Bert van Es

Squared skewness minus kurtosis bounded by 186/125 for unimodal distributions

The sharp inequality for squared skewness minus kurtosis is derived for the class of unimodal distributions.

Survival analysis under cross sectional sampling: length bias and multiplicative censoring

Consider a parametric, nonparametric or semiparametric model for survival times. Interest is in estimation of Euclidean and Banach parameters for these models. However, not the survival times themselves will be observed, since this might be quite time consuming. Instead, cross-sectional sampling is applied: at some point in time one identifies a random sample from the population under study and one registers the survival time up to this time-point. Typically, the resulting reduced survival times do not have the same distributions as the true survival times. On the one hand, longer survival times have a higher probability to be sampled than smaller ones. On the other hand, the observed survival times have been censored multiplicatively. The length bias and multiplicative censoring properties of cross-sectional sampling will be discussed and reviewed as well as estimation in the resulting parametric, nonparametric, and semiparametric models.

Nonparametric volatility density estimation

We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the density of which, at a fixed instant in time, we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the volatility density. An expansion of the bias and a bound on the variance are derived.

Nonparametric volatility density estimation for discrete time models

We consider discrete time models for asset prices with a stationary volatility process. We aim at estimating the multivariate density of this process at a set of consecutive time instants. A Fourier-type deconvolution kernel density estimator based on the logarithm of the squared process is proposed to estimate the volatility density. Expansions of the bias and bounds on the variance are derived.

Deconvolution for an atomic distribution

Let X1, . . . ,Xn be i.i.d. observations, where Xi = Yi + σZi and Yi and Zi are independent. Assume that unobservable Ys are distributed as a random variable UV, where U and V are independent, U has a Bernoulli distribution with probability of zero equal to p and V has a distribution function F with density f. Furthermore, let the random variables Zi have the standard normal distribution and let σ > 0. Based on a sample X1, . . . ,Xn, we consider the problem of estimation of the density f and the probability p. We propose a kernel type deconvolution estimator for f and derive its asymptotic normality at a fixed point. A consistent estimator for p is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem.

Deconvolution for an atomic distribution: Rates of convergence

Let X 1, …, X n be i.i.d. copies of a random variable X=Y+Z, where X i =Y i +Z i , and Y i and Z i are independent and have the same distribution as Y and Z, respectively. Assume that the random variables Y i ’s are unobservable and that Y=AV, where A and V are independent, A has a Bernoulli distribution with probability of success equal to 1−p and V has a distribution function F with density f. Let the random variable Z have a known distribution with density k. Based on a sample X 1, …, X n , we consider the problem of nonparametric estimation of the density f and the probability p. Our estimators of f and p are constructed via Fourier inversion and kernel smoothing. We derive their convergence rates over suitable functional classes. By establishing in a number of cases the lower bounds for estimation of f and p we show that our estimators are rate-optimal in these cases.

Estimation of a multivariate stochastic volatility density by kernel deconvolution

We consider a continuous time stochastic volatility model. The model contains a stationary volatility process. We aim to estimate the multivariate density of the finite-dimensional distributions of this process. We assume that we observe the process at discrete equidistant instants of time. The distance between two consecutive sampling times is assumed to tend to zero. A multivariate Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the multivariate volatility density. An expansion of the bias and a bound on the variance are derived.

Likelihood cross-validation bandwidth selection for nonparametric kernel density estimators †

One of the major problems in kernel density estimation is the choice of bandwidth. We review the first order properties of the likelihood cross-validation bandwidth selection method, introduced by Habbema, Hermans and Van den Broek and Duin. This method was modified by Marron 1985 to obtain an asymptotic optimality property for Lipschitz densities. In particular we show that for densities with jumps the modified method loses this optimality property. Furthermore we establish the asymptotic normality of the distance of the likelihood cross-validation bandwidth to the minimizer of a suitable integrated squared error. It turns out that for certain non smooth densities this distance has a faster rate of convergence to zero than for smooth densities.

A note on the integrated squared error of a kernel density estimator in non-smooth cases

Let X1, ... , Xn be a random sample from a distribution on the real line with an unknown density f. We discuss the performance of the kernel density estimator of the density f. The properties of kernel estimators in cases where the density f to be estimated is sufficiently smooth are well known. Instead we focus on estimation problems where f is non-smooth, i.e. f is allowed to have a finite number of jumps or kinks. Thus the robustness properties of the kernel estimator against unfulfilled smoothness assumptions are illustrated. After a review of properties of the mean integrated squared error we present a central limit theorem for the integrated squared error. This theorem extends results of Bickel, Rosenblatt and Hall. Finally, the distance between the bandwidth minimizing the integrated squared error and the bandwidth which minimizes the mean integrated squared error is discussed.

Basic Proof Skills of Computer Science Students

Computer science students need mathematical proof skills. At our University, these skills are being taught as part of various mathematics and computer science courses. To test the skills of our students, we have asked them to work out a number of exercises. We found that our students are not as well trained in basic proof skills as we would have hoped. The main reason is that proof skills are not emphasized enough. Our findings are the result of a small experiment using a longitudinal measurement of skills. This method gives better insight in the skills of students than more traditional exam-based testing methods. Longitudinal measurement does not allow the students to specifically prepare themselves for particular questions. The measurements thus relate to skills that are retained for a longer period of time.