish the consistency and asymptotic theories of $\left(A{C}^3\right)$. Additionally, we present a novel method, called Iterative Signal Detection Procedure (ISDP), for local signal identification. Our numerical studies and real data application demonstrate that $A{C}^3$ outperforms state-of-the-art methods in terms of general performance and detecting local signals.

Replication is a commonly recommended feature of experimental designs. However, its impact in model-robust design is relatively under-explored; indeed, replication is impossible within the current formulation of random translation designs, which were introduced recently for model-robust prediction. Here we extend the framework of random translation designs to allow replication, and quantify the resulting performance impact. The extension permits a simplification of our earlier heuristic for constructing random translation strategies from a traditional $V$-optimal design. Namely, in the previous formulation any replicates of the $V$-optimal design first had to be split up before a random translation can be applied to the design points. With the new framework we can instead preserve the replicates instead if we so wish. Surprisingly, we find that in low-dimensional problems it is often substantially more efficient to continue to split replicates, while in high-dimensional problems it can be substantially better to retain replicates.

Let $X_1,\dots ,X_n$ be a random sample from a multivariate normal distribution with nonnegative mean $\mu$ and unknown covariance matrix $\Sigma$. The likelihood ratio test of $H_0:\mu =0$ conditional on $V=\sum X_i{X}_i^{\prime }$ is proven to be unbiased. Some related topics are also discussed.