ASYMPTOTIC TESTS FOR THE CONTINUOUS TIME MARKOV BRANCHING PROCESS WITH IMMIGRATION

This paper presents limit distributions for the modified score and the likelihood-ratio (LR) statistic for testing a composite hypothesis involving the split intensity and mean of the offspring distribution of the supercritical continuous time Markov branching process allowing immigration (CBPI). The immigration intensity and mean are treated as nuisance parameters.     

ON THE VARIANCE-COVARIANCE MATRIX IN MARKOVIAN MANPOWER SYSTEMS IN CONTINUOUS TIME

The variability of the grade sizes in Markovian manpower systems in continuous time is considered. The differential equations describing the evolution of the variance-covariance matrix are derived and the associated generator is determined in a closed form. The limiting variability of an expanding system is studied via two theorems which provide conditions for the limit to exist independently of the initial distribution and for the rate of convergence to be exponentially fast. A numerical illustration of the theory is given.     

PROPERTIES OF THE MINIMUM POINT OF AN UNBALANCED TWO-SIDED RANDOM WALK

In this paper, properties of minimum point of a unbalanced two-sided random walk are investigated. Under the condition that the parameters at both sides tend to zero at the same order, probabilities that the minimum point is on which side, and the second order expansions for the first two moments of the minimum point are obtained. Applications of these results are very promising. First, they can be used to study the properties of the maximum likelihood estimator for the change point in the large sample case; second, they can be used to study inference problems after CUSUM test.     

JOINT BEHAVIOUR OF PRECEDENCES AND EXCEEDANCES IN RANDOM THRESHOLD MODELS

This paper concerns the joint behaviour of precedence and exceedance statistics in random threshold models. Joint distributions of precedence and exceedance statistics, both exact and asymptotic, are obtained, and the results are illustrated for random thresholds based on order statistics and record values.     

ESTIMATING POPULATION SIZE FROM CAPTURE-RECAPTURE EXPERIMENTS IN CONTINUOUS TIME

Estimates for the size of a closed population are given for multiple recapture studies in continuous time. The estimates are derived by a method of moments for martingales. An estimate and associated standard error of the population size are derived for a homogeneous population when the capture rates are permitted to depend on time in an unspecified manner. Corresponding results are obtained when the capture rates vary among individuals as well. Explicit expressions are given for these estimates and standard errors which involve only simple computation.     

ON THE CAUSALITY TEST IN TIME SERIES MODELS WITH HEAVY-TAILED DISTRIBUTION

ABSTRACT

In this article, we consider the problem of testing the Granger causality in stationary time series models with non-normal heavy-tailed distributions. We consider a normal mixture model to cover the heavy-tailed distribution, and propose a test statistic based on the partially adaptive estimator proposed by Phillips [1] Phillips, R.F. 1994. Partially Adaptive Estimation via a Normal Mixture. J. Econometics, 64: 123–144. [Crossref], [Web of Science ®] , [Google Scholar]. It is shown that the test statistic asymptotically follows a chi-squared distribution. Simulation results indicate that our test outperforms the conventional test based on the least squares estimator when the observations follow a heavy-tailed distribution.     

AN APPROACH TO THE SAMPLING THEOREM FOR CONTINUOUS TIME PROCESSES

In this paper we extend the sampling theorem to the full spectrum of weakly stationary stochastic processes. The mean square approximation error bound is also given.     

INCOMPLETE DATA IN GENERALIZED LINEAR MODELS WITH CONTINUOUS COVARIATES

This paper proposes a method for estimating the parameters in a generalized linear model with missing covariates. The missing covariates are assumed to come from a continuous distribution, and are assumed to be missing at random. In particular, Gaussian quadrature methods are used on the E-step of the EM algorithm, leading to an approximate EM algorithm. The parameters are then estimated using the weighted EM procedure given in Ibrahim (1990). This approximate EM procedure leads to approximate maximum likelihood estimates, whose standard errors and asymptotic properties are given. The proposed procedure is illustrated on a data set.     

MAXIMUM LIKELIHOOD ESTIMATION IN LINEAR MODELS WITH EQUI-CORRELATED RANDOM ERRORS

Necessary and sufficient conditions for the existence of maximum likelihood estimators of unknown parameters in linear models with equi‐correlated random errors are presented. The basic technique we use is that these models are, first, orthogonally transformed into linear models with two variances, and then the maximum likelihood estimation problem is solved in the environment of transformed models. Our results generalize a result of Arnold, S. F. (1981) [The theory of linear models and multivariate analysis. Wiley, New York]. In addition, we give necessary and sufficient conditions for the existence of restricted maximum likelihood estimators of the parameters. The results of Birkes, D. & Wulff, S. (2003) [Existence of maximum likelihood estimates in normal variance‐components models. J Statist Plann. Inference. 113 , 35–47] are compared with our results and differences are pointed out.     

FIXED VERSUS RANDOM SAMPLING OF CERTAIN CONTINUOUS PARAMETER PROCESSES

Let {Z(t)} be a stochastic point process. When {Z(t)} is Poisson and it is desired to estimate the intensity A, it is shown that the optimal (in terms of Fisher information) discrete sampling scheme is to sample {Z(t)} at predetermined fixed time points. On the other hand, when {Z(t)} is a pure birth process and a maximum likelihood estimator of the birth rate is desired, it is sometimes better to sample at random time points, according to a renewal process. An application of these ideas is given in the estimation of bacterial density in a liquid, by the method of dilutions.     

WAITING TIME IN THE COUPON-COLLECTOR'S PROBLEM1

The coupon collector's problem is generalized by allowing unequal proportions of the various types of coupons. The upper tail probabilities are used to find the probability distribution of the waiting time. The probability generating function is expressed in terms of the hypergeometric functions and therefrom the mean and the variance are derived.     

COMMON FEATURES IN TIME SERIES WITH BOTH DETERMINISTIC AND STOCHASTIC SEASONALITY

This paper extends the notions of common cycles and common seasonal features to time series having deterministic and stochastic seasonality at different frequencies. The conditions under which quarterly time series with these characteristics have common features are investigated, various representations are presented and statistical inference is discussed. Finally, the analysis is applied to study comovements between different components of consumption and income using UK data.     

APPLYING THE SAVAGE-DICKEY DENSITY RATIO TO DEFAULT BAYES FACTORS,WITH AN ILLUSTRATION TO OUTLIER DETECTION IN RANDOM EFFECTS MODELS

When no information is available and hence improper noninformative priors should be used, Bayes factor includes the unspecified constants and can not be calibrated. To solve this problem, we modify the intrinsic Bayes factor (IBF) of Berger and Pericchi 1-2 Berger, J. O. and Pericchi, L. R. 1996. The Intrinsic Bayes Factor for Model Selection and Prediction. Journal of the American Statistical Association, 91: 109–122. Berger, J. O. and Pericchi, L. R. 1998. Accurate and Stable Bayesian Model Selection: The Median Intrinsic Bayes Factor. Sankhya, Series B, 60: 1–18.   and the fractional Bayes factor (FBF) of O'Hagan [3] O'Hagan, A. 1995. Fractional Bayes Factors for Model Comparison. Journal of the Royal Statistical Society, Series B, 57: 99–138.  [Google Scholar] with the generalized Savage-Dickey density ratio of Verdinelli and Wasserman [4] Verdinelli, I. and Wasserman, L. 1995. Computing Bayes Factors Using a Generalization of Savage-Dickey Density Ratio. Journal of the American Statistical Association, 90: 614–618. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]. These modified IBF and FBF are applied to detecting outliers in random effects models with a mean-shift structure. The proposed methodology is exemplified by a simulation experiment with a generated data set and also applied to a real data set, Dyestuff data in Box and Tiao [5] Box, G. E.P. and Tiao, G. C. 1973. Bayesian Inference in Statistical Analysis U.S.A.: Addison-Wesley Publishing Co..  [Google Scholar]     

CATASTROPHE PROCESSES WITH CONTINUOUS STATE-SPACE1

Some continuous state-space models for the size of a population subject to catastrophes are examined. Inputs to the population constitute a compound Poisson process, and catastrophes, which are of random magnitudes, occur at a rate which is dependent on the population size. Necessary and sufficient conditions for the existence of stationary distributions are investigated and in special cases the stationary distributions are found explicitly. The time-dependent behaviour of a process with linear catastrophe rate and truncated exponential catastrophe sizes is also studied.