Information loss on the mean for spatial processes when some values are missing

The loss of information on the mean due to the presence of missing values is discussed for a Gaussian univariate process on a rectangular lattice. The exact as well as the approximate formulae for this loss are given for general conditional autoregressive (CAR) and simultaneous autoregressive (SAR) processes. The formulae are evaluated for some low order CAR and SAR processes. The approximate formula is shown to give a good insight into how the loss varies over the different configurations of missing sites.     

A Bootstrap Algorithm for Data from a Periodic Multiplicative Intensity Function

The periodic multiplicative intensity model is considered. A new bootstrap method for non stationary counting processes which intensity function has some periodicity properties is presented. Its main advantage is that it does not destroy the temporal order and the original periodicity of the underlying counting process. The proposed algorithm is used to construct a bootstrap version of the maximum likelihood hazard function estimator. The consistency of the bootstrap method is shown. A possible modification of the proposed bootstrap method is discussed. The bootstrap simultaneous confidence intervals for the hazard function are presented. The telecommunication network traffic real data example is discussed.     

Stochastic Differential Mixed-Effects Models

Abstract.  Stochastic differential equations have been shown useful in describing random continuous time processes. Biomedical experiments often imply repeated measurements on a series of experimental units and differences between units can be represented by incorporating random effects into the model. When both system noise and random effects are considered, stochastic differential mixed-effects models ensue. This class of models enables the simultaneous representation of randomness in the dynamics of the phenomena being considered and variability between experimental units, thus providing a powerful modelling tool with immediate applications in biomedicine and pharmacokinetic/pharmacodynamic studies. In most cases the likelihood function is not available, and thus maximum likelihood estimation of the unknown parameters is not possible. Here we propose a computationally fast approximated maximum likelihood procedure for the estimation of the non-random parameters and the random effects. The method is evaluated on simulations from some famous diffusion processes and on real data sets.     

Simultaneous confidence bands for growth incidence curves in weighted sup-norm metrics

ABSTRACT

The so-called growth incidence curve (GIC) is a popular way to evaluate the distributional pattern of economic growth and pro-poorness of growth in development economics. The log-transformation of the the GIC is related to the sum of empirical quantile processes which allows for constructions of simultaneous confidence bands for the GIC. However, standard constructions of these bands tend to be too wide at the extreme points 0 and 1 because the estimator of the quantile function can be very volatile at the extreme points. In order to construct simultaneous confidence bands which are narrower at the ends, we consider the convergence of quantile processes with weight functions. In particular, we investigate the asymptotic convergence under specific weighted sup-norm metrics and compare different kinds of qualified weight functions. This implies simultaneous confidence bands that are narrower at the boundaries 0 and 1. We show in simulations that these bands have a more regular shape. Finally, we evaluate real data from Uganda with the improved confidence bands.     

Using multivariate time series methods to estimate location and climate change effects on temperature readings employed in electricity demand simulation

Long‐term historical daily temperatures are used in electricity forecasting to simulate the probability distribution of future demand but can be affected by changes in recording site and climate. This paper presents a method of adjusting for the effect of these changes on daily maximum and minimum temperatures. The adjustment technique accommodates the autocorrelated and bivariate nature of the temperature data which has not previously been taken into account. The data are from Perth, Western Australia, the main electricity demand centre for the South‐West of Western Australia. The statistical modelling involves a multivariate extension of the univariate time series ‘interleaving method’, which allows fully efficient simultaneous estimation of the parameters of replicated Vector Autoregressive Moving Average processes. Temperatures at the most recent weather recording location in Perth are shown to be significantly lower compared to previous sites. There is also evidence of long‐term heating due to climate change especially for minimum temperatures.     

Estimation of cyclic long-memory parameters

This paper studies cyclic long-memory processes with Gegenbauer-type spectral densities. For a semiparametric statistical model, new simultaneous estimates for singularity location and long-memory parameters are proposed. This generalized filtered method-of-moments approach is based on general filter transforms that include wavelet transformations as a particular case. It is proved that the estimates are almost surely convergent to the true values of parameters. Solutions of the estimation equations are studied, and adjusted statistics are proposed. Monte-Carlo study results are presented to confirm the theoretical findings.     

FITTING SPATIAL CORRELATION MODELS: APPROXIMATING A LIKELIHOOD APPROXIMATION

The paper considers a class of spatial correlation models (stationary Gaussian processes) which includes (spatial) conditional autoregressive, simultaneous autoregressive, moving average and direct covariance models. Given observations on a finite rectangular lattice, a likelihood approximation for estimating the parameters in the spectral density of the model is discussed. The approximation consists of applying the trapezoidal rule, with a her grid of frequencies than the usual Fourier frequencies, to compute the integral in an appraximation due to Whittle (1954) and later modified by Guyon (1984). With this approximation, a Fisher scoring type algorithm has a simple form and in some casea reduces to iteratively reweighted least squares. Methods for computing the unbiased two-dimensional periodogram required by the method are presented and the accuracy of the approximation is discussed. The asymptotic distribution of the parameter estimates computed from the likelihood approximation is also given.     

非正态过程能力指数估计与Bootstrap置信区间构建

传统计算非正态分布过程能力指数最经典的方法——Clement方法,其最大的缺点是必须有足够多的观测样本才能得到较为准确的结果。文章利用加权标准差可将非正态过程分解成两个正态过程的思想,结合样本估计相关理论构建了一种基于加权标准差的过程能力指数。新指数无论是在小样本还是大样本的情况下,都比Clement方法估计结果的准确性更高,且在此方法基础上构建的Bootstrap置信区间的真实值覆盖率均远远高于同等条件下Clement方法构建的置信区间。     

Ruin problem of a two-dimensional fractional Brownian motion risk process

This paper investigates ruin probability and ruin time of a two-dimensional fractional Brownian motion risk process. The net loss process of an insurance company is modeled by a fractional Brownian motion. The two-dimensional fractional Brownian motion risk process models the surplus processes of an insurance and a reinsurance company, where the net loss is divided between them in some specified proportions. The ruin problem considered is that of the two-dimensional risk process first entering the negative quadrant, that is, the simultaneous ruin problem. We derive both asymptotics of the ruin probability and approximations of the scaled conditional ruin time as the initial capital tends to infinity.     

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

We consider the problem of parameter estimation for inhomogeneous space‐time shot‐noise Cox point processes. We explore the possibility of using a stepwise estimation method and dimensionality‐reducing techniques to estimate different parts of the model separately. We discuss the estimation method using projection processes and propose a refined method that avoids projection to the temporal domain. This remedies the main flaw of the method using projection processes – possible overlapping in the projection process of clusters, which are clearly separated in the original space‐time process. This issue is more prominent in the temporal projection process where the amount of information lost by projection is higher than in the spatial projection process. For the refined method, we derive consistency and asymptotic normality results under the increasing domain asymptotics and appropriate moment and mixing assumptions. We also present a simulation study that suggests that cluster overlapping is successfully overcome by the refined method.     

Numerical method for means of linear Hawkes processes

Abstract

Linear Hawkes processes are widely used in many fields and means are the basic and critical information of them. However, there is little research on linear Hawkes processes’ means. In this paper, we present a numerical method based on the Laplace transform and inverse Laplace transform for means of linear Hawkes processes. The advantage of this method is that whatever the kernel function is, we can always obtain the numerical solutions of means for a linear Hawkes process. In addition, this numerical method provides the basic information of linear Hawkes processes by means. As an application, the numerical method is applied in a WeChat network model.     

An effective powerful test for one-sided supplier selection problem

In this paper, we consider the supplier selection problem, which deals with comparing two one-sided processes and selecting a better one that has a higher capability. We first review two existing approximation approaches, and an exact approach proposed which we refer to as the division method. We then develop a new exact approach called the subtraction method. We compare the two exact methods on the selection power. The results show that the proposed subtraction method is indeed more powerful than the division method. A two-phase selecting procedure is then developed based on the subtraction method for practical applications. Some computational results are tabulated for practitioners’ convenience.     

Parametric inference in a perturbed gamma degradation process

Abstract

We consider a degradation model which is the sum of two independent processes: an homogeneous gamma process and a Brownian motion. This model is called perturbed gamma process. Based on independent copies of the perturbed gamma process observed at irregular instants we propose to estimate the unknown parameters of the model using the moment method. Some general conditions allow to derive the asymptotic behavior of the estimators. We also show that these general conditions are fulfilled for some specific observation schemes. Finally, we illustrate our method by a numerical study and an application to a real data set.     

Exact simulation of Gaussian Time Series from Nonparametric Spectral Estimates with Application to Bootstrapping

The circulant embedding method for generating statistically exact simulations of time series from certain Gaussian distributed stationary processes is attractive because of its advantage in computational speed over a competitive method based upon the modified Cholesky decomposition. We demonstrate that the circulant embedding method can be used to generate simulations from stationary processes whose spectral density functions are dictated by a number of popular nonparametric estimators, including all direct spectral estimators (a special case being the periodogram), certain lag window spectral estimators, all forms of Welch's overlapped segment averaging spectral estimator and all basic multitaper spectral estimators. One application for this technique is to generate time series for bootstrapping various statistics. When used with bootstrapping, our proposed technique avoids some – but not all – of the pitfalls of previously proposed frequency domain methods for simulating time series.     

Development of a multiattribute synthetic-np chart

Over the last few decades, multiattribute control charts have been widely recommended in practice. They outperform the simultaneous uniattribute charts for monitoring multiattribute processes in many applications. Jolayemi [A statistical model for the design of multiattribute control charts. Indian J Stat. 1999;61:351–365] developed a statistical model for the design of a multiattribute np (Mnp) chart. Based on this model, a multiattribute synthetic (MSyn) chart is proposed in this article. Furthermore, the main features of the MSyn chart and Mnp chart are integrated to build a multiattribute Syn-np (MSyn-np) chart. The results of the comparative studies indicate that the new MSyn-np chart significantly outperforms the Mnp chart and MSyn chart by 83% and 27%, respectively, in terms of the average number of defectives over a wide range of process shifts under different circumstances.     

A Full Bayesian Non-parametric Analysis Involving a Neutral to the Right Process

Implementation of a full Bayesian non-parametric analysis involving neutral to the right processes (apart from the special case of the Dirichlet process) has been difficult for two reasons: first, the posterior distributions are complex and therefore only Bayes estimates (posterior expectations) have previously been presented; secondly, it is difficult to obtain an interpretation for the parameters of a neutral to the right process. In this paper we extend Ferguson & Phadia (1979) by presenting a general method for specifying the prior mean and variance of a neutral to the right process, providing the interpretation of the parameters. Additionally, we provide the basis for a full Bayesian analysis, via simulation, from the posterior process using a hybrid of new algorithms that is applicable to a large class of neutral to the right processes (Ferguson & Phadia only provide posterior means). The ideas are exemplified through illustrative analyses.     

Estimation of Balanced Simultaneous Confidence Sets for SIR Models

Stochastic compartmental (e.g., SIR) models have proven useful for studying the epidemics of childhood diseases while taking into account the variability of the epidemic dynamics. Here, we present a method for estimating balanced simultaneous confidence sets for the mean sample path of a stochastic SIR model, thus providing a simple representation of both the typical behavior and the variability of the epidemic. The confidence sets are estimated by a bootstrap procedure, using asymptotic properties of density dependent jump Markov processes. The method is applied to chickenpox epidemics in France and the coverage probability of the confidence sets is estimated in that context.     

Semiparametric regression for the mean and rate functions of recurrent events

The counting process with the Cox-type intensity function has been commonly used to analyse recurrent event data. This model essentially assumes that the underlying counting process is a time-transformed Poisson process and that the covariates have multiplicative effects on the mean and rate function of the counting process. Recently, Pepe and Cai, and Lawless and co-workers have proposed semiparametric procedures for making inferences about the mean and rate function of the counting process without the Poisson-type assumption. In this paper, we provide a rigorous justification of such robust procedures through modern empirical process theory. Furthermore, we present an approach to constructing simultaneous confidence bands for the mean function and describe a class of graphical and numerical techniques for checking the adequacy of the fitted mean–rate model. The advantages of the robust procedures are demonstrated through simulation studies. An illustration with multiple-infection data taken from a clinical study on chronic granulomatous disease is also provided.     

Bridging the Gap: A Generalized Stochastic Process for Count Data

The Bernoulli and Poisson processes are two popular discrete count processes; however, both rely on strict assumptions. We instead propose a generalized homogenous count process (which we name the Conway–Maxwell–Poisson or COM-Poisson process) that not only includes the Bernoulli and Poisson processes as special cases, but also serves as a flexible mechanism to describe count processes that approximate data with over- or under-dispersion. We introduce the process and an associated generalized waiting time distribution with several real-data applications to illustrate its flexibility for a variety of data structures. We consider model estimation under different scenarios of data availability, and assess performance through simulated and real datasets. This new generalized process will enable analysts to better model count processes where data dispersion exists in a more accommodating and flexible manner.