Martien C. A. van Zuijlen

Maximum likelihood estimator of the volatility of forward rates driven by geometric spatial AR sheet

Discrete-time forward interest rate curve models are studied, where the curves are driven by a random field. Under the assumption of no-arbitrage, the maximum likelihood estimator of the volatility parameter is given and its asymptotic behaviour is studied. First, the so-called martingale models are examined, but we will also deal with the general case, where we include the market price of risk in the discount factor.

ASYMPTOTIC INFERENCE FOR NEARLY UNSTABLE AR(p) PROCESSES

In this paper nearly unstable AR(p) processes (inf other words, models with characteristic roots near the unit circle) are studied. Our main aim is to describe the asymptotic behavior of the least-squares estimators of the coefficients. A convergence result is presented for the general complex-valued case. The limit distribution is given by the help of some continuous time AR processes. We apply the results for real-valued nearly unstable AR(p) models. In this case the limit distribution can be identified with the maximum likelihood estimator of the coefficients of the corresponding continuous time AR processes.

Asymptotic inference for an unstable spatial AR model

The spatial autoregressive process X k ,ℓ = α(X k −1,ℓ + X k ,ℓ−1) + ϵ k ,ℓ, where k, ℓ ⩾ 1 is investigated. We consider the least squares estimator αˆ m , n of α based on the observations {X k ,ℓ: 1 ⩽ k ⩽ m and 1 ⩽ ℓ ⩽ n}. In the stable (i.e. asymptotically stationary) case, when |α| < 1/2, asymptotic normality as m, n → ∞ with m/n → constant > 0 can be derived from the previous more general results due to Basu and Reinsel (1992, 1993, 1994). In the unstable case, when |α| = 1/2, we prove again asymptotic normality, but (in contrast to the doubly geometric spatial model) with a surprising rate of convergence, namely as m, n → ∞ with m/n → constant > 0.

Asymptotic Behaviour of Estimators of the Parameters of Nearly Unstable INAR(1) Models

A sequence of first-order integer-valued autoregressive type (INAR(1)) processes is investigated, where the autoregressive type coefficients converge to 1. It is shown that the limiting distribution of the joint conditional least squares estimators for this coefficient and for the mean of the innovation is normal. Consequences for sequences of Galton-Watson branching processes with unobservable immigration, where the mean of the offspring distribution converges to 1 (which is the critical value), are discussed.

Characterization of weak convergence for smoothed empirical and quantile processes under ϕ-mixing

Let {X,,,n>~ 1} be a sequence of q)-mixing random variables having a smooth common distribution function F. The smoothed empirical distribution function is obtained by integrating a kernel type density estimator. In this paper we provide necessary and sufficient conditions tbr the central limit theorem to hold for smoothed empirical distribution functions and smoothed sample quantiles. Also, necessary and sufficient conditions are given for weak convergence of the smoothed empirical process and the smoothed uniform quantile process. A MS classification: Primary 60F05; secondary 62E20:62G05

Parameter estimation of a shifted Wiener sheet

An explicit form is given for the maximum likelihood estimator of a shift parameter m of a shifted Wiener sheet W(s, t)+mg(s, t), (s, t)∈ℝ+ 2, based on observations {W(s, t)+mg(s, t):(s, t)∈G}, where g:ℝ+ 2→ℝ is a known constraint function and G has a special shape.

Glivenko—Cantelli-type theorems for weighted empirical distribution functions based on uniform spacings

Let U1, U2,... be a sequence of independent r.v.s having the uniform distribution on (0, 1). Let Fn be the empirical distribution based on the transformed uniform spacings Di,n:=G(nDi,n), i = 1, 2,..., n, where G is the exp(1) d.f. and Di,n is the ith spacing based on U1, U2,...,Un-1. The main purpose of this paper is the study of the almost sure behaviour of lim supn --> [infinity] [Delta]n,[alpha](q, ) and lim supn-->[infinity] [Lambda]n,r(q, ), where [Delta]n,[alpha](q, ) = sup0 0 and certain weight functions q and . Moreover, the weak behaviour of the statistics will be examined briefly. It turns out that compared with the uniform empirical process (i.i.d. case) the considered weighted Kolmogorov--Smirnov- and Cramer--von Mises-type statistics behave differently in the right tail only as far as almost sure convergence is concerned. There is no difference in the weak sense. The results can be applied to the study of linear combinations of functions of ordered spacings.

The Limiting Density of Unit Root Test Statistics: A Unifying Technique

In this note we introduce a simple principle to derive a constructive expression for the density of the limiting distribution, under the null hypothesis, of unit root statistics for an AR(1)-process in a variety of situations. We consider the case of unknown mean and reconsider the well-known situation where the mean is zero. For long-range dependent errors we indicate how the principle might apply again. We also show that in principle the method also works for a near unit root case. Weak convergence and subsequent Karhunen-Loeve expansion of the weak limit of the partial sum process of the errors plays an important role, along with the evaluation of a certain normal type integral with complex mean and variance. For independent and long range dependent errors this weak limit is ordinary and fractional Brownian motion respectively. AMS 1991 subject classification. Primary 62M10; secondary 62E20.

On the approximation of an integral by a sum of random variables

We approximate the integral of a smooth function on [0,1], where values are only known at n random points (i.e., a random sample from the uniform-(0,1) distribution), and at 0 and 1. Our approximations are based on the trapezoidal rule and Simpsons rule (generalized to the non-equidistant case), respectively. In the first case, we obtain an n2-rate of convergence with a degenerate limiting distribution; in the second case, the rate of con-vergence is as fast as n3½, whereas the limiting distribution is Gaussian then.