Statistics & Probability Letters最新文献

In this paper, we have obtained sufficient conditions for comparing two multiple-outlier (M-O) finite $\delta$-mixtures based on the usual stochastic order and reversed hazard rate order. We have assumed a general parametric family of distributions for the subpopulations. Many distributions satisfying baseline-related conditions in the established results have also been provided as examples.

The discrete-time Hawkes process (DTHP) is a sub-class of $g$-functions that serves as a discrete-time version of the continuous-time Hawkes process (CTHP). Like the CTHP, the DTHP also has the self-exciting property and its intensity depends on the entire history. In this paper, we study the asymptotic behavior of the DTHP and its compensator. We further analyze the moment generating function (MGF) of the DTHP and obtain some bounds and convergence results on the scaled logarithmic MGF of the DTHP.

In this paper sufficient conditions for the existence and smoothness of the self-intersection local time of a class of Gaussian processes are given in the sense of Meyer–Watanabe through $L^2$ convergence and Wiener chaos expansion. Let $X$ be a centered Gaussian process, whose canonical metric $E\left[(X\left(t\right)-X{\left(s\right)}^2)\right]$ is commensurate with ${\sigma }^2\left(|t-s|\right)$, where $\sigma \left(\cdot \right)$ is continuous, increasing and concave. If ${\int }_0^T\frac{1}{\sigma \left(\gamma \right)}d\gamma <\infty$, then the self-intersection local time of the Gaussian process exists, and if ${\int }_0^T{\left(\sigma \left(\gamma \right)\right)}^{-\frac{3}{2}}d\gamma <\infty$, the self-intersection local time of the Gaussian process is smooth in the sense of Meyer–Watanabe.

This article introduces a new class of stochastic volatility models called $logGARCH$ Stochastic Volatility models ($logGARCH$-$SV$). We establish the strict stationarity and second-order stationarity properties of this model class. Additionally, we provide conditions for the existence of higher-order moments. To estimate the parameters of the proposed model, we utilize a sequential Monte Carlo method. Finally, we assess the performance of the suggested estimation method through a simulation study.

Second order backward stochastic differential equations (2BSDEs, for short) are one of useful tools in solving stochastic control problems with model uncertainty. In this paper, we prove a representation formula for quadratic 2BSDEs with an unbounded terminal value under a convex assumption on the generator. Because of the unboundedness of the terminal value, we are unable to use some fine properties of BMO martingales, which are often employed in the literature to deal with bounded solutions to quadratic backward stochastic differential equations. Instead, we utilize the $\theta$-technique. We also prove an existence result under an additional assumption that the terminal value is of uniformly continuous.

This paper investigates the Bayes estimator of the mean of an elliptically contoured distribution with unknown scale parameter under the quadratic loss. The Laplace transform and inverse Laplace transform of density facilitate us to obtain the expression of Bayes estimator. Then we prove the minimaxity of the Bayes estimator under certain conditions.

In determining the effects of a binary treatment $D$ on an outcome $Y$, a multi-valued instrumental variable (IV) $Z=0,1,\dots ,J$ often appears. Imbens and Angrist (1994, Econometrica) showed that the IV estimator (IVE) of $Y$ on $D$ using $Z$ as an IV is consistent for a non-negatively weighted average of heterogeneous “complier” effects. However, Imbens and Angrist did not include covariates $X$. This paper generalizes their finding by explicitly allowing $X$ to appear in the linear model for the IVE, and shows that the extra condition $E\left(Z\right|X)=L\left(Z\right|X)$ is necessary for generalization, where $L\left(Z\right|X)\equiv E\left(Z{X}^{\prime }\right){\left\{E\left(X{X}^{\prime }\right)\right\}}^{-1}X$ is the linear projection. This paper therefore proposes an alternative IVE using $Z-E\left(Z\right|X)$ as an IV, which is consistent for the same estimand without the restrictive extra condition. A simulation study demonstrates that the extra condition $E\left(Z\right|X)=L\left(Z\right|X)$ is necessary for the usual IVE, but not for the alternative IVE proposed in this paper.

The Frisch–Waugh–Lovell (FWL) theorem shows that for the least squares estimator, parameter estimates from full and partial models are identically same. I show that in linear regression models with a mix of exogenous and endogenous regressors, FWL theorem-type results hold for the k-class estimators (including LIML) and the two-step optimal GMM estimator.

The R-optimality criterion, suggested as a substitute for the widely used D-optimality criteria, is employed in experimental designs when the primary goal is to create a rectangular confidence region. This study explores R-optimal designs concerning third order Becker’s models and calculates weights for values of $3\le q\le 10$. Additionally, it compares and contrasts the D-optimal and R-optimal designs.

We present efficient algorithms for simultaneously computing Kendall’s tau and the jackknife estimator of its variance. For the classical pairwise tau, we describe a modification of Knight’s algorithm (originally designed to compute only tau) that does so while preserving its $O\left(n{log}_{2}n\right)$ runtime in the number of observations $n$. We also introduce a novel algorithm computing a multivariate extension of tau and its jackknife variance in $O\left(n{log}_2^{p}n\right)$ time.