First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties

Some properties of a first-order integer-valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self-decomposability and unimodality of the 1-dimensional marginals of the process {X_n} generated according to the scheme X_n=α° X _n-i +e_n, where α° X _n-1 denotes a sum of X_{n - 1}, independent 0 - 1 random variables Y^(n-1), independent of X _n-1 with Pr -( y ^{(n - 1)}= 1) = 1 - Pr ( y ^(n-i)= 0) =α. The distribution of the innovation process ( e _n) is obtained when the marginal distribution of the process ( X _n) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.     

Efficient parameter estimation for independent and INAR(1) negative binomial samples

We consider moment based estimation methods for estimating parameters of the negative binomial distribution that are almost as efficient as maximum likelihood estimation and far superior to the celebrated zero term method and the standard method of moments estimator. Maximum likelihood estimators are difficult to compute for dependent samples such as samples generated from the negative binomial first-order autoregressive integer-valued processes. The power method of estimation is suggested as an alternative to maximum likelihood estimation for such samples and a comparison is made of the asymptotic normalized variance between the power method, method of moments and zero term method estimators.     

Detecting overdispersion in INARCH(1) processes

A statistical test for the degree of overdispersion of count data time series based on the empirical version of the (Poisson) index of dispersion is considered. The test design relies on asymptotic properties of this index of dispersion, which in turn have been analyzed for time series stemming from a compound Poisson (Poisson‐stopped sum) INAR(1) model. This approach is extended to the popular Poisson INARCH(1) model, which exhibits unconditional overdispersion but has an (equidispersed) conditional Poisson distribution. The asymptotic distribution of the index of dispersion if applied to time series stemming from such a model is derived. These results allow us to investigate the ability of the dispersion test to discriminate between Poisson INAR(1) and INARCH(1) models. Furthermore, the question is considered if the index of dispersion could be used to test the null of a Poisson INARCH(1) model against the alternative of an INARCH(1) model with additional conditional overdispersion.     

Random Sets: Models and Statistics

This paper surveys aspects of the theory of random closed sets, focussing on issues of practical and current interest. First, some historical remarks on this part of probability theory are made, where the important role of Georges Matheron is emphasized. Then, fundamental characteristics of the distribution of random closed sets are introduced. The very important Boolean model serves as an example for discussing mathematical and statistical problems. A number of further models is then considered, namely excursion sets of random fields, the system of edges of the Poisson Voronoi tessellation and various random systems of non-overlapping spheres. Finally, some ideas of particle statistics are presented, including some models of random compact sets.     

组合模型在公路客运量预测中的应用

针对公路客运量预测问题,在建立灰色GM(1,1)模型和三次指数平滑模型的基础上,利用最优加权系数法建立组合模型来对客运量进行预测。以此来避免单一模型的不足,从而提高模型的预测精度。通过2005-2009年安顺到贵阳的贵黄高速客运量模型的检验,证明组合模型是合理、可行的,并且具有较高的预测精度。     

On exact D-optimal designs with 2 two-level factors and n autocorrelated observations

In this paper we consider the exact D-optimal designs for estimation of the unknown parameters in the two factors, each at only two-level, main effects model with autocorrelated errors. The vector of the n random errors in the observed responses is assumed to follow a first-order autoregressive model (AR(1)). The exact D-optimal designs seek the optimal combinations of the design levels as well as the optimal run orders, so that the determinant of the information matrix of BLUEs for the unknown parameters is maximized. Bora-Senta and Moyssiadis (1999) gave some conjectures about the exact D-optimal designs based on their experience of several exhaustive searches. In this paper their conjectures are partially proved to be true.Received: January 2003 / Accepted: October 2003Partially supported by the National Science Council of Taiwan, R.O.C. under grant NSC 91-2115-M-008-013.Supported in part by the National Science Council of Taiwan, R.O.C. under grant NSC 89-2118-M-110-003.     

A new bivariate distribution in natural exponential family

We propose a new bivariate distribution following a GLM form i.e., natural exponential family given the constantly correlated covariance matrix. The proposed distribution can represent an independent bivariate gamma distribution as a special case. In order to derive the distribution we utilize an integrating factor method to satisfy the integrability condition of the quasi-score function. The derived distribution becomes a mixture of discrete and absolute continuous distributions. The proposal of our new bivariate distribution will make it possible to develop some bivariate generalized linear models. Further the discrete correlated bivariate distribution will also arise from an independent bivariate Poisson mass function by compounding our proposed distribution (Iwasaki and Tsubaki, 2002).Received March 2003     

Estimating survivor function using optimally selected order statistics

This paper deals with the estimation of survivor function using optimally selected order statistics when the sample sizen is large. We use the estimates (μ*,σ*) based on the optimum set of order statistics for largen and fixedk (≤n) such that the estimate has optimum variance property. The asymptotic relative efficiency of such an estimator is compared with the one based on the complete sample. The general theory of the problem and specific details with respect to a two-parameter Normal, Logistic, Exponential and Pareto distributions is considered as an example.     

A POISSON PROCESS APPROXIMATION FOR GENERALIZED K–S CONFIDENCE REG

One–sided confidence regions for continuous cumulative distribution functions are constructed using empirical cumulative distribution functions and the generalized Kolmogorov–Smimov distance. The band width of such regions becomes narrower in the right or left tail of the distribution. To avoid tedious computation of confidence levels and critical values, an approximation based on the Poisson process is introduced. This approximation provides a conservative confidence region; moreover, the approximation error decreases monotonically to 0 as sample size increases. Critical values necessary for implementation are given. Applications are made to the areas of risk analysis, investment modelling, and analysis of fault–tolerant systems.