M.C.A. van Zuijlen

Fluctuation limit of branching processes with immigration and estimation of the means

We investigate a sequence of Galton-Watson branching processes with immigration, where the offspring mean tends to its critical value 1 and the offspring variance tends to 0. It is shown that the fluctuation limit is an Ornstein-Uhlenbeck-type process. As a consequence, in contrast to the case in which the offspring variance tends to a positive limit, it transpires that the conditional least-squares estimator of the offspring mean is asymptotically normal. The norming factor is n 3/2, in contrast to both the subcritical case, in which it is n 1/2, and the nearly critical case with positive limiting offspring variance, in which it is n.

Estimation of the mean of stationary and nonstationary Ornstein-Uhlenbeck processes and sheets

Abstract We consider the problem of estimating an unknown parameter m in case one observes in an interval (rectangle) stationary and nonstationary Ornstein-Uhlenbeck processes (sheets), which are shifted by m times a known deterministic function on the interval (rectangle). It turns out that the maximum likelihood estimator (MLE) has a normal distribution and, for instance, in case of the sheet this MLE is a weighted linear combination of the values at the vertices, integrals on the edges, and the integral on the whole rectangle of the weighted observed process. We do not use partial stochastic differential equations; we apply direct discrete time approach instead. To make the transition from the discrete time to the continuous time, a tool is developed, which might be of independent interest.

Forward interest rate curves in discrete time settings driven by random fields

In this paper, we study the term structure of forward interest rates in discrete time settings. We introduce a generalisation of the classical Heath-Jarrow-Morton type models. The forward rates corresponding to different time to maturity values will be equipped with different driving processes. In this way, we use a discrete time random field to drive the forward rates instead of a single process. We assume the existence of a general stochastic (market) discount factor process, which involves market price of risk factors. This way of building the model is motivated by statistical problems, which is the aim of our further studies. Since we are interested only in arbitrage free markets, we derive several sufficient conditions to exclude arbitrage opportunities in the models and we also present examples for the structure of the driving field, in particular, we use Gaussian autoregression fields.

ON CONSERVATIVE CONFIDENCE INTERVALS

AbstractThe subject of the paper – (conservative) confidence intervals – originates in applications to auditing. Auditors are interested in upper confidence bounds for an unknown mean μ for all sample sizes n. The samples are drawn from populations such that often only a few observations are nonzero. The conditional distribution of an observation given that it is nonzero usually has a very irregular shape. However, it can be assumed that observations are bounded. We propose a way to reduce the problem to inequalities for tail probabilities of certain relevant statistics. Note that a traditional approach involving limit theorems forces to impose additional conditions on regularity of samples and leads to approximate or asymptotic bounds. In the case of μ, as a statistic we can use sample mean, say

Limiting connection between discrete and continuous time forward interest rate curve models

\bar \mu

Asymptotic properties of nearly unstable multivariate AR processes

, and we have to use Hoeffding [7] inequalities, since currently they are the best available. This leads to upper confidence bounds for μ which are of (asymptotic) size at most

Bounds for Tail Probabilities of the Sample Variance

b \lesssim \bar \mu + 2.44\sigma /\sqrt n