Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

Abstract. We consider a stochastic process driven by diffusions and jumps. Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred. This allows us to determine a consistent non‐parametric estimator of the integrated volatility when the infinite activity jump component is Lévy. Jump size estimation and central limit results are proved in the case of finite activity jumps. Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data.     

On the use of different metrics for assessing complex pattern reproductions

Nowadays, there is an increasing interest in multi-point models and their applications in Earth sciences. However, users not only ask for multi-point methods able to capture the uncertainties of complex structures and to reproduce the properties of a training image, but also they need quantitative tools for assessing whether a set of realizations have the properties required. Moreover, it is crucial to study the sensitivity of the realizations to the size of the data template and to analyze how fast realization-based statistics converge on average toward training-based statistics. In this paper, some similarity measures and convergence indexes, based on some physically measurable quantities and cumulants of high-order, are presented. In the case study, multi-point simulations of the spatial distribution of coarse-grained limestone and calcareous rock, generated by using three templates of different sizes, are compared and convergence toward training-based statistics is analyzed by taking into account increasing numbers of realizations.     

Further results for parallel systems with two heterogeneous exponential components

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components with respect to likelihood ratio and hazard rate orders. Two sufficient conditions are provided for likelihood ratio and hazard rate orders to hold between the lifetimes of two parallel systems, respectively. Moreover, we extend the results from exponential case to the proportional hazard rate models. The results established here strength some of the results known in the literature. Finally, some numerical examples are given to illustrate the theoretical results derived here as well.     

On dynamic stochastic approximation

The behaviour of the dynamic stochastic approximation algorithm suggested by DUPA[Cbreve] [4, 5] is examined under various conditions. Essentially, convergence in q.m. is proved and the rate of convergence is obtained for three classes of regression functions. Necessary and sufficient conditions are established for the convergence in q.m. of the algorithm in terms of the algorithm parameters for the stationary (nondynamic) case.     

On a statistic useful in dimensionality reduction in multivariable linear stochastic system

In this paper we study the sampling properties of a test statistic which has important applications in the area of linear stochastic control systems with multi-inputs and multi-outputs. The statistic is the ratio of a partial sum of the eigenvalues of a sample covariance matrix and its trace. It turns out that using a method due to Sugiura we may derive a useful approximation for its distribution up to and including terms of order l/n, where n denotes the appropriate size. Numerical illustrations using real data are given.     

Determining Process Death Based on Censored Activity Data

This article addresses the problem of estimating the time of apparent death in a binary stochastic process. We show that, when only censored data are available, a fitted logistic regression model may estimate the time of death incorrectly. We improve this estimation by utilizing discrete-event simulation to produce simulated complete time series data. The proposed methodology may be applied to situations where time of death cannot be formally determined and has to be estimated based on prolonged inactivity. As an illustration, we use observed monthly activity patterns from 300 real Open Source Software development projects sampled from Sourceforge.net.     

A Copula‐Based Non‐parametric Measure of Regression Dependence

Abstract. This article presents a framework for comparing bivariate distributions according to their degree of regression dependence. We introduce the general concept of a regression dependence order (RDO). In addition, we define a new non‐parametric measure of regression dependence and study its properties. Besides being monotone in the new RDOs, the measure takes on its extreme values precisely at independence and almost sure functional dependence, respectively. A consistent non‐parametric estimator of the new measure is constructed and its asymptotic properties are investigated. Finally, the finite sample properties of the estimate are studied by means of a small simulation study.     

Maximum Likelihood Estimation for Stochastic Differential Equations with Random Effects

Abstract. We consider N independent stochastic processes (X _i (t), t ∈ [0,T _i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ _i . The distribution of the random effect φ _i depends on unknown parameters which are to be estimated from the continuous observation of the processes X_i. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ _i and φ _i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T _i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.     

Zenga Distribution and Inequality Ordering

The aim of this article is to establish an ordering related to the inequality for the recently introduced Zenga distribution. In addition to the well-known order based on the Lorenz curve, the order based on I(p) curve is considered. Since the Zenga distribution seems to be suitable to model wealth, financial, actuarial, and, especially, income distributions, these findings are fundamental in the understanding of how parameter values are related to inequality. This investigation shows that for the Zenga distribution, two of the three parameters are inequality indicators.